TITLE: Is the condition number of unitary matrix always equal to 1?
QUESTION [0 upvotes]: I know that the 2-norm condition number $\kappa (\textbf U)={||\textbf U||_2}{||\textbf U^{-1}||_2}$ of a unitary matrix $\textbf U$ is always equal to 1. Is this true for all induced matrix norms, i.e. $\kappa (\textbf U)=||\textbf U||||\textbf U^{-1}||=1$?

REPLY [2 votes]: The condition number with respect to the norm induced by vector norm $\|\cdot\|$ is equal to $1$ precisely when the image of the ball $B_1=\{x:\|x\|\le 1\}$ is another ball $B_r$ of some radius $r>0$.
For any vector norm that is not a constant multiple of the Euclidean one, there is a rotation of its unit ball that maps it into something other than a ball. For example, take the $\ell_\infty$ ball, i.e., a cube, and rotate it slightly. So the answer to your question is negative: up to scaling, the vector $2$-norm is the only norm for which all  unitary matrices have condition number $1$.


TITLE: Fourier transform of partial differential equation
QUESTION [1 upvotes]: How would I find the Fourier Transform of the pde
$$
u_t = u_{xx} + \alpha\cdot u_x,\quad\alpha > 0\,?
$$
I know the definition of the Fourier transform, but I'm not sure how to use it in the context of partial differential equations.

REPLY [0 votes]: If $f : \mathbb R\to\mathbb C$ is a "nice" function (i.e. $f\in H^1$ or $f\in C^1$ with $f(\pm\infty) = 0$) the Fourier transform of its derivative is by partial integration
$$
\widehat{f'}(\omega) = \int f'(x)e^{-2\pi ix\omega}\,dx = 2\pi i\omega\int f(x)e^{-2\pi ix\omega}\,dx = 2\pi i\omega\cdot\hat f(\omega).
$$
So if $U(\omega,t)$ denotes the Fourier transform of $u(x,t)$ with respect to $x$, applying the Fourier transform with respect to $x$ to $u_t = u_{xx} + \alpha u_x$ gives
$$
U_t = -4\pi^2\omega^2 U + 2\pi i\omega\alpha U = (-2\pi + i\alpha\omega)2\pi U.
$$


text:
(*  Title:       CoreC++
    Author:      Tobias Nipkow, Daniel Wasserrab
    Maintainer:  Daniel Wasserrab <wasserra at fmi.uni-passau.de>
*)

section \<open>Definite assignment\<close>

theory DefAss
imports BigStep
begin

subsection \<open>Hypersets\<close>

type_synonym hyperset = "vname set option"

definition hyperUn :: "hyperset \<Rightarrow> hyperset \<Rightarrow> hyperset"   (infixl "\<squnion>" 65) where
  "A \<squnion> B  \<equiv>  case A of None \<Rightarrow> None
                 | \<lfloor>A\<rfloor> \<Rightarrow> (case B of None \<Rightarrow> None | \<lfloor>B\<rfloor> \<Rightarrow> \<lfloor>A \<union> B\<rfloor>)"

definition hyperInt :: "hyperset \<Rightarrow> hyperset \<Rightarrow> hyperset"   (infixl "\<sqinter>" 70) where
  "A \<sqinter> B  \<equiv>  case A of None \<Rightarrow> B
                 | \<lfloor>A\<rfloor> \<Rightarrow> (case B of None \<Rightarrow> \<lfloor>A\<rfloor> | \<lfloor>B\<rfloor> \<Rightarrow> \<lfloor>A \<inter> B\<rfloor>)"

definition hyperDiff1 :: "hyperset \<Rightarrow> vname \<Rightarrow> hyperset"   (infixl "\<ominus>" 65) where
  "A \<ominus> a  \<equiv>  case A of None \<Rightarrow> None | \<lfloor>A\<rfloor> \<Rightarrow> \<lfloor>A - {a}\<rfloor>"

definition hyper_isin :: "vname \<Rightarrow> hyperset \<Rightarrow> bool"   (infix "\<in>\<in>" 50) where
"a \<in>\<in> A  \<equiv>  case A of None \<Rightarrow> True | \<lfloor>A\<rfloor> \<Rightarrow> a \<in> A"

definition hyper_subset :: "hyperset \<Rightarrow> hyperset \<Rightarrow> bool"   (infix "\<sqsubseteq>" 50) where
  "A \<sqsubseteq> B  \<equiv>  case B of None \<Rightarrow> True
                 | \<lfloor>B\<rfloor> \<Rightarrow> (case A of None \<Rightarrow> False | \<lfloor>A\<rfloor> \<Rightarrow> A \<subseteq> B)"

lemmas hyperset_defs =
 hyperUn_def hyperInt_def hyperDiff1_def hyper_isin_def hyper_subset_def

lemma [simp]: "\<lfloor>{}\<rfloor> \<squnion> A = A  \<and>  A \<squnion> \<lfloor>{}\<rfloor> = A"
by(simp add:hyperset_defs)

lemma [simp]: "\<lfloor>A\<rfloor> \<squnion> \<lfloor>B\<rfloor> = \<lfloor>A \<union> B\<rfloor> \<and> \<lfloor>A\<rfloor> \<ominus> a = \<lfloor>A - {a}\<rfloor>"
by(simp add:hyperset_defs)

lemma [simp]: "None \<squnion> A = None \<and> A \<squnion> None = None"
by(simp add:hyperset_defs)

lemma [simp]: "a \<in>\<in> None \<and> None \<ominus> a = None"
by(simp add:hyperset_defs)

lemma hyperUn_assoc: "(A \<squnion> B) \<squnion> C = A \<squnion> (B \<squnion> C)"
by(simp add:hyperset_defs Un_assoc)

lemma hyper_insert_comm: "A \<squnion> \<lfloor>{a}\<rfloor> = \<lfloor>{a}\<rfloor> \<squnion> A \<and> A \<squnion> (\<lfloor>{a}\<rfloor> \<squnion> B) = \<lfloor>{a}\<rfloor> \<squnion> (A \<squnion> B)"
by(simp add:hyperset_defs)

subsection \<open>Definite assignment\<close>

primrec \ :: "expr \<Rightarrow> hyperset" and \s :: "expr list \<Rightarrow> hyperset" where
"\ (new C) = \<lfloor>{}\<rfloor>" |
"\ (Cast C e) = \ e" |
"\ (\<lparr>C\<rparr>e) = \ e" |
"\ (Val v) = \<lfloor>{}\<rfloor>" |
"\ (e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2) = \ e\<^sub>1 \<squnion> \ e\<^sub>2" |
"\ (Var V) = \<lfloor>{}\<rfloor>" |
"\ (LAss V e) = \<lfloor>{V}\<rfloor> \<squnion> \ e" |
"\ (e\<bullet>F{Cs}) = \ e" |
"\ (e\<^sub>1\<bullet>F{Cs}:=e\<^sub>2) = \ e\<^sub>1 \<squnion> \ e\<^sub>2" |
"\ (Call e Copt M es) = \ e \<squnion> \s es" |
"\ ({V:T; e}) = \ e \<ominus> V" |
"\ (e\<^sub>1;;e\<^sub>2) = \ e\<^sub>1 \<squnion> \ e\<^sub>2" |
"\ (if (e) e\<^sub>1 else e\<^sub>2) =  \ e \<squnion> (\ e\<^sub>1 \<sqinter> \ e\<^sub>2)" |
"\ (while (b) e) = \ b" |
"\ (throw e) = None" |

"\s ([]) = \<lfloor>{}\<rfloor>" |
"\s (e#es) = \ e \<squnion> \s es"

primrec \<D> :: "expr \<Rightarrow> hyperset \<Rightarrow> bool" and \<D>s :: "expr list \<Rightarrow> hyperset \<Rightarrow> bool" where
"\<D> (new C) A = True" |
"\<D> (Cast C e) A = \<D> e A" |
"\<D> (\<lparr>C\<rparr>e) A = \<D> e A" |
"\<D> (Val v) A = True" |
"\<D> (e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \ e\<^sub>1))" |
"\<D> (Var V) A = (V \<in>\<in> A)" |
"\<D> (LAss V e) A = \<D> e A" |
"\<D> (e\<bullet>F{Cs}) A = \<D> e A" |
"\<D> (e\<^sub>1\<bullet>F{Cs}:=e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \ e\<^sub>1))" |
"\<D> (Call e Copt M es) A = (\<D> e A \<and> \<D>s es (A \<squnion> \ e))" |
"\<D> ({V:T; e}) A = \<D> e (A \<ominus> V)" |
"\<D> (e\<^sub>1;;e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \ e\<^sub>1))" |
"\<D> (if (e) e\<^sub>1 else e\<^sub>2) A =
  (\<D> e A \<and> \<D> e\<^sub>1 (A \<squnion> \ e) \<and> \<D> e\<^sub>2 (A \<squnion> \ e))" |
"\<D> (while (e) c) A = (\<D> e A \<and> \<D> c (A \<squnion> \ e))" |
"\<D> (throw e) A = \<D> e A" |

"\<D>s ([]) A = True" |
"\<D>s (e#es) A = (\<D> e A \<and> \<D>s es (A \<squnion> \ e))"

lemma As_map_Val[simp]: "\s (map Val vs) = \<lfloor>{}\<rfloor>"
by (induct vs) simp_all

lemma D_append[iff]: "\<And>A. \<D>s (es @ es') A = (\<D>s es A \<and> \<D>s es' (A \<squnion> \s es))"
by (induct es type:list) (auto simp:hyperUn_assoc)

lemma A_fv: "\<And>A. \ e = \<lfloor>A\<rfloor> \<Longrightarrow> A \<subseteq> fv e"
and  "\<And>A. \s es = \<lfloor>A\<rfloor> \<Longrightarrow> A \<subseteq> fvs es"

apply(induct e and es rule: \.induct \s.induct)
apply (simp_all add:hyperset_defs)
apply blast+
done

lemma sqUn_lem: "A \<sqsubseteq> A' \<Longrightarrow> A \<squnion> B \<sqsubseteq> A' \<squnion> B"
by(simp add:hyperset_defs) blast

lemma diff_lem: "A \<sqsubseteq> A' \<Longrightarrow> A \<ominus> b \<sqsubseteq> A' \<ominus> b"
by(simp add:hyperset_defs) blast

(* This order of the premises avoids looping of the simplifier *)
lemma D_mono: "\<And>A A'. A \<sqsubseteq> A' \<Longrightarrow> \<D> e A \<Longrightarrow> \<D> (e::expr) A'"
and Ds_mono: "\<And>A A'. A \<sqsubseteq> A' \<Longrightarrow> \<D>s es A \<Longrightarrow> \<D>s (es::expr list) A'"

apply(induct e and es rule: \<D>.induct \<D>s.induct)
apply simp
apply simp
apply simp
apply simp
apply simp apply (iprover dest:sqUn_lem)
apply (fastforce simp add:hyperset_defs)
apply simp
apply simp
apply simp apply (iprover dest:sqUn_lem)
apply simp apply (iprover dest:sqUn_lem)
apply simp apply (iprover dest:diff_lem)
apply simp apply (iprover dest:sqUn_lem)
apply simp apply (iprover dest:sqUn_lem)
apply simp apply (iprover dest:sqUn_lem)
apply simp
apply simp
apply simp
apply (iprover dest:sqUn_lem)
done

(* And this is the order of premises preferred during application: *)
lemma D_mono': "\<D> e A \<Longrightarrow> A \<sqsubseteq> A' \<Longrightarrow> \<D> e A'"
and Ds_mono': "\<D>s es A \<Longrightarrow> A \<sqsubseteq> A' \<Longrightarrow> \<D>s es A'"
by(blast intro:D_mono, blast intro:Ds_mono)

end

subset_name: curated


TITLE: On finding sites with (difficult) practice problems
QUESTION [0 upvotes]: I am a HS junior taking Calculus BC and have probably one of the most difficult teachers to ever exist ever. Her first test was tear-stimulant and she offers absolutely no practice sheets or problems, other than the relatively simplistic examples she gives in class.
Which leads me to my next point: Are there any particular websites with repositories of these difficult problems? The majority of sites only explain basic concepts and simple problems to apply them to, which is useless for me.  If you know of any very challenging Precalc/Calc websites( OR BOOKS) , please link them here.
Thanks!
(Mods and experienced users: I am still very new to StackExchange in general. Sorry if this question is in the wrong community, and please redirect the question to the correct one if it is.)
EDIT: My little brother is a sophomore in Precalculus and would like the same thing- any other links?

REPLY [2 votes]: For sites, try Google "math problem books". For example, this link, and also this Math.SE post.
For calculus, I personally recommend this book by Demidovich. It is very popular in my country.


text:
\begin{document}
\newtheorem{assumption}{Assumption}
\newtheorem{Remark}{Remark}
\title{Estimating the Most Probable Transition Time for Stochastic Dynamical Systems

}

\titlerunning{Estimating the Most Probable Transition Time}

\author{Yuanfei Huang         \and Ying Chao \and
          Wei Wei \and Jinqiao Duan*
}

\institute{Yuanfei Huang \and Ying Chao \and Wei Wei\at
              School of Mathematics and Statistics \& Center for Mathematical Sciences \& Hubei National Center for Applied Mathematics, Huazhong University of Science and Technology, Wuhan 430074,  China. \\

              \email{yfhuang@hust.edu.cn, yingchao1993@hust.edu.cn, weiw16@hust.edu.cn}

           \and
              Jinqiao Duan \at
              Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA.\\
              \email{duan@iit.edu}
              \and
              $^*$Corresponding author
}

\date{Received: date / Accepted: date}

\maketitle

\begin{abstract}
This work is devoted to the investigation of the most probable transition time between metastable states for stochastic dynamical systems. Such a   system is modeled by a stochastic differential equation  with non-vanishing Brownian noise, and  is restricted in a domain with absorbing boundary. Instead of minimizing the Onsager-Machlup action functional, we examine the maximum probability that the solution process of the  system stays in a neighborhood (or a tube) of a transition path,  in order to characterize the most probable transition path. We first establish  the exponential decay lower bound  and  a power law decay upper bound for the maximum of this probability. Based on these estimates, we further derive the lower and upper bounds for the most probable transition time, under  suitable conditions. Finally, we illustrate our results in simple stochastic dynamical systems,  and highlight the relation with some relevant works.
\keywords{Stochastic differential equations \and  Most probable transition time \and Onsager-Machlup action functional \and Metastable states \and Rare events }

\end{abstract}

\section{Introduction}
\label{intro}

Stochastic differential equations (SDEs) are models for  complex   phenomena in physical, chemical, biological, and  engineering systems under random fluctuations. In particular, transition phenomena between dynamically significant states, such as climate change and gene transcription, occur  under the interaction of nonlinearity and uncertainty.  It has been an interesting issue  to quantify the transition behavior between two metastable states for stochastic dynamical systems. Indeed, this has been actively  investigated  \cite{Durr1978,Kath1981,Machlup1953,Faccioli2006,Zuckerman2000,Wang2006,Gobbo2012,Langouche1982,Schulman1981,Wiegel1986,Wio2013,Khandekar2000,Huang2019,Zeitouni1987,Zeitouni1988,Chao2019}. The Onsager-Machlup method and the related Path Integrals formulation have been set up to study this problem through minimizing a  so-called Onsager-Machlup  action functional.   It is difficult to predict and describe state changes or transitions in, for example,    climate systems and Arctic sea  \cite{Field2012,Ragone2018,Moon2017}. It was reported   \cite{Ditlevsen1999,Lucarini2019} that transition phenomena in climate systems complete in finite (or even relatively short) time scales. In molecule dynamics, the transitions between two molecule states (or protein configurations)  may occur in  finite or  short time \cite{Sturzenegger2018,Hoffer2019}. In gene regulation systems,   transitions between different concentration levels of   transcription factors may happen frequently \cite{Turcotte2008,Stefan2015}.

