TITLE: Comparison of the mod p etale cohomology of special fiber and generic fiber
QUESTION [4 upvotes]: I am new to etale cohomology theory and recently learnt some base change theorems like proper base change and smooth base change. However, I am somehow confused about the etale cohomology group of $\Bbb Z/p\Bbb Z$ after base change.
For example, asssume we have a scheme $X$ proper smooth over $\Bbb Z_p$, where $\Bbb Z_p$ means the $p$-adic integer ring. What is the relation between $H^i_{et}(X_{\Bbb F_p},\Bbb Z/p\Bbb Z)$ and $H^i(X_{\Bbb Q_p},\Bbb Z/ p\Bbb Z)$? Must them have the same dimension as $\Bbb F_p$-vector space?

REPLY [0 votes]: Do you know examples where the dimension of i-th cohomology group of the special fiber is larger than generic fiber?
It's late, so I'll have to think more, but the answer is definitely no for $i=1$, and in fact the proof follows from the above. The key point is to recall that if $X$ is a smooth geometrically connected proper variety over a field, then $H^1(X,\underline{G})$ agrees with $H^1(\text{Alb}(X),\underline{G})$, where $\text{Alb}(X)$ is the Albanese variety of $X$ (an abelian variety!). It's also true for general curves since the only interesting result is for $i=1$.

Of course such examples exist. Let $S$ be the spectrum of a strict Henselian discrete valuation ring, let $f: X \to S$ be a proper morphism such that there exists a finite closed subset $A$ in the special fiber so that $f$ is smooth outside $A$, and $d$ be the relative dimension of $f$. Then the vanishing cycle $R\Phi \mathbb{Q}_l$ is supported on $S$ and its cohomology is concentrated in degree $d$. We have an exact sequence
$$H^d(R\Phi \mathbb{Q}_l) \to H^{d+1}(\text{special fiber}) \to H^{d+1}(\text{generic fiber}) \to 0.$$So $H^{d+1}(\text{special fiber})$ can have higher dimension than $H^{d+1}(\text{generic fiber})$.

This is essentially what I was going to write, but I also wanted to come up with an example that didn't use dimension reasons to deduce the vanishing of the vanishing cycles.

For a smooth morphism the vanishing cycle vanishes. This is proved not by counting dimensions.


TITLE: What forces act on a bob and a rope in circular motion
QUESTION [0 upvotes]: I am trying to understand what forces act on which objects, and which forces are pairs with regards to newtons 3rd law.
So imagine a rope attached to a bob is being spun round in a circle parallel to the ground (I know it can't be perfectly horizontal but just for ease lets say it does). I know the tension acts as the centripetal force on the bob. But does this directly act on the bob because the tension acts in the rope so surely it acts on the rope? And what would be the equal but opposite force be of the centripetal force?

REPLY [0 votes]: Let's say this whole setup lies on a smooth table so we can safely ignore gravity. Now answering your questions:
In reality, every segment of the rope is pulling on the segment attached to it (and vice versa according to the third law). But usually, we assume the mass of the rope to be zero so the tension in the rope is same throughout. You could think of the rope as a long chain of massless blocks each pulling on the adjacent blocks.
The equal and opposite to the centripetal force that the rope applies on the bob is the force which the bob applies on the rope (which equals the tension in the rope). That is the force you will feel when you swing your arm trying to spin a bob very fast.


text:
\subsection{Fiber product notation}
\label{sec:product_notation}

The most important particular notation used in the article is that of fiber products. That is, for a family $f : X \to T$ of varieties we denote the $m$-times fiber product of $X$ with itself over $T$ by $X^{(m)}$. As in our situation the base is always clear, we omit it from the notation. So, for $X^{(m)}$ means $m$-times fiber product over $T$, and $X_t^{(m)}$ means $m$-times fiber product over $t$. In this situation $p_i : X^{(m)} \to X$ denotes the projection onto the $i$-th factor, and we set for any divisor $D$ or line bundle $\sL$:
\begin{equation*}
D^{(m)}:= \sum_{i=1}^m p_i^* D \textrm{, and } \sL^{(m)}:= \bigotimes_{i=1}^m p_i^* \sL.
\end{equation*}

\subsection{General further notation}
\label{sec:general_notation}

A \emph{variety} is an integral, separated scheme of finite type over $k$. A $(X, \Delta)$ is a \emph{pair} if $X$ is a normal variety, and $\Delta$ is an  effective $\bQ$-divisor, called the \emph{boundary}.
A projective pair $(X, \Delta)$ over $k$ is \emph{Fano} if $(X, \Delta)$ has klt singularities, and $-(K_X + \Delta)$ is an ample $\bQ$-Cartier divisor. To avoid confusion, many times we say \emph{klt Fano} instead of \emph{Fano}, nevertheless we mean the same by the two. If there is not boundary, we mean taking the empty boundary. A projective pair $(X,\Delta)$ is a \emph{normal Fano} pair, if the klt condition is not assumed, that is, we only assume that  $-(K_X + \Delta)$ is an ample $\bQ$-Cartier divisor.

A \emph{big open set} $U$ of a variety $ X$ is an open set for which $\codim_X (X \setminus U) \geq 2$.

A \emph{vector bundle} is a locally free sheaf of finite rank.

The \emph{$\bQ$-linear system}  of a $\bQ$-divisor $D$ on a normal variety is $|D|_{\bQ}:=\{L \textrm{ is a $\bQ$-divisor} | \exists m \in \bZ, m>0 : m L \sim m D\}$.

A \emph{geometric fiber} of a morphism $f : X \to T$ is a fiber over a geometric point, that is over a morphism $\Spec K \to T$, where $K$ is an algebraically closed field extension of $k$. We say that a condition holds for a \emph{very general geometric point/fiber}, if there are countably many proper closed sets, outside of which it holds for all geometric points/fibers. \emph{General point/fiber} is defined the same way but with excluding only finitely many proper closed subsets. The \emph{(geometric) generic point/fiber} on the other hand denotes the scheme theoretic (geometric) generic point/generic fiber.

\subsection{Relative canonical divisor}
\label{sec:relative_canonical}

For a flat family $f : X \to T$ the relative dualizing complex  is defined by $\omega_{X/T}^\bullet:=f^! \sO_T$, where $f^!$ is Grothendieck upper shriek functor as defined in \cite{Hartshorne_Residues_and_duality}. If $f$ is also a family of pure dimension $n$, then the relative canonical sheaf is the lowest non-zero cohomology sheaf $\omega_{X/T}:= h^{-n}(\omega_{X/T}^\bullet)$ of the relative dualizing complex. To obtain the absolute versions of these notions one uses the above definition for $T = \Spec k$. The important facts for this article are:
\begin{enumerate}
\item The sheaf $\omega_{X/T}$ is reflexive if the fibers are normal \cite[Prop A.10]{Patakfalvi_Schwede_Zhang_F_singularities_in_families}.
\item If $T$ is Gorenstein, then $\omega_{X/T} \cong \omega_X \otimes f^* \omega_T^{-1}$ \cite[Lemma 2.4]{Patakfalvi_Semi_negativity_of_Hodge_bundles_associated_to_Du_Bois_families}, and then as $\omega_X$ is $S_2$ \cite[Cor 5.69]{Kollar_Mori_Birational_geometry_of_algebraic_varieties}, it is also reflexive  \cite{Hartshorne_Stable_reflexive_sheaves}.
\item By the previous two points, if $f$ is flat and either $T$ is smooth or the fibers are normal, then $\omega_{X/T}$ is reflexive, and then if $X$ is normal, it corresponds to a divisor linear equivalence class which we denote by $K_{X/T}$.
\item \label{itm:Cohen_Macaulay_base_change} On the relative Cohen-Macaulay locus $U \subseteq X$ (that is, on the open set where the fibers are Cohen-Macaulay),  $\omega_{U/T} \cong \omega_{X/T}|_U$ is compatible with base-change \cite[Thm 3.6.1]{Conrad_Grothendieck_duality_and_base_change}.
\end{enumerate}
In particular, by the above we always have the following assumptions on our families: $f : X \to T$ is flat with fibers being of pure dimension $n$, and either $T$ is smooth, or the fibers of $f$ are normal. In both cases  we discuss base-change properties of the relative canonical divisor below.

\subsubsection{Base-change of the relative log-canonical divisor when the fibers are normal}
\label{sec:base_change_relative_canonical_normal_fibers}

Let us assume that $f : X \to T$ is a projective, flat morphism to normal projective variety with normal, connected fibers (in particular $X$ is also normal), and $\Delta$ is an effective $\bQ$-divisor on $X$, such that $\Delta$ does not contain any fiber, and  $K_{X/T} + \Delta$ is a $\bQ$-Cartier divisor. Let $U \subseteq X$ be the smooth locus of $f$, which is an open set, and by the normality assumption on the fibers, $U \cap X_t$ is a big open set on each fiber $X_t$ (see \autoref{sec:general_notation} for the definition of a big open set).

Let $S \to T$ be a morphism from another normal projective variety. Then, we may define a pullback $\Delta_S$ as the unique extension of the pullback of $\Delta|_U$ to $U_S$ (the key here is that $\Delta|_U$ is $\bQ$-Cartier). In particular, $f_S : X_S \to S$ and $\Delta_S$   satisfies all the assumptions we had for $f : X \to T$ and $\Delta$. Moreover, if $\sigma: X_S \to X$ is the induced morphism, then as $\bQ$-Cartier divisors
\begin{equation}
\label{eq:pullback_log_canonical_divisor_normal_fibers}
 K_{X_S/S} +  \Delta_S \sim_{\bQ} \sigma^* (K_{X/T} + \Delta).
\end{equation}
Indeed, it is enough to verify this isomorphism on $U$ (as $U$ is big in $X$, and $U_S$ is big in $X_S$). However, over $U$ the linear equivalence \autoref{eq:pullback_log_canonical_divisor_normal_fibers} holds by the definition of $\Delta_S$ and by the base-change property of point \autoref{itm:Cohen_Macaulay_base_change} above.

\subsubsection{Base-change of the relative log-canonical divisor when the base is smooth}
\label{sec:base_change_relative_canonical_smooth_base}

Let $f : X \to T$ be a flat morphism from a normal projective variety to a smooth, projective variety with connected fibers. Let $\Delta$ be an effective $\bQ$-divisor on $X$ such $K_{X/T} + \Delta$ is $\bQ$-Cartier. Let $T_{\norm} \subseteq T$ be the open set over which the fibers of $X$ are normal.

Note that by the smoothness assumption on $T$, at a point $x \in X$, the fiber $X_{f(x)}$ is Gorenstein if and only if $X$ is relatively Gorenstein if and only if $X$ is Gorenstein.
Let $U \subseteq X$ be the open set of relatively Gorenstein points over $T$. Let $\iota : C \to T$ be a finite morphism from a smooth, projective curve such that $\iota(C) \cap T_{\norm} \neq \emptyset$, and denote by $\sigma: X_C \to X$ the natural morphism.