Therefore, as a part of investigation on the transition pathways, it is crucial   to characterize the transition time.   The purpose of this paper is to estimate the most probable transition time.

We consider the following SDE on a   domain $D$ in Euclidean space $\mathbf{R}^k$:
\makeatletter
\@addtoreset{equation}{section}
\makeatother
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\begin{equation}\label{firstequation}
dX_t=b(X_t)dt+c \; dB_t,  \;\; ~t\geq 0.
\end{equation}
The domain  $D$ is open, bounded, connected  and is taken to have absorbing boundary.  Here $B_t$ is a standard Brownian motion in $\mathbf{R}^k$. The noise intensity $c$ is a positive constant. We assume that the drift term $b(x)$ is  in the   space $C^2(\mathbf{R}^k)$ of functions having all continuous derivatives of order up to 2. Let $x_0$ and $x_f$  be  two   distinct metastable states of the system $(\ref{firstequation})$: $b(x_0)=0$ and $b(x_f)=0$.

For   system $(\ref{firstequation})$ with a given transition time $T$,  this is the usual setup for  studying transition paths between two metastable states \cite{Durr1978,Ikeda1980}:  Among all possible smooth paths connecting two metastable states ($X_0=x_0$ and $X_T=x_f$), which one is the most probable for the solution process of $(\ref{firstequation})$? The solution process of system $(\ref{firstequation})$ is almost surely nowhere differentiable. So to quantify which smooth path is the most probable one, an usual way is to compare the probabilities that the solution process stays in the neighborhood or `tube'  of such a smooth path. It was proved in \cite{Durr1978,Zeitouni1987,Zeitouni1988,Ikeda1980} that, the probability of the solution process of $(\ref{firstequation})$ in a   tube of a smooth path $\psi(t)$,  as the tube size scaling $\delta\rightarrow0$,   is
\begin{equation}\label{RN}
\mathcal{P}^{x_0}\{\|X_t-\psi(t)\|_T< \delta\}\approx\exp(-\frac{S^{OM}_T(\psi)}{c^2})\cdot \mathcal{P}^{x_0}\{\|B^c_t-x_0\|_T< \delta\},
\end{equation}
where $\mathcal{P}^{x_0}$ denotes the probability conditional on the initial position $X_0=x_0$,    $|\cdot|$ is the Euclidean norm, and $\|\cdot\|_T$ is the uniform norm:
\begin{equation}
\|\psi(t)\|_T=\sup_{0\leq t\leq T}|\psi(t)|.
\end{equation}
 Here $B^c_t=c(B_t-B_0)+x_0$ ( a shifted Brownian motion with magnitude $c$).     The Onsager-Machlup (OM) action functional is
\begin{equation}
S^{OM}_T(\psi)=\frac{1}{2}\int_{0}^{T}[\langle \dot{\psi}(t)-b(\psi(t)),(\dot{\psi}(t)-b(\psi(t)))\rangle+c^2\nabla\cdot b(\psi(t))]dt,
\end{equation}
where $\langle \cdot,\cdot\rangle$ denotes the Euclidean scalar product in $\mathbf{R}^k$. The `tube probability'  estimate  $(\ref{RN})$ shows that for a given transition time $T$, in order  to compare the probabilities that the solution process of $(\ref{firstequation})$ stays in the neighborhood of a smooth path, it can be approximately performed by comparing their corresponding OM action functionals. Thus the \textbf{most probable transition path}, when the transition time $T$ is given or known,  is defined as the one which minimizes  the OM action functional among a class of smooth paths \cite{Durr1978,Zeitouni1987,Zeitouni1988,Ikeda1980}:
\begin{equation}
\begin{split}
& \sup_{\psi\in \bar{C}_D([0,T],x_0,x_f)}\mathcal{P}^{x_0}\{\|X_t-\psi(t)\|_T< \delta\}  \\
\approx & \sup_{\psi\in \bar{C}_D([0,T],x_0,x_f)}\exp(-\frac{S^{OM}_T(\psi)}{c^2}) \cdot \mathcal{P}^{x_0}\{\|B^c_t-x_0\|_T< \delta\}     \label{dog}  \\
=& \exp\{-\frac{1}{c^2}\inf_{\psi\in \bar{C}_D([0,T],x_0,x_f)} S^{OM}_T(\psi)\} \cdot \mathcal{P}^{x_0}\{\|B^c_t-x_0\|_T< \delta\},
\end{split}
\end{equation}
where $\bar{C}_D([0,T],x_0,x_f)$ denotes the space of all absolutely continuous functions $g:[0,T]\rightarrow D$ such that $g(0)=x_0$ and $g(T)=x_f$.  As the two metastable states $x_0$ and $x_f$ are given,  we use the notation $\bar{C}_D[0,T]$ to denote $\bar{C}_D([0,T],x_0,x_f)$.   Note that when the transition time $T$ is \emph{known}, the factor  $\mathcal{P}^{x_0}\{\|B^c_t-x_0\|_T< \delta\}$  in   (\ref{dog}) does not affect the minimization.

The aforementioned  works focus on the case that the transition time $T$ is \emph{known}. However,    the transition time $T$  varies or is not known in advance in  many stochastic dynamical systems in mathematical modeling. It is thus desirable to estimate the transition time $T$.  Hence we aim to investigate the following \textbf{double optimization problem on the tube probability},  provided the tube      size $\delta$  satisfies the appropriate condition  $0<\delta<|x_f-x_0|$:
\begin{equation}\label{opt}
\sup_{T>0}\sup_{\psi\in \bar{C}_D([0,T])}\mathcal{P}^{x_0}\{\|X_t-\psi(t)\|_T< \delta\}.
\end{equation}
The \textbf{most probable transition time} for the stochastic dynamical system $(\ref{firstequation})$ transits from metastable state $x_0$ to another metastable state $x_f$ is defined as the time at which this  double optimization problem achieves its double maximum value. The corresponding  \textbf{most probable transition path} may be regarded as global.  We denote the most probable transition time as $T^{\delta}_{x_0\rightarrow x_f}$.

This paper is organized as follows. After recalling some preliminaries (Section $\ref{preliminaries}$), we investigate the probability that the solution process of the SDE stays in a neighborhood of a smooth path (Section $\ref{method}$,). We establish the exponential decay lower bound and power law decay upper bound of the maximum of  the probability for a class of transition paths. Furthermore,  we derive the bounds for the most probable transition time. Then we present two examples to illustrate our results (Section $\ref{example}$).   Finally, in Section $\ref{conclusion}$, we summarize our main results,  and discuss the connection and difference between our work and some   relevant works.

\section{Preliminaries}\label{preliminaries}

For simplicity, we consider one dimensional case (in   real line $\mathbf{R}$) from now on. We first recall some basic concepts about Brownian motion and  Wiener measure induced  by an SDE. Then we  prove results about approximation of this Wiener measure on cylinder sets, and exit properties for the   solution paths of the SDE.

\subsection{Brownian motion}
\begin{definition}
A stochastic process $\{B_t(w):t\geq0\}$ defined on a probability space $(\Omega,\mathcal{F},\mathcal{P})$ is called a Brownian motion or a Wiener process if the following conditions hold:
\begin{enumerate}
\item $B_0$=0 (a.s.);
\item The paths $t\rightarrow B_t(w)$ are continuous, a.s.;
\item $B_t$ has independent increments, i.e., if $0\leq t_1<t_2<\cdots<t_n$, then the random variables $B_{t_2}-B_{t_1},\cdots,B_{t_n}-B_{t_{n-1}}$ are independent.
\item $B_t$ has stationary increments that are Guassian distributed, i.e., $B_t(w)-B_{s}(w)$ has the normal distribution with mean 0 and variance $t-s$. Namely, $B_{t}(w)-B_s(w)\sim \mathcal{N}(0,t-s)$ for any $0\leq s<t$.
    \end{enumerate}
\end{definition}
In particular,   $c\; (B_{t}(w)-B_s(w))\sim \mathcal{N}(0,c^2(t-s))$, for  a positive constant $c$.

\subsection{Measure  induced by  the solution  process}\label{sectionp}

Let $X_t$ denote a nonexploding diffusion process on $[0,T]$ defined by the scalar stochastic differential equation in the probability space $(\Omega,\mathcal{F},\mathcal{P})$
\begin{equation}\label{diffusion1}
dX_t=b(X_t)dt+c\; dB_t,~X_0=x_0\in \mathbf{R}.
\end{equation}
The space of paths of such a diffusion process is the space $C([0,T],x_0)$ of continuous functions
\begin{equation}
C([0,T],x_0)=\{\psi(t)|\psi:[0,T]\rightarrow\mathbf{R},\psi(t)~\mbox{is continuous},\psi(0)=x_0\},
\end{equation}
with the uniform norm $\|\cdot\|_T$
\begin{equation}
\|\psi\|_T=\sup_{t\in[0,T]}|\psi(t)|,~\psi(t)\in C([0,T],x_0).
\end{equation}
In this norm,  we have the Borel field $\mathbf{B}_{[0,T]}^{x_0}$ of $C([0,T],x_0)$.

A subset $I_n$ of $C([0,T],x_0)$ in the following form is called an $n$-dimensional cylinder set:
\begin{equation}
I_n=\{\psi\in C([0,T],x_0)|(\psi(t_1),\cdots,\psi(t_n))\in H\},
\end{equation}
where $0<t_1<\cdots<t_n\leq T$ and $H$ is a Borel set  in $n$-dimensional Euclidean space. The collection of all $n$-dimensional cylinder sets is a $\sigma$-field and the class of all finite-dimensional cylinder sets is a field, which is denoted by $I$. It is known that the $\sigma$-field $\sigma(I)$, generated by $I$,  is the Borel field $\mathbf{B}_{[0,T]}^{x_0}$. That is,  $\sigma(I)=\mathbf{B}_{[0,T]}^{x_0}$.

The measure $\mu_X$ on $\mathbf{B}_{[0,T]}^{x_0}$ induced by the solution  process $X_t$ of the SDE  ($\ref{diffusion1}$) is defined by
\begin{equation} \label{Wiener}
\mu_X(B)=\mathcal{P}^{x_0}(\{w\in \Omega|~X_t(w)\in B\}),~B\in\mathbf{B}_{[0,T]}^{x_0}.
\end{equation}
Recall that such a measure induced by Brownian motion $B_t$ is the Wiener measure. For convenience, we also  call $\mu_X$ the Wiener measure induced by  solution process $X_t$.
Let $K_T(\psi,\delta)=\{x\in C([0,T],x_0)|~\|x-\psi\|_T<\delta\}$ denote the tube of a path $\psi$ with tube size $\delta$ ($i.e.$ neighborhood size),  for  $\psi\in C([0,T],x_0)$. Thus the probability that the solution process of $(\ref{diffusion1})$ stays in the $\delta$-tube of a smooth path $\psi\in \bar{C}([0,T],x_0)$ is     $\mu_X(K_T(\psi,\delta))$.

\begin{Remark}
By  the definition  (\ref{Wiener}),   the tube probability in the double optimization problem (\ref{opt}) will be estimated via  the Wiener measure $\mu_X$ for the rest of this paper.
\end{Remark}

We start with  the following lemma.
\begin{lemma}\label{lemma}
(Approximation on  cylinder sets for the measure  $\mu_X$  of a tube) \\
For a fixed transition time $T$,  a smooth path $\psi\in C([0,T],x_0)$ and a   tube size $\delta$, there exists a sequence of sets $\{J_n\}_{n=1}^{\infty}$ in the field $I$,  such that
\begin{equation}
\mu_{X}(K_T(\psi,\delta))=\lim_{n\rightarrow\infty}\mathcal{P}^{x_0}(J_n).
\end{equation}
\end{lemma}
{\bf Proof.}
Let $\mathbf{Q}$  be  the  countable set of rational numbers in  $\mathbf{R}$.   So for a fixed transition time $T>0$, the set $\mathbf{Q}\cap[0,T]$ is also countable. Define
\begin{equation}
\mathbf{Q}\cap[0,T]=\{q_0,q_1,\cdots,q_n,\cdots\}.
\end{equation}
If $T$ is a rational number, we switch the positions of 0 and $T$ with the ones of $q_0$ and $q_1$ respectly and still represent the sequence as
$\{q_0=0,q_1=T,q_2\cdots,q_n,\cdots\}$. If $T$ is an irrational number we switch the position of 0 with the one of $q_0$ and add $T$ into the sequence and denote the sequence as the same representation $\{q_0=0,q_1=T,q_2\cdots,q_n,\cdots\}$.

Introduce the subsets
\begin{equation}
\begin{split}
O_n&=\{q_0,q_1,\cdots,q_n\},\\
J_n&=\{X_t \in C([0,T],x_0)|~|X_{q}-\psi(q)|<\delta,\forall q\in O_n\}\subset I.
\end{split}
\end{equation}
Note  that $\{J_n\}_{n=1}^{\infty}$ is a decreasing sequence. Recall that the solution process of $(\ref{diffusion1})$ is almost surely continuous. So
\begin{equation}
\begin{split}
&\mu_X(K_T(\psi,\delta))\\
=&\mathcal{P}^{x_0}(\{w\in \Omega|\sup_{t\in[0,T]}|X_t(w)-\psi(t)|<\delta\})\\
=&\mathcal{P}^{x_0}(\{w\in \Omega|\sup_{t\in \mathbf{Q}\cap[0,T]}|X_t(w)-\psi(t)|<\delta\})\\
=&\mathcal{P}^{x_0}(\cap_{n=1}^{\infty}J_n)\\
=&\lim_{n\rightarrow\infty}\mathcal{P}^{x_0}(J_n).
\end{split}
\end{equation}
This completes the proof of this lemma. $\hfill\blacksquare$\\

\subsection{Non-exit probability and mean exit time of a diffusion process}\label{subsection}

In this subsection, we discuss the probability that a diffusion process $B^c_t=c(B_t-B_0)$ (where $B_t$ is a standard Brownian motion and c is a positive constant) stays in the $\delta$-tube of the origin 0 during the time period $[0,T]$. This probability was studied in   Lemma 8.1 of \cite{Ikeda1980} with unit noise intensity.  We need  to obtain the probability for different noise intensity.

Define  $G=\{x\in \mathbf{R}|~|x|<1\}$ and   $\tau^x_G(B^c_t)=\inf\{t|~B^c_t\notin G,~B^c_0=x\in G\}$. By \cite{Oksendal2003}, it is known that $\tau^x_G(B^c_t)$ is a stopping time. Then $u(t,x)=\mathcal{E}^{x}[f(B^c_t)I_{\{\tau^x_G(B^c_t)>t\}}]$ (here $\mathcal{E}^x$  denotes the expectation with respect to the probability $\mathcal{P}^x$), $x\in G$, $t>0$, is the solution of the initial value problem
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}=\frac{1}{2}c^2\Delta u,~~u\in G,~~\\
u|_{\partial G}=0,~~u|_{t=0}=f.
\end{cases}
\end{equation}
Consequently
\begin{equation}
u(t,x)=\sum_{n=1}^{\infty}e^{-\lambda_n t}\varphi_n(x)\int_{G}\varphi_n(y)f(y)dy,
\end{equation}
where $0<\lambda_0<\lambda_1\leq \lambda_2\leq\cdots$ are eigenvalues and $\{\varphi_n(x)\}$ are corresponding eigenfunctions of the eigenvalue problem
\begin{equation}
\begin{cases}
\frac{1}{2}c^2\triangle \varphi+\lambda\varphi=0~~\mbox{in G},\\
\varphi|_{\partial G}=0.
\end{cases}
\end{equation}
The eigenfunctions (which need to be nonzero) are $\{\sin(m\pi x)\}_{m=1}^{\infty}$ and $\{\cos((m+\frac{1}{2})\pi x)\}_{m=0}^{\infty}$,  with the corresponding eigenvalues
$$\{\frac{(m\pi)^2}{2}c^2\}_{m=1}^{\infty}, \;\;\;     ~\{\frac{[(m+\frac{1}{2})\pi]^2}{2}c^2\}_{m=0}^{\infty}.$$
The set of normalized eigenfunctions form an orthonormal basis for the Hilbert space $H=L^2(-1,1)$.