\emph{We claim that $\sigma^{-1} U$ is big in $X_C$.} This is equivalent  to show that for each $c \in C$, $X_c$ is Gorenstein at some point, and that for general $c \in C$, there is a big open set of $X_c$ where $X_c$ is Gorenstein. The former is true for all schemes of finite type over $k$ (hence also for $X_c$), and the latter is true by the $\iota(C) \cap T_{\norm} \neq \emptyset$ assumption. This concludes our claim.

Now, let $\pi : Z \to X_C$ be the normalization of $X_C$, $\rho: Z \to X$ and $g : Z \to C$ the induced morphisms and set $W:= \rho^{-1} U$. The notations are summarized in the following diagram:
\begin{equation*}
\xymatrix{
&   W \ar@{^(->}[ld] \ar[r] & \sigma^{-1} U \ar@{^(->}[ld] \ar[r]  & U \ar@{^(->}[ld] & \\

Z \ar[dr]_g
\ar@/^1.3pc/[rr]|!{[rru];[r]}\hole^\rho

\ar[r]_{\pi} & X_C \ar[r]_{\sigma} \ar[d]^{f_C} & X \ar[d]^f \\
& C \ar[r]_{\iota} & T & T_{\norm} \ar@{_(->}[l]
}
\end{equation*}
Then, \cite[Lem 9.13]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} tells us that there is a natural injection $\omega_{W/C} \to \left( \pi|_W \right)^* \omega_{\sigma^{-1} U/C}$. To be percise, \cite[Lem 9.13]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} assumes $\sigma^{-1} U$ to be normal, but as the proof does not use it, this is an unnecessary assumption.  Combining this injection with the isomorphism
$\left( \sigma|_{\sigma^{-1}U} \right)^* \omega_{U/T} \cong \omega_{\sigma^{-1} U/C}$ given by point \autoref{itm:Cohen_Macaulay_base_change} above we obtain
\begin{equation}
\label{eq:relative_canonical_base_change}
\omega_{W/C}  \hookrightarrow \left( \pi|_W \right)^* \omega_{\sigma^{-1} U/C} \cong  \left( \pi|_W \right)^* \left( \sigma|_{\sigma^{-1}U} \right)^* \omega_{U/T} \cong \left( \rho|_W \right)^* \omega_{U/T} ,
\end{equation}
which is an isomorphism over the locus $T_{\red}$ over which the fibers of $f$ are reduced. Indeed, over $T_{\red}$ the fibers of $X_C \to C$ are all reduced, and by the $\iota(C) \cap T_{\norm} \neq \emptyset$ assumption the general fiber of $X_C \to C$ is normal. In particular, over $T_{\red}$, $X_C$ is $R_1$ and $S_2$,  and hence normal. So, $\pi$ is the identity over $T_{\red}$.

Let $m>0$ be then an integer such that $m(K_{X/T}+ \Delta)$ is Cartier. That is, $\sL:=\sO_X(m(K_{X/T}+ \Delta))$ is a line bundle, and furthermore, $m\Delta$ yields an embedding $\omega_{U/T}^{\otimes m} \hookrightarrow \sL|_U$. Composing this with the $m$-th power of the homomorphism of \autoref{eq:relative_canonical_base_change} we obtain:
\begin{equation}
\label{eq:relative_canonical_base_change_final}
 \omega_{W/C}^{\otimes m} \to (\rho|_W)^* \sL \cong \sO_W(m \rho^*(K_{X/T} + \Delta)|_W),
\end{equation}
which map over $T_{\red}$ is given by ``multiplying with $\left(\rho|_{g^{-1}\iota^{-1} T_{\red}}\right)^* m\Delta$''. Indeed, for the latter remark, the main thing to note is that the regular locus of $X$, over which   $m \Delta$ is necessarily Cartier, pulls back to a big open set of  $g^{-1}\iota^{-1}T_{\red}$ (as general fiber of $f_C$ is normal and special fiber of $f_C$ over $T_{\red}$ are reduced). Hence $\pi$ is an isomorphism over $g^{-1}\iota^{-1}T_{\red}$ and also the pullback $\left(\rho|_{g^{-1}\iota^{-1} T_{\red}}\right)^* m\Delta$ is sensible the usual way: restricting to the regular locus, performing the pullback there, and then taking divisorial extension using bigness of the open set.

Lastly, the map \autoref{eq:relative_canonical_base_change_final} is given by an effective divisor $D$. If we set $\Delta_Z:= \frac{D}{m}$, using that $W$ is big in $Z$, we obtain:

\begin{proposition}
\label{rem:reduced_fibers_no_boundary}
\label{prop:relative_canonical_base_change_normal}
In the above situation, there is an effective $\bQ$-divisor $\Delta_Z$ on $Z$ such that:
\begin{enumerate}
\item \label{itm:relative_canonical_base_change:base_change} $K_{Z/C} + \Delta_Z \sim_{\bQ} \rho^*( K_{X/T} + \Delta)$,
\item \label{itm:relative_canonical_base_change:red_fibers} $X_C$ is normal over $T_{\red}$ and $\Delta_Z|_{g^{-1}\iota^{-1}T_{\red}} = \left(\rho|_{g^{-1}\iota^{-1} T_{\red}}\right)^* \Delta$, and
\item $\Delta_Z|_{g^{-1}\iota^{-1}T_{\norm}}$ agrees with the pullback of $\Delta|_{f^{-1}T_{\norm}}$ in the sense of \autoref{sec:base_change_relative_canonical_normal_fibers}.
\end{enumerate}
\end{proposition}

config: arxiv


text:
\begin{document}

\title[The M\"obius disjointness conjecture for distal flows]
{The M\"obius disjointness conjecture \\ for distal flows}
\author{Jianya Liu \& Peter Sarnak}
\address{School of Mathematics
\\
Shandong University
\\
Jinan
\\
Shandong 250100
\\
China}
\email{jyliu@sdu.edu.cn}