In particular,
\begin{equation}\label{probability}
\begin{split}
&\mu_{B^c}(K_T(0,\delta))\\
=&\mathcal{P}^{x_0}(\{w\in \Omega|~B^c_t(w)\in K_T(0,\delta)\})\\
=&\mathcal{P}^{x_0}(\|B^c_t-0\|_T<\delta)\\
=&\sum_{n=0}^{\infty}e^{-\lambda_n T/\delta^2}\varphi_n(0)\int_{G}\varphi_n(y)dy\\
=&\sum_{n=0}^{\infty}\frac{(-1)^n4}{(2n+1)\pi}\exp\{-\frac{(2n+1)^2\pi^2c^2T}{8\delta^2}\}.
\end{split}
\end{equation}

Now we turn to discuss the mean exit time of system $(\ref{firstequation})$. The following lemma from \cite{Duan2015,Oksendal2003} is needed later.
\begin{lemma}\label{lemma2}
(Mean exit time) The mean exit time $u(x)=\mathcal{E}\tau^x_{D}(X_t)$ of the stochastic system $(\ref{firstequation})$ with $c=1$, for an orbit ($i.e.$, a trajectory) starting at $x\in D$, satisfies the following elliptic partial differential equation
\begin{equation}
Au=-1,u|_{\partial D}=0,
\end{equation}
where $\partial D$ is the boundary of $D$ and $A$ is the generator
\begin{equation}
Au=b\cdot\nabla u+\frac{1}{2}\triangle u.
\end{equation}
Moreover, if the domain $D$ has $C^{2,\gamma}$ boundary and the drift $b$ is in $C^{\gamma}(D)$ for some $\gamma\in(0,1)$, then the mean exit time $u$ uniquely exists and is in $C^{2,\gamma}(D)$.
\end{lemma}
  Recall that $C^{\gamma}(D)$ is the H\"{o}lder space consisting of functions in $D$ which are locally H\"{o}lder continuous with exponent $\gamma$. In particular,   H\"{o}lder space $C^{2,\gamma}(D)$ is the subspace of $C^2(D)$ consisting of functions whose second order derivatives are locally H\"{o}lder continuous with exponent $\gamma$. A bounded domain $D$   is called a $C^{2,\gamma}$ domain if each point of its boundary $\partial D$ has a neighborhood in which $\partial D$ is the graph of a $C^{2,\gamma}$ function. We also say that $D$ has a $C^{2,\gamma}$ boundary.

\section{Bounds for the Most Probable Transition Time} \label{method}

In this section, without loss of generality,  we take domain  $D=\{x\in \mathbf{R}|~|x-x_0|<l\}$,  with $l$   a large enough positive constant (such that $|x_f-x_0|\ll l$).  For a given tube size $\delta$, the upper and lower bounds of $\sup_{\psi\in \bar{C}_D[0,T]}\mu_X(K_T(\psi,\delta))$  are estimated  in the following two subsections. Then we   establish the bounds for the most probable transition time in Subsections 3.3.  Finally, we examine    the connection between the double optimization problems  on the tube probability  $(\ref{opt})$ and on the  OM action functional $S_T^{OM}(\psi)$, in Subsection 3.4.

\subsection{Upper bound of $\sup_{\psi\in \bar{C}_D[0,T]}\mu_X(K_T(\psi,\delta))$   }

Define an `enlarged'  set      $D^*=\{x\in \mathbf{R}|~|x-x_0|<l+\delta\}$ and let $\tau^{x_0}_{D^*}$ be the first time that the solution process $X_t$ of system $(\ref{diffusion1})$ escapes from the domain $D^*$:
\begin{equation}
\tau^{x_0}_{D^*}=\inf\{t>0|X_0=x_0,X_t\in\partial D^*\}.
\end{equation}

For every time instant     $t>0$,
\begin{equation}
\begin{split}
\mathcal{E}\tau^{x_0}_{D^*}=&\int_{\Omega}\tau^{x_0}_{D^*}(w)d\mathcal{P}^{x_0}(w)\\
=&\int_{\{\tau^{x_0}_{D^*}(w)\leq t\}}\tau^{x_0}_{D^*}(w)d\mathcal{P}^{x_0}(w)+\int_{\{\tau^{x_0}_{D^*}(w)> t\}}\tau^{x_0}_{D^*}(w)d\mathcal{P}^{x_0}(w)\\
\geq&\int_{\{\tau^{x_0}_{D^*}(w)\leq t\}}\tau^{x_0}_{D^*}(w)d\mathcal{P}^{x_0}(w)+\int_{\{\tau^{x_0}_{D^*}(w)> t\}}td\mathcal{P}^{x_0}(w)\\
=&\int_{\{\tau^{x_0}_{D^*}(w)\leq t\}}\tau^{x_0}_{D^*}(w)d\mathcal{P}^{x_0}(w)+t\mathcal{P}^{x_0}(w|\tau^{x_0}_{D^*}(w)>t).
\end{split}
\end{equation}
According to   Lemma $\ref{lemma2}$ and the regularities of the domain $D^*$ and the drift term $b(x)$,  we know that $\mathcal{E}\tau^{x_0}_{D^*}$ is finite.    Letting $t$ tend to infinity, we obtain that
\begin{equation}
\lim_{t\rightarrow\infty}t\mathcal{P}^{x_0}(w|\tau^{x_0}_{D^*}(w)>t)=0.
\end{equation}
So for a positive constant $\epsilon$, there exists a positive constant $t_{\epsilon}$ such that for any $t>t_{\epsilon}$:
\begin{equation}
t\mathcal{P}^{x_0}(w|\tau^{x_0}_{D^*}(w)>t)<\epsilon\Rightarrow  \mathcal{P}^{x_0}(w|\tau^{x_0}_{D^*}(w)>t)<\frac{\epsilon}{t}.
\end{equation}
Notice that for every  $\psi\in \bar{C}_D([0,T])$ with  $T>t_{\epsilon}$,
\begin{equation}
\begin{split}
&\mu_X(K_T(\psi,\delta))\\
=&\mathcal{P}^{x_0}(w|~|X_t(w)-\psi(t)|<\delta,\forall t\in[0,T])\\
\leq&\mathcal{P}^{x_0}(w|~|X_t(w)-x_0|<|\psi(t)-x_0|+\delta,\forall t\in[0,T])\\
\leq&\mathcal{P}^{x_0}(w|~|X_t(w)-x_0|<l+\delta,\forall t\in[0,T])\\
=&\mathcal{P}^{x_0}(w|~X_t(w)\in D^*,\forall t\in[0,T])\\
\leq&\mathcal{P}^{x_0}(w|~\tau^{x_0}_{D^*}(w)>T)\\
<&\frac{\epsilon}{T}.
\end{split}
\end{equation}
Thus for $T>t_{\epsilon}$,
\begin{equation}
\sup_{\psi\in \bar{C}_D[0,T]}\mu_X(K_T(\psi,\delta))\leq\frac{\epsilon}{T}.
\end{equation}

\subsection{Lower bound of $\sup_{\psi\in \bar{C}_D[0,T]}\mu_X(K_T(\psi,\delta))$   }

Now we discretize $X_t$ of $(\ref{diffusion1})$ in the following way:
\begin{equation}\label{descreteversion}
\begin{cases}
X_{i}-X_{i-1}=[\kappa b(X_{i})+(1-\kappa)b(X_{i-1})]\Delta t+c(B_i-B_{i-1}),i=1,\cdots n,\\
X_0=x_0,
\end{cases}
\end{equation}
For simplicity we consider the following discretized version which has the same distribution of $(\ref{descreteversion})$:
\begin{equation}
\begin{cases}
X_{i}-X_{i-1}=[\kappa b(X_{i})+(1-\kappa)b(X_{i-1})]\Delta t+B^c_i-B^c_{i-1},i=0,1,\cdots n,\\
X_0=x_0,
\end{cases}
\end{equation}
where $(B^c_{t}(w)-B^c_s(w))\sim \mathcal{N}(0,c^2(t-s))$ and the time partition is $\Pi_n$: $0=t_0<t_1=\cdots<t_n=T$ and $\kappa\in[0,1]$. For every  $\psi\in C^2_D[0,T]:=\{\psi\in C_D[0,T]|\mbox{$\psi$ has all continuous derivatives of order no more than $2$}\}$,   and using    Lemma $\ref{lemma}$, the probability $\mu_X(K_T(\psi,\delta))$ is calculated in the following way (for every $O_n$ in Lemma $\ref{lemma}$, we reorder the elements of $O_n$ from small to large and use the new sequence for the time partition $\Pi_n$):
{\small\begin{equation}\label{pathprobability}
\begin{split}
&\mu_X(K_T(\psi,\delta))\\
=&\mu_X(\sup_{t\in [0,T]}|X_t-\psi_t|<\delta)\\
=&\lim_{n\rightarrow\infty}\mathcal{P}^{x_0}(w|\{X_{t_0},X_{t_1},\cdots,X_{t_n}\}\in \mathcal{I}_n)\\
=&\lim_{n\rightarrow\infty}\int_{\{\{B^c_{i}\}~s.t.~\{X_{t_0},X_{t_1},\cdots,X_{t_n}\}\in \mathcal{I}_n\}}\mathcal{D}^c_nB^c \exp\{-\frac{1}{2}\sum_{i=1}^{n}\frac{(B^c_i-B^c_{i-1})^2}{c^2\Delta t}\}\\
=&\lim_{n\rightarrow\infty}\int_{\{|x_i-\psi_i|<\delta\}}\mathcal{D}^c_nx\exp\{-\sum_{i=1}^{n}[\frac{(x_i-x_{i-1}-[\kappa b(x_{i})+(1-\kappa)b(x_{i-1})]\Delta t)^2}{2c^2\Delta t}]\}\\
&\cdot\prod_{i=1}^n(1+\kappa b'(x_i)\Delta t)\\
=&\lim_{n\rightarrow\infty}\int_{\{|y_i|<\delta\}}\mathcal{D}^c_ny\prod_{i=1}^n(1+\kappa b'(y_i+\psi_i)\Delta t)\\
&\cdot\exp\{-\frac{1}{2}\sum_{i=1}^{n}[\frac{(y_i-y_{i-1}+\psi_i-\psi_{i-1}-[\kappa b(y_{i}+\psi_i)+(1-\kappa)b(y_{i-1}+\psi_{i-1})]\Delta t)^2}{c^2\Delta t}]\}\\
=&\int_{|y|<\delta}\mathcal{D}^cy\exp\{-\frac{1}{2}\int_{0}^{T}[\frac{(\dot{y}+\dot{\psi}-b(y+\psi))^2}{c^2}+2\kappa b'(y+\psi)]dt\}\\
=&\int_{|y|<\delta}\mathcal{D}^cy\exp\{\frac{\mathcal{U}(y+\psi)|_{0}^{T}}{c^2}-\frac{1}{2}\int_{0}^{T}[\frac{\dot{y}^2+2\dot{y}\dot{\psi}+\dot{\psi}^2+b^2(y+\psi)}{c^2}+2\kappa b'(y+\psi)]dt\}\\
=&\exp\{-\frac{1}{2c^2}\int_{0}^{T}[(\dot{\psi}-b(\psi))^2+2\kappa c^2b'(\psi)]dt\}\int_{|y|<\delta}\mathcal{D}^cy\exp\{\frac{(\mathcal{U}(y(T)+x_f)-\mathcal{U}(x_f))}{c^2}\\
&-\int_{0}^{T}[\frac{\dot{y}^2+2\dot{y}\dot{\psi}+b^2(y+\psi)-b^2(\psi)}{2c^2}+\kappa (b'(y+\psi)-b'(\psi))]dt\},
\end{split}
\end{equation}}
where $\mathcal{I}_n=(\psi_{t_0}-\delta,\psi_{t_0}+\delta)\times(\psi_{t_1}-\delta,\psi_{t_1}+\delta)\times\cdots\times(\psi_{t_n}-\delta,\psi_{t_n}+\delta)$ and $\mathcal{D}^c_nB^c=\prod_{i=1}^n\frac{dB^c_i}{\sqrt{2\pi c^2\Delta t}}$, $\mathcal{D}^cx=\lim_{n\rightarrow\infty}\prod_{i=1}^n\frac{dx_i}{\sqrt{2\pi c^2\Delta t}}$
(the notation $\mathcal{D}_n^cx$ and $\mathcal{D}^cy$ are defined in a similar way) and
\begin{equation}
\mathcal{U}(x)=\int_{x_0}^xb(y)dy.
\end{equation}

\begin{Remark}
In $(\ref{pathprobability})$ we have used the results of  the path integrals method. For more mathematical details, such as the existence of the limitation and the links between discrete approximation and continuous functionals,       see   \cite{Chaichian2001}.
\end{Remark}

For a path $\psi\in C_D([0,T])$, denote that
\begin{equation}\label{notation}
\begin{split}
&\gamma(\psi,\delta):=\bigcup_{t\in[0,T]}\{x\in \mathbf{R}|~|x-\psi(t)|<\delta\}\subseteq\mathbf{R},\\
&h_0:=\sup_{|x-x_f|<\delta}|\mathcal{U}(x)-\mathcal{U}(x_f))|,\\
&h_1(\psi,\delta):=\sup_{x\in \gamma(\psi,\delta)}|b(x)b'(x)|,\\
&h_2(\psi,\delta):=\sup_{x\in \gamma(\psi,\delta)}|b''(x)|.
\end{split}
\end{equation}

Now we give the following estimations:
{\small\begin{equation}\label{estimation}
\begin{split}
|\int_{0}^T\dot{y}\dot{\psi}dt|=&|y(T)\dot{\psi}(T)-y(0)\dot{\psi}(0)-\int_{0}^Ty\ddot{\psi}dt|\\
=&|y(T)\dot{\psi}(T)-\int_{0}^Ty\ddot{\psi}dt|\leq|\dot{\psi}(T)|\delta+\|\ddot{\psi}\|T\delta,\\
|\int_{0}^Tb^2(y+\psi)-b^2(\psi)dt|=&|\lim_{n\rightarrow\infty}\sum_{i=1}^{n}[b^2(y_{i-1}+\psi_{i-1})-b^2(\psi_{i-1})]\Delta t|\\
=&|\lim_{n\rightarrow\infty}\sum_{i=1}^{n}[2b(\theta^1_{i-1}y_{i-1}+\psi_{i-1})b'(\theta^1_{i-1}y_{i-1}+\psi)y_{i-1}]\Delta t|\\
\leq&\lim_{n\rightarrow\infty}\sum_{i=1}^{n}|2b(\theta^1_{i-1}y_{i-1}+\psi_{i-1})b'(\theta^1_{i-1}y_{i-1}+\psi)y_{i-1}|\Delta t\\
\leq& 2h_1\delta\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\Delta t\\
=&2h_1\delta T,\\
|\int_{0}^Tb'(y+\psi)-b'(\psi)dt|=&|\lim_{n\rightarrow\infty}\sum_{i=1}^{n}[b'(y_{i-1}+\psi_{i-1})-b'(\psi_{i-1})]\Delta t|\\
=&|\lim_{n\rightarrow\infty}\sum_{i=1}^{n}[b''(\theta^2_{i-1}y_{i-1}+\psi_{i-1})y_{i-1}]\Delta t|\\
\leq&\lim_{n\rightarrow\infty}\sum_{i=1}^{n}|b''(\theta^2_{i-1}y_{i-1}+\psi_{i-1})y_{i-1}|\Delta t\\
\leq& h_2\delta\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\Delta t\\
=&h_2\delta T,\\
\end{split}
\end{equation}}
where $\theta^j_{i-1}\in[0,1]$ for $j=1,2$ and $i=1,2,\cdots,n$,  and  we have used the well known mean value theorem.