\address{Department of Mathematics
\\
Princeton University \& Institute for Advanced Study
\\
Princeton, NJ 08544-1000
\\
USA}
\email{sarnak@math.princeton.edu}

\date{\today}

\subjclass[2000]{11L03, 37A45, 11N37}
\keywords{The M\"obius function, distal flow,  affine linear map,
skew product, nilmanifold}

\maketitle

\addtocounter{footnote}{1}

\section{The M\"obius  disjointness conjecture}
\setcounter{equation}{0}
Let $\mathscr{X}=(T, X)$ be a flow, namely $X$ is a compact topological space
and $T: X\to X$ a continuous map. The sequence $\xi(n)$
is observed in $\mathscr{X}$ if there is an $f\in C(X)$ and an $x\in X$,
such that $\xi(n)=f(T^n x)$. Let $\mu(n)$ be the M\"obius function, that
is $\mu(n)$  is $0$ if $n$ is not square-free, and is $(-1)^t$ if  $n$ is a
product of $t$ distinct primes. We say that $\mu$ is linearly
disjoint from $\mathscr{X}$ if
\begin{eqnarray}\label{def/DISJO}
\f{1}{N}\sum_{n\leq N} \mu(n)\xi(n) \to 0,  \quad \mbox{as } N\to\infty,
\end{eqnarray}
for every observable $\xi$ of $\mathscr{X}$.
The M\"obius Disjointness Conjecture of
the second author  (\cite{Sar}, \cite{Sar1}) states the following.

\begin{conjecture} [The M\"obius Disjointness Conjecture]
The M\"obius function $\mu$ is linearly disjoint from
every $\mathscr{X}$ whose entropy is $0$.
\end{conjecture}

This Conjecture has been established for many flows
$\mathscr{X}$ (see \cite{Dav}, \cite{MauRiv}, \cite{GreTao2}, \cite{BouSarZie}, \cite{Bou})
however all of these flows are quasi-regular (or rigid) in the sense that
the Birkhoff averages
\begin{eqnarray}\label{def/Birkh}
\f{1}{N}\sum_{n\leq N} \xi(n)
\end{eqnarray}
exist for every  $\xi$ observed in $\mathscr{X}$.
In \cite{LiuSar} we establish some new cases of the Disjointness Conjecture and
in particular for irregular flows $\mathscr{X}$, that is ones for which  (\ref{def/Birkh})
fails.  These flows are complicated in terms of the behavior of their individual
orbits but they are distal and of zero entropy, so that the
disjointness is still expected to hold.

\section{Results}
\setcounter{equation}{0}

In this section we summarize the results we have established
in \cite{LiuSar}. The first result in \cite{LiuSar}
is concerned with certain regular flows, namely affine linear
maps  of a compact abelian group $X$.  Such a flow $(T, X)$
is given by
\begin{eqnarray}\label{def/AFF}
T(x)=Ax+b
\end{eqnarray}
where $A$ is an automorphism of $X$ and $b\in X$ (see \cite{Hah},  \cite{HoaPar}).

\begin{theorem}\label{thm1}
Let $\mathscr{X}=(T, X)$ be an affine linear flow on a compact
abelian group which is of zero entropy. Then $\mu$
is linearly disjoint from $\mathscr{X}$.
\end{theorem}

Theorem~\ref{thm1} actually holds with a rate of convergence.
We first reduce to the torus case and then handle
the torus case by Fourier analysis and classical results of Davenport \cite{Dav}
and Hua \cite{Hua} on exponential sums concerning the
M\"{o}bius function.