Substituting the estimations $(\ref{estimation})$ into $(\ref{pathprobability})$,  we obtain that
\begin{equation}
\begin{split}
\mu_X(K_T(\psi,\delta))\geq\exp\{-\frac{1}{c^2}[|\dot{\psi}(T)|\delta+h_0+(h_1+\kappa c^2h_2+\|\ddot{\psi}\|)T\delta]\}\\
\cdot\exp\{-\frac{1}{c^2}S^{\kappa,c}_T(\psi)\}\mu_{B^c}(K_T(0,\delta)),
\end{split}
\end{equation}
where $S^{\kappa,c}_T(\psi)=\int_{0}^TL^{\kappa,c}(\psi,\dot{\psi})dt=\int_{0}^T\frac{1}{2}[(\dot{\psi}-b(\psi))^2+2\kappa c^2b'(\psi)]dt$.

\begin{Remark}
When $\kappa=\frac{1}{2}$, the   functional $S^{\kappa,c}_T(\psi)$ is the OM action functional. The parameter $\kappa$ sometimes plays an important role in studying the dynamical behavior of stochastic systems. For example, to describe time-reversible dynamics, the effective action based on the symmetrical (Stratonovich's, $\kappa=\frac{1}{2}$) interpretation is applied to the system with additive noise \cite{Faccioli2006,Zuckerman2000}. In \cite{Volpe2010}, an experiment on a Brownian particle near a wall suggests that the system favors the anti-It\^{o}'s ($\kappa=1$) interpretation rather than the Stratonovich's, to ensure the Boltzmann-Gibbs distribution for the final steady state.
\end{Remark}

To give a clear and intuitive approximate form of the lower bound, we consider the following family of paths $\{\phi_T(t)\}_{T>0}$:
\begin{equation}
\phi_T(t)=\frac{x_f-x_0}{T}t+x_0,~t\in[0,T].
\end{equation}
Then we have
\begin{equation}
\begin{split}
&\sup_{\psi\in \bar{C}_D[0,T]}\mu_X(K_T(\psi,\delta))\\
\geq&\mu_X(K_T(\phi_T,\delta))\\
\geq &\exp\{-\frac{1}{c^2}[|\dot{\phi}_T(T)|\delta+h_0+(h_1(\phi_T,\delta)+\kappa c^2h_2(\phi_T,\delta)+\|\ddot{\phi}_T\|)T\delta]\}\\
&\cdot\exp\{-\frac{1}{c^2}S^{\kappa,c}_T(\phi_T,\dot{\phi}_T)\}\mu_{B^c}(K_T(0,\delta))\\
=&\exp\{-\frac{h_0}{c^2}\}\exp\{-\frac{1}{c^2}[\frac{|x_f-x_0|}{T}\delta+(h_1(\phi_T,\delta)+\kappa c^2h_2(\phi_T,\delta))T\delta]\}\\
&\cdot\exp\{-\frac{1}{c^2}S^{\kappa,c}_T(\phi_T,\dot{\phi}_T)\}\exp\{\ln(\mu_{B^c}(K_T(0,\delta)))\}\\
:=&c_0\exp\{-\frac{1}{c^2}S^{\kappa,c}_T(\phi_{T},\dot{\phi}_{T})-c_1T+\ln(\mu_{B^c}(K_T(0,\delta)))-\frac{|x_f-x_0|\delta}{Tc^2}\},
\end{split}
\end{equation}
where
\begin{equation}\label{coefficient}
\begin{split}
c_0=&\exp\{-\frac{h_0}{c^2}\},\\
c_1=&\frac{1}{c^2}[(h_1(\phi_T,\delta)+\kappa c^2h_2(\phi_T,\delta))\delta],
\end{split}
\end{equation}
are positive constants depending on the tube $K_T(\phi_T,\delta)$. Notice that all the paths $\{\phi_T(t)\}_{T>0}$ share the same curve in the state space $\mathbf{R}$ which means the two constants $c_0,c_1$ are independent of time $T$.  Here we have used the fact  that $x_f$ and $x_0$ are distinct metastable states.  Furthermore, we obtain that
{\small\begin{equation}\label{exponent}
\begin{split}
&c_0\exp\{-\frac{1}{c^2}S^{\kappa,c}_T(\phi_{T},\dot{\phi}_{T})-c_1T+\ln(\mu_{B^c}(K_T(0,\delta)))-\frac{|x_f-x_0|\delta}{Tc^2}\}\\
=&c_0\exp\{-\frac{1}{2}\int_0^T[\frac{(\dot{\phi}_T-b(\phi_T))^2}{c^2}+2\kappa b'(\phi_T)]dt-c_1T+\ln(\mu_{B^c}(K_T(0,\delta)))\\
&-\frac{|x_f-x_0|\delta}{Tc^2}\}\\
=&c_0\exp\{-\frac{1}{2}\int_0^T[\frac{((\frac{x_f-x_0}{T})-b(\phi_T))^2}{c^2}+2\kappa b'(\phi_T)]dt-c_1T+\ln(\mu_{B^c}(K_T(0,\delta)))\\
&-\frac{|x_f-x_0|\delta}{Tc^2}\}\\
=&c_0\exp\{-\frac{(x_f-x_0)^2}{2Tc^2}+\frac{(x_f-x_0)}{Tc^2}\int_0^Tb(\phi_T)dt-\frac{1}{2}\int_0^T[\frac{b^2(\phi_T)}{c^2}+2\kappa b'(\phi_T)]dt-c_1T\\
&+\ln(\mu_{B^c}(K_T(0,\delta)))-\frac{|x_f-x_0|\delta}{Tc^2}\}\\
=&c_0\exp\{-\frac{(x_f-x_0)^2}{2Tc^2}+\frac{(x_f-x_0)}{c^2}\int_0^1b(\phi_1)dt-\frac{T}{2}\int_0^1[\frac{b^2(\phi_1)}{c^2}+2\kappa b'(\phi_1)]dt-c_1T\\
&+\ln(\mu_{B^c}(K_T(0,\delta)))-\frac{|x_f-x_0|\delta}{Tc^2}\}\\
=&c_0\exp\{-\frac{(x_f-x_0)^2}{2Tc^2}+\frac{(x_f-x_0)}{c^2}\int_0^1b(\phi_1)dt-\frac{T}{2}\int_0^1\frac{b^2(\phi_1)}{c^2}dt-\frac{\kappa T (b(x_f)-b(x_0))}{x_f-x_0}\\
&-c_1T+\ln(\mu_{B^c}(K_T(0,\delta)))-\frac{|x_f-x_0|\delta}{Tc^2}\}\\
=&c_0\exp\{-\frac{(x_f-x_0)^2+2|x_f-x_0|\delta}{2Tc^2}+\frac{(x_f-x_0)}{c^2}\int_0^1b(\phi_1)dt-(\frac{1}{2}\int_0^1\frac{b^2(\phi_1)}{c^2}dt+c_1)T\\
&+\ln(\mu_{B^c}(K_T(0,\delta)))\}.
\end{split}
\end{equation}}
Here we use the fact that
\begin{equation}
\phi_T(t)=\phi_1(\frac{t}{T}),~t\in[0,T],
\end{equation}
and
\begin{equation}
\int_{0}^Tb(\phi_T(t))dt=T\int_{0}^Tb(\phi_1(\frac{t}{T}))d\frac{t}{T}=T\int_{0}^1b(\phi_1(t))dt.
\end{equation}

Notice that the integral term $\int_0^1b(\phi_1)dt$ is a constant. This lower bound can be rewritten as
\begin{equation}
k_0\exp\{-\frac{k_1}{T}-k_2T+\ln(\mu_{B^c}(K_T(0,\delta)))\},
\end{equation}
where
\begin{equation}
\begin{split}
k_0=&c_0\exp\{\frac{(x_f-x_0)}{c^2}\int_0^1b(\phi_1)dt\},\\
k_1=&\frac{(x_f-x_0)^2+2|x_f-x_0|\delta}{2c^2},\\
k_2=&(\frac{1}{2}\int_0^1\frac{b^2(\phi_1)}{c^2}dt+c_1).
\end{split}
\end{equation}

\subsection{Estimation of the most probable transition time}

We now summarise the estimation on the most probable transition time  in the following theorem.
\begin{theorem}\label{theorem1}
(Upper and lower bounds of the most probable transition time)\\
 For the stochastic dynamical system (\ref{diffusion1})   in a bounded domain $D$   with $C^{2,\gamma}$ boundary,  assume that the drift   $b$ is of $C^{2}(D)$, and $x_0$ and $x_f$ are two distinct metastable states. Then for every $\delta$ with  $0<\delta<|x_f-x_0|$, there exist   strictly positive upper and lower bounds for the most probable transition time $T^{\delta}_{x_0\rightarrow x_f}$.
\end{theorem}
{\bf Proof.} We derive the upper and lower bounds for the most probable transition time  separately.

\textbf{Step 1: Upper bound.}

 Combining the upper and lower bounds in previous sections  and  for every positive constant $\epsilon$,  we have
\begin{equation}
k_0\exp\{-\frac{k_1}{T}-k_2T+\ln(\mu_{B^c}(K_T(0,\delta)))\}\leq\sup_{\psi\in \bar{C}_D[0,T]}\mu_X(K_T(\psi,\delta))\preceq \frac{\epsilon}{T},
\end{equation}
where the notation $\preceq$ means the inequality holds when $T>t_{\epsilon}$ and $\{\phi_T\}_{T>0}$ is the family of paths which have been introduced earlier.  Recalling the infinite series representation of $\mu_{B^c}(K_T(0,\delta))$ in $(\ref{probability})$, and denoting that
\begin{equation}
\begin{split}
\mu_1(t):=\sum_{n=0}^{1}\frac{(-1)^n4}{(2n+1)\pi}\exp\{-\frac{(2n+1)^2\pi^2c^2t}{8\delta^2}\}),
\end{split}
\end{equation}
we have
\begin{equation}\label{inequality}
k_0\exp\{-\frac{k_1}{T}-k_2T+\ln(\mu_1(t))\}\leq\sup_{\psi\in \bar{C}_D[0,T]}\mu_X(K_T(\psi,\delta))\preceq \frac{\epsilon}{T}.
\end{equation}
Note that the function $\Theta(t)$ defined by
\begin{equation}
\begin{split}
\Theta(t):=k_0\exp\{-\frac{k_1}{t}-k_2t+\ln(\mu_1(t))\},
\end{split}
\end{equation}
is continuous and differentiable in $(0,\infty)$. Moreover,
\begin{equation}
\begin{split}
\frac{d\Theta(t)}{dt}=\Theta(t)[\frac{k_1}{t^2}-k_2+\frac{1}{\mu_1(t)}\frac{d\mu_1(t)}{dt}].
\end{split}
\end{equation}
It can be checked that  $\frac{d\mu_1(t)}{dt}$ is uniformly bounded for all  $t\in[0,\infty)$.

So we know that there exist constants $ T'_1$ and $ T'_2$, with $0<T'_1\leq T'_2<\infty$,  such that $\Theta(t)$ monotonically increases in $[0,T_1']$ starting at 0,  and  monotonically   decreases in $[T_2',+\infty)$ tending to 0 as $t\rightarrow\infty$. Thus the function $\Theta(t)$ achieves its maximum value for some $T'\in[T'_1,T'_2]$.

Define a function $\Xi(t)$:
\begin{equation}
\Xi(t)=\sup_{\psi\in \bar{C}_D[0,t]}\mu_X(K_t(\psi,\delta)).
\end{equation}
Since for every  path $\psi\in \bar{C}_D[0,t]$, the probability $\mu_X(K_t(\psi,\delta))$ is well defined (this has been introduced in Subsection $\ref{sectionp}$). Thus for any $t\in[0,\infty)$, the function $\Xi(t)$ is well defined and it is easy to see  that $0\leq\Xi(t)\leq1$.

Therefore
\begin{equation}
\Theta(t)\leq\Xi(t)\preceq \frac{\epsilon}{t},~t\in[0,\infty).
\end{equation}

For a given constant $\epsilon>0$, denote   $T^*=\max\{\frac{\epsilon}{\Theta(T')},t_\epsilon\}$. Then for any $t\in[T^*,\infty)$,  we have
\begin{equation}
\Xi(t)\leq\Theta(T')\leq\Xi(T').
\end{equation}
Thus the function $\Xi(t)$ achieves its maximum value in finite interval $[0,T^*]$,  and the  upper bound $T^*$  for the most probable transition path   is thus established.

\medskip

\textbf{Step 2: Lower bound.}

We now prove that the most probable transition time has a   positive lower bound.
Notice that $|x_f-x_0|>\delta$ and the solution process $X_t$ of $(\ref{diffusion1}$) is almost surely continuous. If such a  lower bound does not exist, then
\begin{equation}
\begin{split}
&\sup_{T>0}\sup_{\psi\in \bar{C}_D[0,T]}\mathcal{P}^{x_0}\{\|X_t-\psi(t)\|_T< \delta\}\\
=&\lim_{T\downarrow0}\sup_{\psi\in \bar{C}_D[0,T]}\mathcal{P}^{x_0}\{\|X_t-\psi(t)\|_T< \delta\}\\
\leq&\lim_{T\downarrow0}\mathcal{P}^{x_0}(\{w|~|X_T(w)-x_f|<\delta\})\\
=&0.
\end{split}
\end{equation}
This contradicts   the fact that the double supreme value is strictly positive. So the proof is complete.$\hfill\blacksquare$\\

Theorem $\ref{theorem1}$   provides   a rough range for the most probable transition time:
\begin{equation}  \label{bounds}
  0<\varrho\leq T^{\delta}_{x_0\rightarrow x_f}\leq T^*<\infty.
 \end{equation}
 Figure $\ref{demonstration}$ offers an intuitive representation of our idea. The left graph of Figure $\ref{demonstration}$ shows a schematic plot of the relationship $(\ref{inequality})$. The right graph shows the   logarithm form of the relationship.

\begin{figure}
  \centering

  \includegraphics[width=1\textwidth]{demonstration}\\
  \caption{A schematic plot:  (Left)  Relationship  (\ref{inequality}).  (Right)  Logarithm of  relationship  (\ref{inequality}).}\label{demonstration}.
\end{figure}

\subsection{Modified action functional for estimating the most probable transition time      }

In this subsection, we try to find the connection between the double optimization problems on the tube probability  $(\ref{opt})$ and on the Onsager-Machlup action functional $S_T^{OM}(\psi)$.  Based on this connection, we will present an estimation  method for the most probable transition time, under   additional assumptions. To achieve this we start with the framework and assumptions of \cite{Du2020}.

We focus on a gradient system with the drift term $b(x)=-V'(x)$,  for a  potential energy $V(x)\in C^3(\mathbf{R})$, although some results can be readily extended to the non-gradient case. The path potential $U(x)$ is given by
\begin{equation}
U(x)=c^2V''(x)-\frac{1}{2}(V'(x))^2.
\end{equation}
Then $U(x)\in C^1(\mathbf{R})$ by the smoothness assumption on $V$.  We make the following assumptions on the potentials $V$ and $U$.
\begin{assumption}
There exists a local minimizer $x_m$ of $V(x)$, such that $V'(x_m)=0$, and $V''(x_m)$ is strictly positive.
\end{assumption}
\begin{assumption}
The   maximizers  of $U(x)$ are contained in a bounded domain.
\end{assumption}
\begin{assumption}
For every  $E\in\mathbf{R}$, the level set $\mathcal{L}_E=\{x\in \mathbf{R}|~U(x)=E\}$ can be decomposed into a finite number of closed and connected subsets, $i.e.$
\begin{equation}
\mathcal{L}_E=\bigcup_{k=1}^NB_k,
\end{equation}
where the subsets $B_k$ are closed and connected, and (pair-wise disjoint) $B_j\bigcap B_k=\emptyset$ if $j\neq k.$
\end{assumption}
The critical point is defined    \cite{Du2020} as follows:
 The point  $x^*\in \mathbf{R}$ is called a critical point if
\begin{equation}
U(x^*)=\max_{y\in \mathbf{R}}U(y).
\end{equation}

Denoting the set of critical points by $\Lambda$, we adopt the following   assumption.
\begin{assumption}
The set of critical points, $\Lambda$,  is discrete and has no accumulation points.
\end{assumption}

Note that
\begin{equation}
\begin{split}
S_T^{OM}(\psi)=&\int_0^T(\frac{1}{2}(\dot{\psi})^2+V'(x)\dot{\psi}-U(\psi))dt\\
=&\int_0^T(\frac{1}{2}(\dot{\psi})^2-U(\psi))dt+V(x_f)-V(x_0).\\
\end{split}
\end{equation}
We make the following assumptions on the minimizer of $S_T^{OM}(\psi)$.
\begin{assumption}\label{unique}
For    given  metastable states $x_0, x_f $ and final time $T$, the OM action functional  $S_T^{OM}(\psi)$ has a unique minimizer $\psi_T$.
\end{assumption}
\begin{assumption}\label{uniformbound}
For    given  metastable states $x_0, x_f $ and final  time $T$, there exists a positive constant $M=M(x_0,x_f)$ such that the minimizer $\psi_T$ of the  OM action functional $S_T^{OM}(\psi)$ satisfies an integral condition, i.e., the `speed' of the minimizing transition path has bounded integral:
$$\int_{0}^T|\dot{\psi}_T|dt\leq M.$$
\end{assumption}

Under   Assumption $\ref{uniformbound}$, the minimizer $\psi_T \in C^2([0,T],x_0,x_f)$ and satisfies the following Euler-Lagrangian equation:
\begin{equation}\label{ELequation}
\begin{cases}
\ddot{\psi}_T+U'(\psi_T)=0,\\
\psi_T(0)=x_0, \; \;  \psi_T(T)=x_f.
\end{cases}
\end{equation}
The proof of the smoothness of the minimizer can be found in \cite{Giaquinta2004}. It is a classical result that the energy of this Euler-Lagrangian equation is conserved along the path $\psi_T$, $i.e.$
\begin{equation}\label{conserved}
\frac{1}{2}(\dot{\psi}_T)^2+U(\psi_T)\equiv E,  \;    t\in[0,T].
\end{equation}
With   Assumption $\ref{unique}$, the value  $E$ (or denoted by $E(T)$) is uniquely determined by the initial and terminal states $x_0,x_f$ and the transition time $T$. For fixed $x_0$ and $x_f$, the value $E$ is a function of $T$ only. In this case, $E$ and $T$ are related  by the equation
\begin{equation}
T=\int_{\gamma(\psi_T)}\frac{|d\psi_T|}{\sqrt{2E-2U(\psi_T)}},
\end{equation}
where $\gamma(\psi_T)$ is the graph of $\psi_T$.

It was proved in Proposition 3 of \cite{Du2020} that for every  $T>0$,  there exists a $t_c\in[0,T]$ such that
\begin{equation}\label{ET}
U(\psi_T(t_c))+\frac{|x_0-x_f|^2}{2T^2}\leq E(T)\leq U(\psi_T(t_c))+\frac{M^2}{2T^2}.
\end{equation}
As also shown in \cite{Du2020},   $\{\psi_T\}_{T>0}$ is uniformly bounded. Since $U(x)$
and $U'(x)$ are continuous,  $\{U(\psi_T)\}_{T>0}$ and $\{U'(\psi_T)\}_{T>0}$ are also uniformly bounded.
Therefore, by further combining $(\ref{ELequation})$, $(\ref{conserved})$ and $($\ref{ET}$)$, we have the following result.

\begin{theorem}\label{dotpsi}
(Uniform boundedness for the most probable transition path)\\
 (i) The velocity for the most probable transition path is uniformly bounded after a positive time: For every positive $\varrho$,  the set of `speed' $\{\|\dot{\psi}_T\|_T\}_{T>\varrho}$ is uniformly  bounded by a constant $M_{1,\varrho}$.\\
 (ii)  The acceleration for the most probable transition path is uniformly bounded: The   set of magnitude for the acceleration $\{\|\ddot{\psi}_T\|_T\}_{T>0}$ is uniformly bounded by a constant $M_{2}$.
\end{theorem}

\medskip

Recalling the results from the  previous section,  we can further provide  an exact  lower bound of $\sup_{\psi\in \bar{C}_D[0,T]}\mu_X(K_T(\psi,\delta))$ by introducing a family of paths $\{\phi_T\}_{T>0}$.  We use the property that $\{\phi_T\}_{T>0}$, $\{\dot{\phi}_T\}_{T>0}$ and $\{\ddot{\phi}_T\}_{T>0}$ (in fact $\ddot{\phi}_T\equiv0$) are uniformly bounded. Thus, according to Theorem $\ref{theorem1}$ and Theorem $\ref{dotpsi}$,   we   give a lower bound  by  replacing $\{\phi_T\}_{T>\varrho}$ by $\{\psi_T\}_{T>\varrho}$ for some $\varrho>0$:
\begin{equation}\label{boundpsi}
\begin{split}
&\sup_{T>0}\sup_{\psi\in \bar{C}_D[0,T]}\mathcal{P}^{x_0}\{\|X_t-\psi(t)\|_T< \delta\}\\
=&\sup_{T>\varrho}\sup_{\psi\in \bar{C}_D[0,T]}\mathcal{P}^{x_0}\{\|X_t-\psi(t)\|_T< \delta\}\\
\geq&\sup_{T>\varrho}\mathcal{P}^{x_0}\{\|X_t-\psi_T(t)\|_T< \delta\}\\
\geq&\sup_{T>\varrho}\tilde{c}_0\exp\{-M_{1,\varrho}\}\exp\{-\frac{1}{c^2}S^{OM}_T(\psi_{T})-\tilde{c}_1T+\ln(\mu_{B^c}(K_t(0,\delta)))\}.
\end{split}
\end{equation}
where the coefficients $\tilde{c}_0,\tilde{c}_1$ can be determined in a similar way like $c_0,c_1$ in $(\ref{estimation})$ and ($\ref{coefficient}$).

Recall from  references \cite{Durr1978,Zeitouni1987,Zeitouni1988,Ikeda1980}, the estimation of probability $\mu_X(K_T(\psi,\delta))$ is
\begin{equation}\label{muestimation}
\begin{split}
\mu_X(K_T(\psi,\delta))\sim&\exp\{-\frac{1}{2}\int_{0}^{T}[\frac{(\dot{\psi}-b(\psi))^2}{c^2}+b'(\psi)]dt\}\mu_{B^c}(K_T(0,\delta))\\
\sim&\frac{4}{\pi}\exp\{-\frac{1}{2}\int_{0}^{T}[\frac{(\dot{\psi}-b(\psi))^2}{c^2}+b'(\psi)]dt-\frac{\pi^2 c^2T}{8\delta^2}\},~\delta\downarrow0.
\end{split}
\end{equation}
This approximation has the similar form with the lower bound in $(\ref{boundpsi})$. Note that $\frac{4}{\pi}\exp\{-\frac{\pi^2 c^2T}{8\delta^2}\}$ is the first term of the infinite series representation of the probability $\mu_{B^c}(K_T(0,\delta))$ in $(\ref{probability})$.

Moreover,  for a fixed constant $\varrho>0$ and when $T>\varrho$, we have
\begin{equation}
\begin{split}
S_T^{OM}(\psi_T)=&\int_0^T(\frac{1}{2}(\dot{\psi}_T)^2-U(\psi_T))dt+V(x_f)-V(x_0)\\
=&\int_0^T(\dot{\psi}_T)^2dt+V(x_f)-V(x_0)-E(T)T.
\end{split}
\end{equation}
It was proved in   Lemma 2 of \cite{Du2020} that
\begin{equation}
\liminf_{T\rightarrow+\infty}E(T)>0.
\end{equation}
Hence,  if
\begin{equation}\label{condition}
\frac{\pi^2 c^4}{8\delta^2}>\limsup_{T\rightarrow+\infty}E(T),
\end{equation}
then for any $\epsilon>0$ and when $T$ is large enough we have
\begin{equation}
\begin{split}
&\sup_{\psi\in \bar{C}[0,T]}\frac{4}{\pi}\exp\{-\frac{1}{c^2}S_T^{OM}(\psi)-\frac{\pi^2 c^2T}{8\delta^2}\}\\
=&\frac{4}{\pi}\exp\{-\frac{1}{c^2}S_T^{sOM}(\psi_T)-\frac{V(x_f)-V(x_0)}{c^2}-\frac{\pi^2 c^2T}{8\delta^2}\}\\
\leq& \frac{\epsilon}{T},
\end{split}
\end{equation}
and the following inequality holds for every $T>0$:
\begin{equation}
\begin{split}
&k_0\exp\{-\frac{k_1}{T}-k_2T+\ln(\mu_{B^c}(K_T(0,\delta)))\}\\
\leq&\frac{4}{\pi}\exp\{-\frac{1}{c^2}S_T^{OM}(\phi_T)-\frac{\pi^2 c^2T}{8\delta^2}\}\\
\leq&\sup_{\psi\in \bar{C}[0,T]}\frac{4}{\pi}\exp\{-\frac{1}{c^2}S_T^{OM}(\psi)-\frac{\pi^2 c^2T}{8\delta^2}\}.
\end{split}
\end{equation}
That is
\begin{equation}
k_0\exp\{-\frac{k_1}{T}-k_2T\}\mu_{B^c}(K_T(0,\delta))\leq\sup_{\psi\in \bar{C}[0,T]}\frac{4}{\pi}\exp\{-\frac{S_T^{OM}(\psi)}{c^2}-\frac{\pi^2 c^2T}{8\delta^2}\}\preceq\frac{\epsilon}{T}.
\end{equation}
This implies  that although the accuracy of the estimation $(\ref{muestimation})$ is different when $\psi$ is different,   the upper bound for this estimation,  for all paths $\psi\in\bar{C}[0,T]$, is  controlled uniformly by $\frac{\epsilon}{T}$. So the error of the estimation is also uniformly controlled.

This inspires us to use the estimation $(\ref{muestimation})$ to approximately calculate the probability of the solution process $X_t$ staying in the neighborhood of a transition path (although this estimate is quite rough for a fixed $\delta$):
\begin{equation}
\begin{split}
\mathcal{P}^{x_0}\{\|X_t-\psi(t)\|< \delta\}
\approx\frac{4}{\pi}\exp\{-\frac{1}{2}\int_{0}^{T}[\frac{(\dot{\psi}-b(\psi))^2}{c^2}+b'(\psi)]dt-\frac{\pi^2 c^2T}{8\delta^2}\}.
\end{split}
\end{equation}
We now use it to find the most probable transition time.   Define a  modified Lagrangian functional $(L^{mOM})$  by
\begin{equation}\label{lagrangian}
L^{mOM}(\psi):=\frac{1}{2}[(\dot{\psi}-b(\psi))^2+c^2b'(\psi)+\frac{\pi^2 c^4}{4\delta^2}]
\end{equation}
and the corresponding modified action functional is
\begin{equation}
S^{mOM}_T(\psi)=\frac{1}{2}\int_{0}^{T}[(\dot{\psi}-b(\psi))^2+c^2b'(\psi)+\frac{\pi^2 c^4}{4\delta^2}]dt.
\end{equation}

Thus the double optimization problem  on the tube probability  $(\ref{opt})$  in this case   is approximately equivalent to the following {\bf double optimization problem  on the modified Onsager-Machlup action functional}
\begin{equation}
\begin{split}
\inf_{T>0}\inf_{\psi\in \bar{C}[0,T]} S^{mOM}_T(\psi).   \label{newdouble}
\end{split}
\end{equation}

\begin{Remark}
We should notice that the original OM action functional comes from the path density functions, while  the modified OM action functional is derived from the   estimation of the probability that the diffusion process stays in a tube surrounding the transition path.
\end{Remark}

\begin{Remark}
The condition $(\ref{condition})$  is indeed valid for some noise intensity $c$ and tube size $\delta$. This is shown as follows,

Recall the inequalities $(\ref{ET})$:
\begin{equation}
U(\psi_T(t_c))+\frac{|x_0-x_f|^2}{2T^2}\leq E(T)\leq U(\psi_T(t_c))+\frac{M^2}{2T^2},
\end{equation}
thus
\begin{equation}
E(T)\leq c^2V''(\psi_T(t_c))-\frac{1}{2}(V'(\psi_T(t_c)))^2+\frac{M^2}{2T^2}.
\end{equation}
Since $\{\psi_T\}_{T>0}$ are uniformly bounded and the potential $V(x)$ is smooth enough, thus when $T$ is large enough, there exist some $c$ and $\delta$ such that
\begin{equation}
E(T)\leq c^2V''(\psi_T(t_c))-\frac{1}{2}(V'(\psi_T(t_c)))^2+\frac{M^2}{2T^2}< \frac{\pi^2 c^4}{8\delta^2}.
\end{equation}

Furthermore we have
\begin{equation}
\begin{split}
&\inf_{T>0}\inf_{\psi\in \bar{C}[0,T]} S^{mOM}_T(\psi)=\inf_{T>0} S^{mOM}_T(\psi_T)=\inf_{T>0} (S^{OM}_T(\psi_T)+\frac{\pi^2 c^4T}{8\delta^2}),
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\frac{d}{dT}(S^{OM}_T(\psi_T)+\frac{\pi^2 c^4T}{8\delta^2})=\frac{dS^{OM}_T(\psi_T)}{dT}+\frac{\pi^2 c^4}{8\delta^2}=-E(T)+\frac{\pi^2 c^4}{8\delta^2},
\end{split}
\end{equation}
here the relation $\frac{dS^{OM}_T(\psi_T)}{dT}=-E(T)$ is a classical result in Hamilton-Jacobi theory whose proof can be found in \cite{Du2020,Giaquinta2004}. Thus if $E(T)$ is monotonic, then $T_{x_0\rightarrow x_f}^\delta=E^{-1}(\frac{\pi^2 c^4T}{8\delta^2})$. This implies that the global most probable transition path roughly lies on the energy shell $E=\frac{\pi^2 c^4T}{8\delta^2}$ in phase space.
\end{Remark}