The flows in Theorem~\ref{thm1} are distal,
and our main
result in \cite{LiuSar} is concerned with nonlinear distal flows on such spaces.
We restrict to $X={\Bbb T}^2$ the two dimensional torus
${\Bbb R}^2/{\Bbb Z}^2$ and consider nonlinear smooth (or even analytic)
skew products as discussed in Furstenberg \cite{Fur61}. $T: {\Bbb T}^2\to {\Bbb T}^2$
is given by
\begin{eqnarray}\label{def/SKEW}
T(x, y)= (ax+\a, cx+dy+ h(x))
\end{eqnarray}
where $a, c, d\in \Bbb Z, ad=\pm 1,  \a\in \Bbb R$
and $h$ is a smooth periodic function of period $1$.
The affine
linear part is in the form
$$
\left[
\begin{array}{ccc}
a & 0\\
c & d
\end{array}
\right] \in GL_2 (\Bbb Z),
$$
ensuring that $T$ has zero entropy (and it can always be brought into
this form). The flow $(T, {\Bbb T}^2)$ is distal and this skew product
is a basic building block (with $e(h(x))$ continuous) in Furstenberg's
classification theory of minimal distal flows \cite{Fur63}. If
$\a$ is diophantine, that is
$$
\bigg|\a-\f{a}{q}\bigg|\geq \f{c}{q^B}
$$
for some $c>0, B<\infty$ and all $a/q$ rational,  then $T$ can be conjugated
by a smooth map of ${\Bbb T}^2$ to its affine linear part
\begin{eqnarray}\label{DIO/conjugate}
(x, y)\mapsto (ax+\a, cx+dy+ \b)
\end{eqnarray}
where
$$
\b=\int_0^1 h(x)dx
$$
(see \cite{SanUrz}).
Hence the disjointness of $\mu$ from
$\mathscr{X}=(T, {\Bbb T}^2)$ for a $T$ with a diophantine
$\a$, follows from Theorem~\ref{thm1}.
However if $\a$ is not diophantine the dynamics of the
flow $(T, {\Bbb T}^2)$ can be very different from an affine
linear flow. For example, as Furstenberg shows it may be
irregular (i.e. the limits in (\ref{def/Birkh}) fail to exist  for certain observables).
Our main result is that these nonlinear skew products
are linearly disjoint from $\mu$, at least if $h$ satisfies some further
technical hypothesis. Firstly we assume that $h$ is analytic, namely that
if
\begin{eqnarray}\label{h=/FOUR}
h(x)=\sum_{m\in \Bbb Z} \hat{h}(m) e(mx)
\end{eqnarray}
then
\begin{eqnarray}\label{hhat/UPP}
\hat{h}(m)\ll e^{-\tau |m|}
\end{eqnarray}
for some $\tau>0$. Secondly we assume that there is
$\tau_2<\infty$ such that
\begin{eqnarray}\label{hhat/LOW}
|\hat{h}(m)|\gg e^{-\tau_2 |m|}.
\end{eqnarray}
This is not a very natural condition being an artifact of our proof.
However it is not too restrictive and the following applies rather
generally (and most importantly there is no condition on $\a$).

\begin{theorem}\label{thm2}
Let $\mathscr{X}=(T, {\Bbb T}^2)$ be of the form (\ref{def/SKEW}),
with $h$ satisfying (\ref{hhat/UPP}) and (\ref{hhat/LOW}).
Then $\mu$ is linearly disjoint from $\mathscr{X}$.
\end{theorem}

The assertion of
Thereom~\ref{thm2} holds for all $\a$, and so we have to consider all
diophantine possibilities of $\a$. The tool is the Bourgain-Sarnak-Ziegler
\cite{BouSarZie}
finite version of the Vinogradov method, incorporated with various analytic methods
such as Poisson's summation and stationary phase.
Furstenberg \cite{Fur61} gives examples of skew product transformations
of the form (\ref{def/SKEW}) which are not regular in the sense of (\ref{def/Birkh}).
Many of the flows ${\mathscr X}$
in Theorem~\ref{thm2} have this property and we show in \cite{LiuSar}
that Furstenberg's
examples are smoothly conjugate to such ${\mathscr X}$'s. In particular his examples are linearly
disjoint from $\mu$.

Theorem~\ref{thm1} deals with the affine
linear distal flows on the $n$-torus. A different source
of  homogeneous distal flows are the affine linear flows on nilmanifold
$X=G/\G$ where $G$ is a nilpotent Lie group and
$\G$ a lattice in $G$.  For
$\mathscr{X}=(T, G/\G)$ where $T(x)=\a x\G$
with $\a\in G$, i.e. translation on $G/\G$, the
linear disjointness of $\mu$ and $\mathscr{X}$ is proven in
\cite{GreTao1} and \cite{GreTao2}. Using the classification of zero entropy (equivalently distal)
affine linear flows on nilmanifolds \cite{Dan}, and
Green and Tao's results we establish
in \cite{LiuSar} the following.

\begin{theorem}\label{thm3}
Let $\mathscr{X}=(T, G/\G)$ where $T$ is an affine
linear map of the nilmanifold  $G/\G$ of zero entropy.
Then $\mu$ is linearly disjoint from $\mathscr{X}$.
\end{theorem}

The above results for $\mu(n)$ can be
proved in the same way for similar multiplicative functions such as
$\l(n)=(-1)^{\tau(n)}$ where $\tau(n)$ is the number of
prime factors of $n$.

\section{Disjointness of $\mu$ from Furstenberg's systems}
\setcounter{equation}{0}

As a consequence of Theorem~\ref{thm2},
it is proved in \cite{LiuSar} that $\mu$ is linearly
disjoint from Furstenberg's systems.
But no rate of convergence is obtained there
since Theorem~\ref{thm2} in general offers no rate.
In this
section we show that, for Furstenberg's systems, rate of convergence is actually
available if we work on these systems directly rather than appeal to
Theorem~\ref{thm2}.

\subsection{The continued fraction expansion of $\a$.}
We assume that $\a$ is irrational, and our argument will
depend on the continued fraction expansion of $\a$.
Every real number $\a$ has its continued fraction
representation
\begin{eqnarray}\label{a=a0a1+}
\a=a_0+\f{1}{a_1+\f{1}{a_2+\cdots}}
\end{eqnarray}
where $a_0=[\a]$ is the integral part of $\a$,
and $a_1, a_2, \ldots$ are positive integers.
The expression (\ref{a=a0a1+}) is infinite since $\a\not\in \Bbb Q$.
We write $[a_0; a_1, a_2, \ldots]$ for the
expression on the right-hand side of (\ref{a=a0a1+}), which is the limit of
the finite continued expressions
\begin{eqnarray}\label{a=a0a1F}
[a_0; a_1, a_2, \ldots, a_k]=a_0+\f{1}{a_1+\f{1}{a_2+\cdots+\f{1}{a_k}}}
\end{eqnarray}
as $k\to \infty$.
Writing
\begin{eqnarray*}
\f{l_k}{q_k}=[a_0; a_1, a_2, \ldots, a_k],
\end{eqnarray*}
we have $l_0=a_0, l_1=a_0a_1+1, q_0=1, q_1=a_1, $
and for $k\geq 2$,
\begin{eqnarray*}
l_k=a_k l_{k-1}+l_{k-2}, \quad
q_k=a_k q_{k-1}+q_{k-2}.
\end{eqnarray*}
Since $\a$ is irrational we have $q_{k+1}\geq q_k+1$ for all $k\geq 1$.
An induction argument gives the stronger
assertion that
$q_k\geq 2^{(k-1)/2}$
for all $k\geq 2$, and thus $q_k$ increases at least like an exponential function of $k$.
The irrationality of $\a$ also implies that, for all $k\geq 2$,
\begin{eqnarray}\label{ratAPP}
\f{1}{2 q_k q_{k+1}}<\bigg|\a-\f{l_k}{q_k}\bigg| <\f{1}{q_kq_{k+1}}.
\end{eqnarray}