\section{Examples}\label{example}
In this section we present two examples to illustrate our results.
\begin{example}
One-dimensional Brownian motion
\end{example}
\noindent Consider a scalar SDE without drift:
\begin{equation}
dX_t=c\; dB_t,~0\leq t\leq T,
\end{equation}
where $c$ is a positive constant. The modified Lagrange function is
\begin{equation}
L^{mOM}(\dot{\psi},\psi)=\frac{1}{2}(\dot{\psi})^2+\frac{\pi^2 c^4}{8\delta^2}.
\end{equation}
and the modified OM action functional is
\begin{equation}
S^{mOM}_T(\psi_T)=[\frac{x_f-x_0}{T}]^2\frac{T}{2}+\frac{\pi^2 c^4T}{8\delta^2}.
\end{equation}
So by minimizing this functional (setting its first derivative with respect time $T$ to be zero), we obtain  the estimation for the most probable transition time $T^{\delta}_{x_0\rightarrow x_f}$
\begin{equation}
T^{\delta}_{x_0\rightarrow x_f}=\frac{2\delta|x_f-x_0|}{\pi c^2}.
\end{equation}

\begin{example}
A stochastic double well system
\end{example}
\noindent Consider a nonlinear scalar SDE:
\begin{equation}\label{doublewellSDE}
dX_t=(X_t-X_t^3)dt+cdB_t,~0\leq t\leq T,
\end{equation}
with $c$ is a positive constant (without loss of generality we set $c=1$). The corresponding undisturbed system has three equilibrium points: -1, 0, 1 ( we know that -1 and 1 are stable equilibrium points,  and 0 is an unstable equilibrium point). In Figure $\ref{doublewell1}$, the middle graph shows two sample paths of system $(\ref{doublewellSDE})$ with initial position $x_0=-1$.   The top graph of Figure $\ref{doublewell1}$ shows two sample paths of Brownian motion   staring at 0 as a contrast. From the comparison of these two graphs it is easy to see the difference of the behavior of a diffusion process with different drift terms. The diffusion process $(\ref{doublewellSDE})$ fluctuates between two metastable states $-1$ and $1$. The bottom graph of Figure $\ref{doublewell1}$ shows the most probable transition path  (in fact it is a local minimizer of the OM action functional) calculated by a shooting method with  $T=10$. In Figure $\ref{doublewell2}$ there are two sample transition paths of this stochastic double well system. The red and yellow curves are the corresponding most probable transition paths (MPTPs), and the dash curves are the boundaries of the transition tube.

The corresponding modified Lagrangian $L^{mOM}$ is
\begin{equation}
L^{mOM}(\dot{\psi},\psi)= \frac{1}{2}[(\dot{\psi}-\psi+\psi^3)^2+c^2(1-3\psi^2)+\frac{\pi^2 c^4}{4\delta^2}],
\end{equation}
and the modified OM action functional is
\begin{equation}
S^{mOM}_{T}(\psi)=\frac{1}{2}\int_{0}^{T}[(\dot{\psi}-\psi+\psi^3)^2+c^2(1-3\psi^2)+\frac{\pi^2 c^4}{4\delta^2}]ds.
\end{equation}

We  use  Euler method to generate  sample solution paths with time step size $\Delta t=10^{-4}$. To solve the optimization problem,   we assume that the minimizer is twice differentiable and thus we obtain the Euler-Lagrange equation:
\begin{equation}
\frac{d}{dt}\frac{\partial L^{mOM}(\dot{\psi},\psi)}{\partial \dot{\psi}}=\frac{\partial L^{mOM}(\dot{\psi},\psi)}{\partial \psi}.
\end{equation}
Hence the optimization problem turns into a second order ordinary differential equation with two boundary values,  which can be solved numerically by a shooting method:
\begin{equation}
\begin{cases}
\ddot{\psi}=b'(\psi)b(\psi)+\frac{c^2}{2}b''(\psi),\\
\psi(0)=x_0, \;\;  \psi(T)=x_f.
\end{cases}
\end{equation}
We should note that the Euler-Lagrange equation determines the local minimizer (if the joint mapping $(\dot{\psi},\psi)\mapsto OM(\dot{\psi},\psi)$ is convex then the local minimizer actually is a global minimizer which has been shown in \cite{Chao2019}). A shortcoming
 of   the shooting method is restricted by the selection of the time $T$. For instance in this example, if we choose $T$=3, the shooting method fails to find the local minimizer.

So we restrict our attention in relative short time interval $[0,1.5]$ and the first transition behavior. Figure $\ref{doublewell3}$ shows the graphs of $S^{mOM}_{T}(\psi_T)$. Figure $\ref{doublewell4}$ shows the most probable transition paths for different tube sizes $\delta$ and transition time $T$. These paths are uniformly bounded in domain $[-1,1]$, so in this example we use modified action functional to characterize the transition behavior. Thus the numerical result of the most probable transition times $T^{\delta}_{x_0\rightarrow x_f}$ for different $\delta$ is obtained which are shown in Figure $\ref{doublewell5}$ in red spots.

Furthermore,  we simulated 30000 sample paths starting at -1 and there are 3165 transition paths (the first transition occurred before time 1.5). For every transition path we recorded its transition time $T$ and calculated its corresponding most probable transition path and tube size $\delta$ (i.e. this transition path can be contained in the tube of this MPTP with this tube size).   All the pairs $(T,\delta)$ were draw in Figure $\ref{doublewell5}$ by blue stars. The yellow squares are the mean values of the tube sizes. We separated the time interval [0,1.5] into subintervals $[0,0.25),[0.25,0.3),[0.3,0.35),[0.35,0.4),\cdots,[1.45,1.5]$ and calculated the mean tube size values in every subinterval. As shown in Figure $\ref{doublewell5}$, this numerical simulation shows that our results   characterize the transition behavior.
\begin{figure}
  \centering

  \includegraphics[width=1\textwidth]{doublewell}\\
  \caption{Top: Sample paths of a Brownian motion. Middle: Sample paths of the stochastic double well system (\ref{doublewellSDE}).  Bottom: The solution of the Euler-Lagrange equation for  the stochastic double well system   by a shooting method,  with $x_0=-1,x_f=1$ and $T=10.$}\label{doublewell1}
\end{figure}
\begin{figure}
\centering

\includegraphics[width=0.7\textwidth]{doublewell1}\\
\caption{Two sample transition paths of the stochastic double well system  (\ref{doublewellSDE}). Black: The transition time of sample path 1 is 0.676 and the tube size is $\delta=0.47$; Blue:  the transition time of sample path 2 is 1.522 and the tube size is  $\delta=1.20$.}\label{doublewell2}
\end{figure}

\begin{figure}
 \centering

 \includegraphics[width=8cm,height=6cm]{size1.jpg}
 \caption{ The graphs of $S^{mOM}_{T}(\psi_T)$ with different tube sizes $\delta$.}\label{doublewell3}

\end{figure}
\begin{figure}
 \centering

 \includegraphics[width=8cm,height=6cm]{size2.jpg}
 \caption{ The `most probable transition paths' with different transition time calculated by shooting method.}\label{doublewell4}

 \end{figure}
\begin{figure}
\centering

\includegraphics[width=1\textwidth]{doublewell2}\\
\caption{The transition times and tube sizes of 3165 transition paths.}\label{doublewell5}
\end{figure}

\newpage

\section{Discussion}\label{conclusion}

We now summarize our work and highlight the differences with relevant works.

\subsection{Our contribution}
Under some mild assumptions, we have   estimated   the  most probable transition time between metatstable states, for stochastic dynamical systems   with  non-vanishing Brownian noise. The problem is represented by a double optimization on the probability that sample paths staying in a tube surrounding the most probable transition pathway.   We have provided estimates for  the most probable transition time.

In our framework, we have adopted the original idea of Onsager-Machlup framework \cite{Machlup1953}, using the concept of tubes surrounding the transition path to study the transition time. Instead of letting the tube size $\delta$ tending to 0, we require this tube size $\delta$ to be a positive constant. This promises the double optimization problem is well-defined. Since when $\delta\rightarrow0$ all tube probabilities are 0. This requirement makes sense as  the noise intensity is non-vanishing,  which is different from the case in Freidlin-Wentzell's large deviation theory \cite{Friedlin1998,Heymann2008}.

Our method can be extended to higher dimensional systems. Similarly, Lemma $\ref{lemma2}$ shows that  the probability that the solution process staying in the tube of a transition path has a power decay law upper bound. It is similar to the one-dimensional case that the most probable transition time has upper and lower bounds. The probability of the solution process staying in the tube of a transition path can be computed approximately in higher dimensional case:
\begin{equation}
\mathcal{P}^{x_0}\{\|X_t-\psi(t)\|_T< \delta\}\approx\exp(-\frac{S^{OM}_T(\psi)}{c^2})\cdot \mathcal{P}^{x_0}\{\|B^c_t-x_0\|_T< \delta\},~\delta\downarrow0.
\end{equation}
  The probability $\mathcal{P}^{x_0}\{\|B^c_t-x_0\|_T< \delta\}$   monotonically decreases  in  $T$. So  we could define a modified OM action functional,  if we have an appropriate analytical estimation for the probability $\mathcal{P}^{x_0}\{\|B^c_t-x_0\|_T< \delta\}$.

  There are some works related to the heuristic discussions and measurements  of transition time. These include    the transition time distribution  \cite{Malinin2010,Carlon2018,Laleman12017,Janakiraman2018} and   the expected transition time     under small noise intensity \cite{Barret2015}.

\subsection{The difference between   double optimizations of  tube probability and of Onsager-Machlup action functional}

A  \textbf{double optimization problem on the Onsager-Machlup action functional} was investigated in \cite{Du2020}:
\begin{equation}\label{optLD}
\inf_{T>0}\inf_{\psi\in \bar{C}[0,T]} S^{OM}_T(\psi).
\end{equation}
This problem aims to study the property of the OM action functional.
Although \cite{Du2020} and we focus on the same stochastic dynamical systems, our result is different.   Du  et al.  \cite{Du2020} focused on the OM action functional in the $\delta\rightarrow0$ scaling, and the effect of the probability $\mu_{B^c}(K_T(0,\delta))$ is ignored when $T$ varies.

In   the estimation $(\ref{muestimation})$, it can be seen that the OM action functional term   characterizes the geometrical features of a transition path, while  the term
$\mu_{B^c}(K_T(0,\delta))$ characterizes the diffusion ability of the path. So for a stochastic system with  non-vanishing Brownian noise, the most probable transition path and the most probable transition time are supposed to be determined by these two terms (in the original Onsager-Machlup context). Although \cite{Du2020} ignored the effect of $\mu_{B^c}(K_T(0,\delta))$, it provides     valuable insights on  the OM action functional.

\subsection{The difference between our work and the large deviation theory}

Transition phenomena have been treated in the large deviation theory (i.e., under sufficiently small noise). The large deviation theory focuses on the following system \cite{Friedlin1998}:
\begin{equation}\label{ldp}
dX^{\varepsilon}_t=b(X^{\varepsilon}_t)dt+\sqrt{\varepsilon} \; dB_t,~t\geq0,~X_0^{\varepsilon}=x_0\in \mathbf{R}^k,
\end{equation}
where $\varepsilon\rightarrow0$. The Freidlin-Wentzell  (FW)  action functional $S^{FW}_T(\psi)$ is
\begin{equation}
S^{FW}_T(\psi)=
\int_{0}^{T}L^{FW}(\dot{\psi},\psi)dt,
\end{equation}
if $\psi\in C(0,T)$ is absolutely continuous and the integral converges, otherwise denote that $S^{FW}_T(\psi)=\infty$. Here the Lagrangian $L^{FW}(x,y)$ is given by
\begin{equation}
L^{FW}(x,y)=\langle y-b(x),y-b(x)\rangle.
\end{equation}
When the transition time was considered as a factor in transition phenomena, the quasi-potential in \cite{Heymann2008} was defined as
\begin{equation}\label{potential}
\mathbf{V}(x_0,x_f)=\inf_{T>0}\inf_{\psi\in \bar{C}[0,T]} S^{FW}_T(\psi),
\end{equation}
which is used to find the `global' most probable transition path and the corresponding most probable transition time. The large derivation theory asserts that, for $\delta$ and $\varepsilon$ sufficiently small,
\begin{equation}
\mathcal{P}^{x_0}\{\sup_{0\leq t\leq T}|X^{\varepsilon}_t-\psi(t)|\leq \delta\}\approx\exp(-\varepsilon^{-1}S^{FW}_T(\psi)).
\end{equation}
It was shown in \cite{Heymann2008} that the most probable transition time between two metastable states of system ($\ref{ldp}$) is infinite. Since the noise intensity $\sqrt{\varepsilon}$ tends to 0, the behavior of the system is closely  influenced by that of  the deterministic system. Hence the transition between two distinct  metastable states needs infinite time to make it happen most likely.

The modified OM functional in  this small  noise case is
\begin{equation}
\lim_{c\rightarrow0} S^{mOM}_T(\psi)=\lim_{c\rightarrow0}\frac{1}{2}\int_{0}^{T}[(\dot{\psi}-b(\psi))^2+c^2b'(\psi)+\frac{\pi^2 c^4}{4\delta^2}]dt=S^{FW}_T(\psi).
\end{equation}
In particular, the estimation of the most probable transition time $T_{x_0\rightarrow x_f}$ for     Example 1 in  Section $\ref{example}$ is thus \begin{equation}
T^{\delta}_{x_0\rightarrow x_f}=\frac{2\delta|x_f-x_0|}{\pi c^2}.
\end{equation}
When $c\rightarrow0$ then $T^{\delta}_{x_0\rightarrow x_f}\rightarrow\infty$ for any $\delta>0$. This result is consistent with the large deviation theory.

\section*{Acknowledgements}
The authors would like to thank   Jianyu Hu,   Pingyuan Wei,   Xiujun Cheng,   Qiao Huang,   Ao Zhang and Yancai Liu for helpful discussions. This work was partly supported by the  NSFC grants 11531006 and 11771449.

config: arxiv


TITLE: What am I doing (simple uniform converge problem) $f_n(x)=n^2x^2e^{-xn}$
QUESTION [0 upvotes]: consider the function $f_n(x)=n^2x^2e^{-xn}$ I am asked if it uniform converge on $A=(a,\infty) \quad a>0$
So it easy to see that in converge to $0$, but when I wanted to check uniform converge I noticed that the function getting maximum on $x=\frac{2}{n}$ pluging this value of $x$ to the function will give:
$f_n(\frac{2}{n})=\frac{4}{e^2}$, from here I concluded that $\sup|f_n(x)-f(x)|>0$
but this solution is wrong according to their solution, they actually used the same value of $x$ that I found to show it is uniform converge.they say for every $n>\frac{2}{a}$ u get that $\sup |f_n|=f_n(a)$ wich can indicates that is uniform converge, I actually not understand why we looking on $n>\frac{2}{a}$
I found specific $x$ that getting a value for the function so the supremum cannot be zero

REPLY [1 votes]: It is false that $\sup_{x\in[a,\infty)}|f_n(x)|=4e^{-2}$. This is true if $\frac2n\geqslant a$, but this only occurs for finitely many $n$'s. If $\frac2n<a$, then $f_n$ is strictly decreasing and it attains its maximum at $a$. That maximum is $n^2a^2a^{-na}$ and, since $\lim_{n\to\infty}n^2a^2a^{-na}=0$, your sequence converges uniformly to $0$ indeed.