\subsection{Furstenberg's examples.}
Furstenberg \cite{Fur61}
gave examples of smooth transformation $T: {\Bbb T}^2\to {\Bbb T}^2$
such that the ergodic averages do not all exist.
Let $\a$ be as above such that
\begin{eqnarray}\label{qk+1>eqk}
q_{k+1}\geq e^{q_k}
\end{eqnarray}
for all positive $k$.
Define $q_{-k}= - q_k$ and set
\begin{eqnarray}
h(x)=\sum_{k\not=0}\f{e(q_k\a)-1}{|k|} e(q_k x).
\end{eqnarray}
It follows from (\ref{ratAPP}) and (\ref{qk+1>eqk}) that
$h(x)$ is a smooth function. We also have
$h(x)=g(x+\a)-g(x)$
where
\begin{eqnarray}
g(x)=\sum_{k\not=0}\f{1}{|k|}e(q_k x)
\end{eqnarray}
so that $g(x)\in L^2(0,1)$ and in particular
defines and measurable function. But $g(x)$ cannot
correspond to a continuous function, as shown in Furstenberg \cite{Fur61}.

\subsection{Disjointness of $\mu$ from Furstenberg's systems.}
In the following we consider slightly more general $h$'s with
\begin{eqnarray}\label{def/hx=sum}
h(x)=\sum_{k\not=0} c_k (1-e(q_k\a))e(q_k x)
\end{eqnarray}
where $\a$ satisfies (\ref{qk+1>eqk}) and, for all positive $k$, the coefficients
$c_k$ satisfy
\begin{eqnarray}
c_k= c_{-k}, \quad |c_k|\leq C
\end{eqnarray}
for some positive constant $C$ which we may
assume to be greater than $1$.
We note that if $c_k$ is not bounded then $g(x)$ will not be $L^2$.
Now what we want to estimate is essentially
\begin{eqnarray}\label{defSN}
S(N):=\sum_{n\leq N} \mu(n) e\bigg(\sum_{j=0}^{n-1}h(j\a)\bigg),
\end{eqnarray}
and our result is the following.

\begin{proposition}\label{thm:2-2}
Let $S(N)$ be as in (\ref{defSN}). Then
\begin{eqnarray}\label{S2est}
S(N)\ll_{} N\log^{-A}N
\end{eqnarray}
where the implied constant depends on $A$,
but is independent of $\a$.
\end{proposition}

\begin{proof}
By (\ref{def/hx=sum}) we have
\begin{eqnarray*}
\sum_{j=0}^{n-1}h(j\a)
=\sum_{k\not=0} c_k (1-e(q_k\a))\sum_{j=0}^{n-1}e(q_k j\a)
=\sum_{k\not=0} c_k (1-e(n q_k\a)).
\end{eqnarray*}
We should cut the last sum at some point. Let $K$ be such that
$q_{K-1}< 2\log N \leq q_{K}$.
Then for $k\geq K$ we have
\begin{eqnarray}\label{qk-lk}
|q_k\a-l_k|
\leq \f{1}{q_{k+1}}
\leq e^{-q_k},
\end{eqnarray}
so that
\begin{eqnarray*}
\sum_{|k|\geq K} |c_k (1-e(n q_k\a))|
\ll C\sum_{|k|\geq K} n e^{-q_{k}}
\ll  CN e^{-q_{K}}
\ll  CN^{-1}
\end{eqnarray*}
where we the implied constant does not depend on $C$.
It follows that
\begin{eqnarray*}
S(N)=\sum_{n\leq N} \mu(n)
e\bigg\{\sum_{1\leq |k|\leq K-1} c_k (1-e(n q_k\a))+O\bigg(\f{C}{N}\bigg)\bigg\}.
\end{eqnarray*}
The $O$-term as well as $\sum\limits_{1\leq |k|\leq K-1} c_k $
above are harmless,
and so from now on we concentrate on
\begin{eqnarray}\label{TilS=}
\widetilde{S}(N)
&:=&\sum_{n\leq N} \mu(n)
e\bigg\{\sum_{1\leq |k|\leq K-1} c_k e(n q_k\a)\bigg\} \nonumber\\
&=&\sum_{n\leq N} \mu(n)
e\bigg\{\sum_{1\leq |k|\leq K-1} c_k e(n \th_k)\bigg\}
\end{eqnarray}
on writing  $\th_k=\|q_k\a\|$. Note that by
(\ref{qk-lk}) we have $\th_k\leq e^{-q_k}$.

We have a sequence
$$
q_1<\exp(q_1) \leq q_2 < \exp(q_2) \leq q_3< \ldots,
$$
and therefore
$q_k> \exp \cdots\exp (q_2)$
with $k-2$ repeated $\exp$'s.
Taking $k=K-1$ and noting that $q_2\geq 2$
give
\begin{eqnarray}\label{repEXP}
\exp \cdots\exp (2) \leq \exp \cdots\exp (q_2) < q_{K-1}\leq 2\log N
\end{eqnarray}
where on the left-hand side there are $K-3$ $\exp$'s.

Now let
\begin{eqnarray}\label{DEFfi}
\phi(x)
= 2c_1\cos (2\pi x), \quad x\in [\th_1,\th_1 N].
\end{eqnarray}
Then $e(\phi(x))$ is a smooth periodic function
and hence can expanded into
Fourier series
\bea
e(\phi(x))=\sum_{l\in \Bbb Z} a_l(c_1)e(lx),
\eea
where
\bea
a_l(c_1)=\int_0^1 e(\phi(x))e(-lx)dx.
\eea
We must compute the dependence of $a_l(c_1)$ on $c_1$.
By partial integration for $l\not=0$ we have
\bna
a_l(c_1)
&=& -\f{1}{2\pi i l}\int_0^1 e(\phi(x))d e(-lx) \\
&=& -\f{1}{l}\int_0^1 e(\phi(x))\phi'(x) e(-lx) dx\\
&=& \f{1}{2\pi i l^2}\int_0^1 \f{d[e(\phi(x))\phi'(x)]}{dx} e(-lx) dx.
\ena
Since
\bna
\phi'(x)= - 4\pi c_1\sin (2\pi x), \quad
\phi''(x)= - 8\pi^2 c_1\cos (2\pi x),
\ena
we have
\bna
\bigg|\f{d[e(\phi(x))\phi'(x)]}{dx}\bigg|
=|e(\phi(x)) [\phi''(x)+2\pi i\phi'(x)\phi'(x)]|
\ll |c_1|+|c_1|^2\ll C^2
\ena
and consequently
\bea\label{al<<}
a_l(c_1)
\ll \f{C^2}{l^2}
\eea
for all $l\not=0$.
The above $\ll$-constant is absolute. Of course the above $l^2$
can be improved to $l^A$ for arbitrary $A>0$, but we are not going to
use this.

The sum $\widetilde{S}(N)$ in (\ref{TilS=}) is of the form
\begin{eqnarray*}
\widetilde{S}(N)
&=&\sum_{n\leq N} \mu(n)e(\phi(n\th_1)+F(n))\\
&=& \sum_{n\leq N} \mu(n) e(F(n))\sum_{l\in \Bbb Z} a_{l}(c_1) e(ln\th_1)\\
&=& \sum_{l\in \Bbb Z} a_{l}(c_1)
\sum_{n\leq N} \mu(n) e(ln\th_1+F(n)),
\end{eqnarray*}
where $F(n)$ stands for the remaining terms in the $e(\cdot)$ in (\ref{TilS=}).
It follows from this and (\ref{al<<}) that
\begin{eqnarray*}
|\widetilde{S}(N)|
&\leq & \sum_{l\in \Bbb Z} |a_{l}(c_1) |
\bigg|\sum_{n\leq N} \mu(n) e(ln\th_1+F(n))\bigg|\\
&\ll & C^2 \sum_{l\in \Bbb Z}\f{1}{l^2+1}
\bigg|\sum_{n\leq N} \mu(n) e(ln\th_1+F(n))\bigg|\\
&\ll & C^2
\sup_{l_1} \bigg|\sum_{n\leq N} \mu(n) e(n l_1\th_1 +F(n))\bigg|.
\end{eqnarray*}
Repeating this procedure in (\ref{TilS=}), we get
\begin{eqnarray*}
|\widetilde{S}(N)|
\ll  C^{2(K-1)}
\sup_{l_1, \ldots, l_{K-1}} \bigg|\sum_{n\leq N}
\mu(n) e(n l_1\th_1+ \ldots+ n l_{K-1}\th_{K-1}) \bigg|.
\end{eqnarray*}
The inner sum involving $\mu$ can be estimated by classical results of
Davenport \cite{Dav} and Hua \cite{Hua} as
$$
\ll N\log^{-A}N
$$
where $A>0$ is arbitrary, and the implied constant
depends at most on $A$, i.e. it does not depend on the coefficients
$l_1, l_2, \ldots, l_{K-1}$ or $\th_1, \th_2, \ldots, \th_{K-1}$.
By (\ref{repEXP}) we have $C^{2(K-1)}\leq \log N$. The proposition is proved.
\end{proof}

\medskip
\noindent
{\bf Acknowledgements.}
The first author is supported by the 973 Program,
NSFC grant 11031004, and IRT1264 from the Ministry of Education.
The second author is supported by an NSF grant.


text:
\begin{document}

\title{A local large deviation principle\\ for inhomogeneous birth-death processes}

\author{N.D. Vvedenskaya~$^{1}$, A.V. Logachov~$^{2,3,4}$, Y.M. Suhov~$^{1,5}$,
 A.A. Yambartsev~$^{6}$ }

\maketitle

 {\footnotesize

 \noindent $^1$ Dobrushin Laboratory, Institute for Information Transmission Problems, RAS, 19 Bol'shoi
Karetnyi per, Moscow 127051, RF\\
E-mail:\ \ ndv@iitp.ru\footnote{The work by N.D. Vvedenskaya had been conducted at the IITP RAS
under the support of the RSF (Project  No14-50-00 150).}