TITLE: Help showing aMarkov chain with a doubly-stochastic matrix has uniform limiting distribution
QUESTION [1 upvotes]: I have a lot of difficulty with proofs; could someone help me with this question that really can not solve? I would also like some indication of material to get through with this kind of question and some hint of material about Markov chain. Thanks in advance.
"A stochastic matrix is called doubly stochastic if its columns sum to 1. Let
$X_0
, X_1, \dots$ be a Markov chain on $\{1,\dots, k\}$ with a doubly stochastic transition
matrix and initial distribution that is uniform on $\{1, \dots, k\}.$ Show that the distribution of $X_n$ is uniform on $\{1.\dots, k\},$ for all $n \ge 0."$

REPLY [1 votes]: If a Markov chain with state space $\{1, 2, \dots, k\}$ is ergodic then its stationary distribution is its limiting distribution. As I read your question, it seems you are trying to show that
a chain with a doubly stochastic matrix has a uniform stationary distribution on
the state space.
You are using the convention that element $p_{ij}$ for $1 \le i,j \le k,$ of the transition matrix $\mathbf{P}$ has $p_{ij} = P(X_n = j | X_{n-1} = i).$
Let vector $\sigma = (\sigma_1, \sigma_2, \dots, \sigma_k).$ If $\sigma$ is uniform, then $\sigma_i = 1/k,$ for $i = 1, \dots, k.$
$$\sigma_i\mathbf{P} = \sum_{j=1}^k \frac 1 k p_{ij} =
\frac 1 k\sum_{j=1}p_{ij} = \frac 1 k(1) = \frac 1 k,$$
where the last equality is due to the doubly stochastic property of the transition matrix. The argument for all $i = 1, \dots k$ is the same so
$\sigma\mathsf{P} = \sigma,$ and $\sigma$ is a stationalry distribution.
If your question is not covered by this answer and the Answer of @Momo (+1), please
explain what part you don't understand, and maybe someone here can help.


TITLE: Does every mon-small submodule contain a nonzero supplement submodule of the module.
QUESTION [0 upvotes]: Definition: Let $M$ be an $R$-module. Then

A submodule $N$ is called small (in $M$) if $N+K=M\implies K=M,\forall K\leq M$.
A submodule $N$ is called a supplement of a submodule $K$ (in $M$) if $N$ is minimal with respect to the property $N+K=M$.
A submodule $N$ is called a supplement submodule if it is a supplement of some submodule $K$.

My Question: Does every mon-small submodule contain a nonzero supplement submodule of the module.

REPLY [1 votes]: False.
You can take example of $\mathbb{Z}$-module $\mathbb{Z}$. Its every nonzero submodule is nonsmall while it has no nonzero supplement submodule.


TITLE: Are connection forms tensorial objets?
QUESTION [2 upvotes]: I think the standard answer is they are not, for example, from wiki. Under a coordinate transformation of $e' =eA$ where $e$ is a set of frames and $A$ the transformation matrix, connection form $\omega$ transform as
$$\omega'=A^{-1}\omega A + A^{-1}\mathrm{d}A$$
Apperatently apart from the usual tensoril transformation term $A^{-1}\omega A$, there is an additional $A^{-1}\mathrm{d}A$ term.
However, due to the $\mathcal{F}$-linear w.r.t. $W$ as in $\nabla_W V$, one can consider $\nabla_{(\cdot)}V$ with a fixed vector field $V$ a $(1, 1)$ tensor field (reference: pg. 204 of [1]).
Aren't connection forms the same construction as $\nabla_{(\cdot)}V$? i.e.
$$\nabla_Ve_a = \omega^b_a(V)e_b$$
This is just picking a set of frame fields $e_a$ in place of $V$ in $\nabla_{(\cdot)} V$, and we decompose the result on to the same set of frames?
How do I reconcile these two understandings?
[1]: Schutz, Bernard - Geometrical methods of mathematical physics. Cambridge university press 1980

REPLY [2 votes]: Say that $U$ is the domain of the frame $e$. In parts:

Each $\omega^a_{~b}$ is a $1$-form on $U$ precisely due to $\mathcal{F}$-linearity of the connection $\nabla$ in the lower argument, yes.
The whole matrix $(\omega^a_{~b})_{a,b=1}^n$ is a $\mathfrak{gl}_n(\Bbb R)$-valued $1$-form on $U$.
However, $(U,\{\omega^a_{~b} \mid a,b=1,\ldots,n\})$ are not the local components of any tensor on $M$, because whenever $(\widetilde{U} ,\{\widetilde{\omega}^a_{~b}\mid a,b=1,\ldots, n\})$ corresponds to a second frame $\widetilde{e}$ on $\widetilde{U}$ (with $\widetilde{e} = eA$ on $U\cap\widetilde{U}\neq\varnothing$), they do not satisfy the tensor transformation law (due to $A^{-1}{\rm d}A$).


text:
\chapter{Affine varieties}
In this chapter we introduce affine varieties.
We introduce them in the context of coordinates,
but over the course of the other chapters
we'll gradually move away from this perspective to
viewing varieties as ``intrinsic objects'',
rather than embedded in coordinates.

For simplicity, we'll do almost everything over the field of complex numbers,
but the discussion generalizes to any algebraically closed field.

\section{Affine varieties}
\prototype{$\VV(y-x^2)$ is a parabola in $\Aff^2$.}
%An \vocab{affine variety} is just the zero locus of a set of polynomials.
%We think of it as living in the $n$-dimensional space $\Aff^n$.

\begin{definition}
	Given a set of polynomials $S \subseteq \CC[x_1, \dots, x_n]$
	(not necessarily finite or even countable),
	we let $\VV(S)$ denote the set of points vanishing on \emph{all}
	the polynomials in $S$.
	Such a set is called an \vocab{affine variety}.
	It lives in \vocab{$n$-dimensional affine space}, denoted $\Aff^n$
	(to distinguish it from projective space later).
\end{definition}
For example, a parabola is the zero locus of the polynomial $\VV(y-x^2)$. Picture:
\begin{center}
	\begin{asy}
		import graph;
		size(5cm);

		real f(real x) { return x*x; }
		graph.xaxis("$x$");
		graph.yaxis("$y$");
		draw(graph(f,-2,2,operator ..), blue, Arrows);
		label("$\mathcal V(y-x^2)$", (0.8, f(0.8)), dir(-45), blue);
		label("$\mathbb A^2$", (2,3), dir(45));
	\end{asy}
\end{center}

\begin{example}[Examples of affine varieties]
	These examples are in two-dimensional space $\Aff^2$,
	whose points are pairs $(x,y)$.
	\begin{enumerate}[(a)]
	\ii A straight line can be thought of as $\VV(Ax + By + C)$.
	\ii A parabola as above can be pictured as $\VV(y-x^2)$.
	\ii A hyperbola might be the zero locus of the polynomial $\VV(xy-1)$.
	\ii The two axes can be thought of as $\VV(xy)$; this is the set of points
	such that $x=0$ \emph{or} $y=0$.
	\ii A point $(x_0, y_0)$ can be thought of as $\VV(x-x_0, y-y_0)$.
	\ii The entire space $\Aff^2$ can be thought of as $\VV(0)$.
	\ii The empty set is the zero locus of the constant polynomial $1$, that is $\VV(1)$.
	\end{enumerate}
\end{example}

\section{Naming affine varieties via ideals}
\prototype{$\VV(I)$ is a parabola, where $I=(y-x^2)$.}
As you might have already noticed, a variety can be named by $\VV(-)$ in multiple ways.
For example, the set of solutions to
\[ x=3 \text{ and } y=4 \]
is just the point $(3,4)$.
But this is also the set of solutions to
\[ x=3 \text{ and } y=x+1. \]
So, for example
\[ \{(3,4)\}
	= \VV(x-3, y-4)
	= \VV(x-3, y-x-1).
	\]
That's a little annoying, because in an ideal\footnote{Pun not intended
	but left for amusement value.}
world we would have \emph{one} name
for every variety.
Let's see if we can achieve this.

A partial solution is to use \emph{ideals} rather than small sets.
That is, consider the ideal
\[
	I = \left( x-3, y-4 \right)
	= \left\{ p(x,y) \cdot (x-3) + q(x,y) \cdot (y-4)
	\mid p,q \in \CC[x,y] \right\}
\]
and look at $\VV(I)$.
\begin{ques}
	Convince yourself that $\VV(I) = \{(3,4)\}$.
\end{ques}
So rather than writing $\VV(x-3, y-4)$ it makes sense to
think about this as $\VV\left( I \right)$, where $I = (x-3,y-4)$ is the \emph{ideal}
generated by the two polynomials $x-3$ and $y-4$.
This is an improvement because
\begin{ques}
	Check that $(x-3, y-x-1) = (x-3, y-4)$.
\end{ques}

Needless to say, this pattern holds in general.
\begin{ques}
	Let $\{f_i\}$ be a set of polynomials, and consider
	the ideal $I$ generated by these $\{f_i\}$.
	Show that $\VV(\{f_i\}) = \VV(I)$.
\end{ques}

Thus we will only consider $\VV(I)$ when $I$ is an ideal.
Of course, frequently our ideals are generated by one or two polynomials,
which leads to:
\begin{abuse}
	Given a set of polynomials $f_1, \dots, f_m$
	we let $\VV(f_1, \dots, f_m)$ be shorthand for
	$\VV\left( \left( f_1, \dots, f_m \right) \right)$.
	In other words we let $\VV(f_1, \dots, f_m)$
	abbreviate $\VV(I)$, where $I$ is the \emph{ideal} $I=(f_1, \dots, f_m)$.
\end{abuse}

This is where the Noetherian condition really shines:
it guarantees that every ideal $I \subseteq \CC[x_1, \dots, x_n]$
can be written in the form above with \emph{finitely} many polynomials,
because it is \emph{finitely generated}.
(The fact that $\CC[x_1, \dots, x_n]$ is Noetherian follows from the Hilbert basis theorem,
which is \Cref{thm:hilbert_basis}).
This is a relief, because dealing with infinite sets of polynomials is not much fun.

\section{Radical ideals and Hilbert's Nullstellensatz}
\prototype{$\sqrt{(x^2)} = (x)$ in $\CC[x]$, $\sqrt{(12)} = (6)$ in $\ZZ$.}
You might ask whether the name is unique now:
that is, if $\VV(I) = \VV(J)$, does it follow that $I=J$?
The answer is unfortunately no: the counterexample can be found in just $\Aff^1$.
It is
\[ \VV(x) = \VV(x^2). \]
In other words, the set of solutions to $x=0$
is the same as the set of solutions to $x^2=0$.

Well, that's stupid.
We want an operation which takes the ideal $(x^2)$ and makes it into the ideal $(x)$.
The way to do so is using the radical of an ideal.

\begin{definition}
	Let $R$ be a ring.
	The \vocab{radical} of an ideal $I \subseteq R$, denoted $\sqrt I$,
	is defined by
	\[ \sqrt I = \left\{ r \in R
			\mid r^m \in I \text{ for some integer $m \ge 1$} \right\}. \]
	If $I = \sqrt I$, we say the ideal $I$ itself is \vocab{radical}.
\end{definition}
For example, $\sqrt{(x^2)} = (x)$.
You may like to take the time to verify that $\sqrt I$ is actually an ideal.

\begin{remark}
	[Number theoretic motivation]
	This is actually the same as the notion of ``radical'' in number theory.
	In $\ZZ$, the radical of an ideal $(n)$ corresponds to just
	removing all the duplicate prime factors, so for example
	\[ \sqrt{(12)} = (6). \]
	In particular, if you try to take $\sqrt{(6)}$,
	you just get $(6)$ back;
	you don't squeeze out any new prime factors.

	This is actually true more generally,
	and there is a nice corresponding alternate definition:
	for any ideal $I$, we have
	\[ \sqrt I = \bigcap_{I \subseteq \kp \text{ prime}} \kp. \]
	Although we could prove this now,
	it will be proved later in \Cref{thm:radical_intersect_prime},
	when we first need it.
\end{remark}

Here are the immediate properties you should know.
\begin{proposition}
	[Properties of radical]
	\label{prop:radical}
	In any ring:
	\begin{itemize}
		\ii If $I$ is an ideal, then $\sqrt I$ is always a radical ideal.
		\ii Prime ideals are radical.
		\ii For $I \subseteq \CC[x_1, \dots, x_n]$
		we have $\VV(I) = \VV(\sqrt I)$.
	\end{itemize}
\end{proposition}
\begin{proof}
	These are all obvious.
	\begin{itemize}
		\ii If $f^m \in \sqrt I$ then $f^{mn} \in I$, so $f \in \sqrt I$.
		\ii If $f^n \in \kp$ for a prime $\kp$,
		then either $f \in \kp$ or $f^{n-1} \in \kp$,
		and in the latter case we may continue by induction.
		\ii We have $f(x_1, \dots, x_n) = 0$
		if and only if $f(x_1, \dots, x_n)^m = 0$ for some integer $m$.
		\qedhere
	\end{itemize}
\end{proof}

The last bit makes sense: you would never refer to $x=0$ as $x^2=0$,
and hence we would always want to call $\VV(x^2)$ just $\VV(x)$.
With this, we obtain a theorem called Hilbert's Nullstellensatz.
\begin{theorem}[Hilbert's Nullstellensatz]
	\label{thm:hilbert_null}
	Given an affine variety $V = \VV(I)$,
	the set of polynomials which vanish
	on all points of $V$ is precisely $\sqrt I$.
	Thus if $I$ and $J$ are ideals in $\CC[x_1, \dots, x_n]$, then
	\[ \VV(I) = \VV(J) \text{ if and only if $\sqrt I = \sqrt J$}. \]
\end{theorem}
In other words
\begin{moral}
	Radical ideals in $\CC[x_1, \dots, x_n]$ correspond
	exactly to $n$-dimensional affine varieties.
\end{moral}
The proof of Hilbert's Nullstellensatz will be given in
\Cref{prob:hilbert_from_weak}; for now it is worth remarking that
it relies essentially on the fact that $\CC$ is
\emph{algebraically closed}.
For example, it is false in $\RR[x]$,
with $(x^2+1)$ being a maximal ideal with empty vanishing set.

\section{Pictures of varieties in $\Aff^1$}
\prototype{Finite sets of points (in fact these are the only nontrivial examples).}
Let's first draw some pictures.
In what follows I'll draw $\CC$ as a straight line\dots sorry.

First of all, let's look at just the complex line $\Aff^1$.
What are the various varieties on it?
For starters, we have a single point $9 \in \CC$,
generated by $(x-9)$.

\begin{center}
	\begin{asy}
		size(6cm);
		pair A = (-9,0); pair B = (9,0);
		draw(A--B, Arrows);
		label("$\mathcal V(x-9)$", (0,0), 2*dir(-90), blue);
		dot("$9$", (3,0), dir(90), blue);
		label("$\mathbb A^1$", A+(2,0), dir(90));
	\end{asy}
\end{center}

Another example is the point $4$.
And in fact, if we like we can get an ideal consisting of just these two points;
consider $\VV\left( (x-4)(x-9) \right)$.

\begin{center}
	\begin{asy}
		size(6cm);
		pair A = (-9,0); pair B = (9,0);
		draw(A--B, Arrows);
		label("$\mathcal V( (x-4)(x-9) )$", (0,0), 2*dir(-90), blue);
		dot("$4$", (-1,0), dir(90), blue);
		dot("$9$", (3,0), dir(90), blue);
		label("$\mathbb A^1$", A+(2,0), dir(90));
	\end{asy}
\end{center}

In general, in $\Aff^1$ you can get finitely
many points $\left\{ a_1, \dots, a_n \right\}$ by
just taking \[ \VV\left( (x-a_1)(x-a_2)\dots(x-a_n) \right). \]
On the other hand, you can't get the set $\{0,1,2,\dots\}$ as an affine variety;
the only polynomial vanishing
on all those points is the zero polynomial.
In fact, you can convince yourself that these
are the only affine varieties, with two exceptions:
\begin{itemize}
	\ii The entire line $\Aff^1$ is given by $\VV(0)$, and
	\ii The empty set is given by $\VV(1)$.
\end{itemize}
\begin{exercise}
	Show that these are the only varieties of $\Aff^1$.
	(Let $\VV(I)$ be the variety and pick a $0 \neq f \in I$.)
\end{exercise}

As you might correctly guess, we have:
\begin{theorem}[Intersections and unions of varieties]
	\label{thm:many_aff_variety}
	\listhack
	\begin{enumerate}[(a)]
		\ii The intersection of affine varieties
		(even infinitely many) is an affine variety.
		\ii The union of finitely many affine varieties
		is an affine variety.
	\end{enumerate}
	In fact we have
	\[ \bigcap_\alpha \VV(I_\alpha)
		= \VV\left( \sum_\alpha I_\alpha \right)
		\qquad\text{and}\qquad
		\bigcup_{k=1}^n \VV(I_k)
		= \VV\left( \bigcap_{k=1}^n I_k \right). \]
\end{theorem}
You are welcome to prove this easy result yourself.
\begin{remark}
	Part (a) is a little misleading in that the sum $I+J$ need not be radical:
	take for example $I = (y-x^2)$ and $J = (y)$ in $\CC[x,y]$,
	where $x \in \sqrt{I+J}$ and $x \notin I+J$.
	But in part (b) for radical ideals $I$ and $J$,
	the intersection $I \cap J$ is radical.
\end{remark}

\section{Prime ideals correspond to irreducible affine varieties}
\prototype{$(xy)$ corresponds to the union of two lines in $\Aff^2$.}

Note that most of the affine varieties of $\Aff^1$, like $\{4,9\}$,
are just unions of the simplest ``one-point'' ideals.
To ease our classification,
we can restrict our attention to the case of \emph{irreducible} varieties:
\begin{definition}
	A variety $V$ is \vocab{irreducible} if it cannot be written
	as the union of two proper sub-varieties $V = V_1 \cup V_2$.
\end{definition}
\begin{abuse}
	Warning: in other literature,
	irreducible is part of the definition of variety.
\end{abuse}

\begin{example}
	[Irreducible varieties of $\Aff^1$]
	The irreducible varieties of $\Aff^1$ are:
	\begin{itemize}
		\ii the empty set $\VV(1)$,
		\ii a single point $\VV(x-a)$, and
		\ii the entire line $\Aff^1 = \VV(0)$.
	\end{itemize}
\end{example}
\begin{example}
	[The union of two axes]
	Let's take a non-prime ideal in $\CC[x,y]$, such as $I = (xy)$.
	Its vanishing set $\VV(I)$ is the union of two lines $x=0$ and $y=0$.
	So $\VV(I)$ is reducible.
\end{example}

%We have already seen that the radical ideals
%are in one-to-one correspondence with affine varieties.
%In the next sections we answer the two questions:
%\begin{itemize}
%	\ii What property of $\VV(I)$ corresponds to $I$ being prime?
%	\ii What property of $\VV(I)$ corresponds to $I$ being maximal?
%\end{itemize}
%The first question is easier to answer.

In general:
\begin{theorem}[Prime $\iff$ irreducible]
	Let $I$ be a radical ideal, and $V = \VV(I)$ a nonempty variety.
	Then $I$ is prime if and only if $V$ is irreducible.
\end{theorem}
\begin{proof}
	First, assume $V$ is irreducible; we'll show $I$ is prime.
	Let $f,g \in \CC[x_1, \dots, x_n]$ so that $fg \in I$.
	Then $V$ is a subset of the union $\VV(f) \cup \VV(g)$;
	actually, $V = \left( V \cap \VV(f) \right) \cup \left( V \cap \VV(g) \right)$.
	Since $V$ is irreducible, we may assume $V = V \cap \VV(f)$,
	hence $f$ vanishes on all of $V$. So $f \in I$.

	The reverse direction is similar.
\end{proof}

%\begin{remark}
%	The above proof illustrates the following principle:
%	Let $V$ be an irreducible variety.
%	Suppose that $V \subseteq V_1 \cup V_2$;
%	this implies $V = (V_1 \cap V) \cup (V_2 \cap V)$.
%	Recall that the intersection of two varieties is a variety.
%	Thus an irreducible variety can't even be \emph{contained}
%	in a nontrivial union of two varieties.
%\end{remark}

\section{Pictures in $\Aff^2$ and $\Aff^3$}
\prototype{Various curves and hypersurfaces.}

With this notion, we can now draw pictures in
``complex affine plane'', $\Aff^2$.
What are the irreducible affine varieties in it?

As we saw in the previous discussion,
naming irreducible affine varieties in $\Aff^2$
amounts to naming the prime ideals of $\CC[x,y]$.
Here are a few.
\begin{itemize}
	\ii The ideal $(0)$ is prime. $\VV(0)$ as usual corresponds to the entire plane.
	\ii The ideal $(x-a, y-b)$ is prime,
	since $\CC[x,y] / (x-a, y-b) \cong \CC$ is an integral domain.
	(In fact, since $\CC$ is a field, the ideal $(x-a,y-b)$ is \emph{maximal}).
	The vanishing set of this is $\VV(x-a, y-b) = \{ (a,b) \} \in \CC^2$,
	so these ideals correspond to a single point.
	\ii Let $f(x,y)$ be an irreducible polynomial, like $y-x^2$.
	Then $(f)$ is a prime ideal! Here $\VV(I)$ is a ``degree one curve''.
\end{itemize}

By using some polynomial algebra
(again you're welcome to check this; Euclidean algorithm),
these are in fact the only prime ideals of $\CC[x,y]$.
Here's a picture.

\begin{center}
	\begin{asy}
		import graph;
		graph.xaxis("$x$", -4, 4);
		graph.yaxis("$y$", -4, 4);

		real f (real x) { return x*x; }
		draw(graph(f,-2,2,operator ..), blue);
		label("$\mathcal V(y-x^2)$", (1,1), dir(-45), blue);
		dot("$\mathcal V(x-1,y+2)$", (1,-2), dir(-45), red);
	\end{asy}
\end{center}

As usual, you can make varieties which are just unions of these irreducible ones.
For example, if you wanted the variety consisting of a parabola $y=x^2$
plus the point $(20,15)$ you would write
\[ \VV \left( (y-x^2)(x-20), (y-x^2)(y-15) \right). \]

The picture in $\Aff^3$ is harder to describe.
Again, you have points $\VV(x-a, y-b, z-c)$ corresponding to
be zero-dimensional points $(a,b,c)$, and two-dimensional surfaces
$\VV(f)$ for each irreducible polynomial $f$ (for example, $x+y+z=0$ is a plane).
But there are more prime ideals, like $\VV(x,y)$, which corresponds to the
intersection of the planes $x=0$ and $y=0$: this is the one-dimensional $z$-axis.
It turns out there is no reasonable way to classify the ``one-dimensional'' varieties;
they correspond to ``irreducible curves''.

Thus, as Ravi Vakil \cite{ref:vakil} says:
the purely algebraic question
of determining the prime ideals of $\CC[x,y,z]$
has a fundamentally geometric answer.

\section{Maximal ideals}
\prototype{All maximal ideals are $(x_1-a_1, \dots, x_n-a_n)$.}
We begin by noting:
\begin{proposition}
	[$\VV(-)$ is inclusion reversing]
	If $I \subseteq J$ then $\VV(I) \supseteq \VV(J)$.
	Thus $\VV(-)$ is \emph{inclusion-reversing}.
\end{proposition}
\begin{ques}
	Verify this.
\end{ques}
Thus, bigger ideals correspond to smaller varieties.
As the above pictures might have indicated,
the smallest varieties are \emph{single points}.
Moreover, as you might guess from the name,
the biggest ideals are the \emph{maximal ideals}.
As an example, all ideals of the form
\[ \left( x_1-a_1, \dots, x_n-a_n \right) \]
are maximal, since the quotient
\[ \CC[x_1, \dots, x_n] / \left( x_1-a_1, \dots, x_n-a_n \right) \cong \CC \]
is a field.
The question is: are all maximal ideals of this form?

The answer is in the affirmative.
%It's equivalent to:
%\begin{theorem}
%	[Weak Nullstellensatz, phrased as nonempty varieties]
%	Let $I \subsetneq \CC[x_1, \dots, x_n]$ be a proper ideal.
%	Then the variety $\VV(I) \neq \varnothing$.
%\end{theorem}
% From this we can deduce that all maximal ideals are of the above form.
\begin{theorem}
	[Weak Nullstellensatz, phrased with maximal ideals]
	Every maximal ideal of $\CC[x_1, \dots, x_n]$
	is of the form $(x_1-a_1, \dots, x_n-a_n)$.
\end{theorem}
The proof of this is surprisingly nontrivial,
so we won't include it here yet; see \cite[\S7.4.3]{ref:vakil}.
%% TODO we might include this eventually
%\begin{proof}
%	[WN implies MI]
%	Let $J$ be a maximal ideal, and consider the corresponding variety $V = \VV(J)$.
%	By WN, it contains some point $p=(a_1, \dots, a_n)$.
%	Now, define $I = (x_1-a_1, \dots, x_n-a_n)$; this ideal contains all polynomials
%	vanishing at $p$, so necessarily $J \subseteq I \subsetneq \CC[x_1, \dots, x_n]$.
%	Then by maximality of $J$ we have $J=I$.
%\end{proof}
Again this uses the fact that $\CC$ is algebraically closed.
(For example $(x^2+1)$ is a maximal ideal of $\RR[x]$.)
Thus:
\begin{moral}
	Over $\CC$, maximal ideals correspond to single points.
\end{moral}

Consequently, our various ideals over $\CC$ correspond to various flavors
of affine varieties:
\begin{center}
	\begin{tabular}[h]{|cc|}
		\hline
		Algebraic flavor & Geometric flavor \\ \hline
		radical ideal & affine variety \\
		prime ideal & irreducible variety \\
		maximal ideal & single point \\
		any ideal & (scheme?) \\ \hline
	\end{tabular}
\end{center}
There's one thing I haven't talked about: what's the last entry?

\section{Motivating schemes with non-radical ideals}
One of the most elementary motivations for the scheme
is that we would like to use them to count multiplicity.
That is, consider the intersection
\[ \VV(y-x^2) \cap \VV(y) \subseteq \Aff^2 \]
This is the intersection of the parabola with the tangent $x$-axis,
this is the green dot below.

\begin{center}
	\begin{asy}
		import graph;
		size(5cm);

		real f(real x) { return x*x; }
		graph.xaxis("$x$", red);
		graph.yaxis("$y$");
		draw(graph(f,-2,2,operator ..), blue, Arrows);
		label("$\mathcal V(y-x^2)$", (0.8, f(0.8)), dir(-45), blue);
		label("$\mathbb A^2$", (2,3), dir(45));
		dotfactor *= 1.5;
		dot(origin, heavygreen);
	\end{asy}
\end{center}

Unfortunately, as a variety, it is just a single point!
However, we want to think of this as a ``double point'':
after all, in some sense it has multiplicity $2$.
You can detect this when you look at the ideals:
\[ (y-x^2) + (y) = (x^2,y) \]
and thus, if we blithely ignore taking the radical, we get
\[ \CC[x,y] / (x^2,y) \cong \CC[\eps] / (\eps^2). \]
So the ideals in question are noticing the presence of a double point.

In order to encapsulate this, we need a more refined object than
a variety, which (at the end of the day) is just a set of points;
it's not possible using topology along to encode more information
(there is only one topology on a single point!).
This refined object is the \emph{scheme}.

\section\problemhead
\todo{some actual computation here would be good}

\begin{problem}
	Show that a \emph{real} affine variety $V \subseteq \Aff_\RR^n$
	can always be written in the form $\VV(f)$.
	\begin{hint}
		Squares are nonnegative.
	\end{hint}
	\begin{sol}
		If $V = \VV(I)$ with $I = (f_1, \dots, f_m)$
		(as usual there are finitely many polynomials since $\RR[x_1, \dots, x_n]$ is Noetherian)
		then we can take $f = f_1^2 + \dots + f_m^2$.
	\end{sol}
\end{problem}

\begin{problem}
	[Complex varieties can't be empty]
	\label{prob:complex_variety_nonempty}
	Prove that if $I$ is a proper ideal in $\CC[x_1, \dots, x_n]$
	then $\VV(I) \ne \varnothing$.
	\begin{hint}
		This is actually an equivalent formulation
		of the Weak Nullstellensatz.
	\end{hint}
	\begin{sol}
		Let $I$ be an ideal, and let $\km$ be a maximal ideal contained in it.
		(If you are worried about the existence of $\km$,
		it follows from Krull's Theorem, \Cref{prob:krull_max_ideal}).
		Then $\km = (x_1 - a_1, \dots, x_n - a_n)$ by Weak Nullstellensatz.
		Consequently, $(a_1, \dots, a_n)$ is the unique point of $\VV(\km)$,
		and hence this point is also in $\VV(I)$.
	\end{sol}
\end{problem}

\begin{problem}
	\yod
	\label{prob:hilbert_from_weak}
	Show that Hilbert's Nullstellensatz in $n$ dimensions
	follows from the Weak Nullstellensatz.
	(This solution is called the \vocab{Rabinowitsch Trick}.)
	\begin{hint}
		Use the weak Nullstellensatz on $n+1$ dimensions.
		Given $f$ vanishing on everything,
		consider $x_{n+1}f-1$.
	\end{hint}
	\begin{sol}
		The point is is to check that if $f$ vanishes on all of $\VV(I)$,
		then $f \in \sqrt I$.

		Take a set of generators $f_1, \dots, f_m$,
		in the original ring $\CC[x_1, \dots, x_n]$;
		we may assume it's finite by the Hilbert basis theorem.

		We're going to do a trick now:
		consider $S = \CC[x_1, \dots, x_n, x_{n+1}]$ instead.
		Consider the ideal $I' \subseteq S$ in the bigger ring
		generated by $\{f_1, \dots, f_m\}$ and the polynomial $x_{n+1} f - 1$.
		The point of the last guy is that its zero locus
		does not touch our copy $x_{n+1}=0$ of $\Aff^n$
		nor any point in the ``projection'' of $f$ through $\Aff^{n+1}$
		(one can think of this as $\VV(I)$ in the smaller ring
		direct multiplied with $\CC$).
		Thus $\VV(I') = \varnothing$, and by the weak Nullstellensatz
		we in fact have $I' = \CC[x_1, \dots, x_{n+1}]$.
		So
		\[ 1 = g_1f_1 + \dots + g_mf_m + g_{m+1} \left( x_{n+1}f-1 \right). \]
		Now the hack: \textbf{replace every instance of $x_{n+1}$ by $\frac 1f$},
		and then clear all denominators.
		Thus for some large enough integer $N$ we can get
		\[ f^N = f^N(g_1f_1 + \dots + g_mf_m) \]
		which eliminates any fractional powers of $f$ in the right-hand side.
		It follows that $f^N \in I$.
	\end{sol}
\end{problem}