\noindent $^2$ Laboratory of Applied Mathematics, Novosibirsk State University, 2 Pirogov ul,
Novosibirsk 630090, RF\\

\noindent $^3$ Laboratory of Probability Theory and Mathematical Statistics, Sobolev Institute of Mathematrics,
Siberiian Branch of the RAS, 4 Koptyug ul, Novosibirsk 630090, RF

\noindent $^4$ Statistics Division, Novosibirsk State University of Economics and Management,
56 Kamenskaya ul, Novosibirsk 630099, RF\\
E-mail:\ \ omboldovskaya@mail.ru\footnote{The work by A.V. Logachov was supported by the RSF Grant
(Project No18-11-00129). }

\noindent $^5$  Math Dept, Penn State University, University Park, State College, PA 16802, USA.\\
E-mail:\ \ yms@statslab.cam.ac.uk

\noindent $^6$ Department of Statistics, Institute of Mathematics and Statistics, University of S\~ao Paulo,
1010 Rua do Mat\~ao, CEP 05508--090, S\~ao Paulo SP, Brazil \\
E-mail:\ \ yambar@gmail.com}

\vspace{1cm}

\begin{abstract}
The paper considers a continuous-time birth-death process where the jump rate has an asymptotically
polynomial dependence on the process position. We obtain a rough exponential asymptotics
for the probability of excursions of a re-scaled process contained within a neighborhood of a given
continuous non-negative function.
\end{abstract}

\section{Introduction} \def\lam{\lambda}

In the modern literature on the large deviation principle, one considers various conditions for random
processes guaranteeing a rough exponential asymptotics for probabilities of rare events. See,
for example, \cite{DZ} -- \cite{Var}.
In this paper we deal with birth-and-death Markov processes that are inhomogeneous in the state space:
the rates of jumps are polynomially dependent on the position of the process. For these processes we obtain
exponential asymptotics for the probabilities the normalized process to be in a neighborhood of a continuous
function. Moreover, we provide this asymptotics both for ergodic processes and for transient (even exploding)
processes.

The study of birth-and-death processes is of a certain mathematical interest and, moreover, is important
for a number of applications. As examples, we can cite the information theory (encoding and storage of
information, see \cite{SS1}), biology and chemistry (models of growth and extinction in systems with multiple
components,  see \cite{SS2},  \cite{MSSZ}), and economics (models of competitive production and pricing,
\cite{MPY}, \cite{VSB}).

Consider a continuous-time Markov process $\xi (t)$, $t\geq 0$, with state space $\mathbb{Z}^+\cup\{\infty\}$,
where $\mathbb{Z}^+=\{0\}\cup\mathbb{N}$. Let us assume that the process starts at 0.

The evolution of the process $\xi (\,\cdot\,)$ is described as follows. For a given $t\ge 0$, let
$\xi (t)=x\in\mathbb{Z}^+$. The state of the random process does not change during the random time
$\tau_x$ with exponential distribution  with parameter $h(x)>0$. At the moment $t+\tau_x$ the process
jumps to the states $x\pm 1$ with probabilities
\beq
{\bf P}(\xi (t+\tau_x)=x+1)=\frac{\lambda(x)}{h(x)},\ \ \ {\bf P}(
  \xi(t+\tau_x)=x-1)=\frac{\mu(x)}{h(x)}, \label{2}
\eeq
correspondingly, where $\lambda(x)+\mu(x)=h(x)$, $\lambda(x)> 0$ when $x\in \mathbb{Z}^+$, and
$\mu(x)> 0$ for $x\in \mathbb{N}$.

Suppose that for $x=0$ the rates $\mu (x)=0$, $\lambda(x)=\lambda_0>0$ (i.e. the process cannot take
negative values), and the following asymptotics hold true
\beq
\lim_{x\to
\infty}\frac{\lambda(x)}{P_lx^l}= \lim_{x\to
\infty}\frac{\mu(x)}{Q_mx^m}=1, \label{1}
\eeq
where $P_l$ and $Q_m$ are positive constants, and $l\geq 0$,
$m\geq 0$, $l\vee m>0$.

When $l\leq 1$, the  existence of a Markov process with the above properties
is established in the standard way, see for example \cite{Fell}, Ch. 17, \S 4, 5, and also  \cite{KS},
Ch. 2, $\S$ 5, Theorem 2.5.5, \cite{KT}, Ch. 6, 7. When $l>1$ the process $\xi(\cdot)$, generally
speaking, can go to infinity ("explode") during a random time, finite with probability one. There are
two approaches to construct such processes. (1) One can stop the process at the random time point
(the time of explosion); viz., see \cite{Korol}, Ch. 15, \S 4,
\cite{Dyn},  vol. {\bf 1}, Ch 6, P. 365; vol. {\bf 2},  P. 274.  \  (2)\ One can extend the phase space
$\mathbb{Z}^+$ by adding an
absorbing state (denoted by $\infty$); see, e.g., \cite{Ito}, Ch. 4, \S 48, \cite{Fell}, Ch. 17, $\S$ 10.
In this paper we use the second approach.

The above class of random processes has been given the name birth-and-death processes; see,
for example,  \cite{Fell},  \cite{Ito}.

There exist conditions on $l$ and $m$ which are sufficient for explosion and
non-explosion. For example, when $l>1$ and $m<l$, the process $\xi (\,\cdot\,)$ explodes, while if
$m>l$ it does not.  As references, cf. original papers
\cite{KMcG}, \cite{LR} and references within. See also \cite{No}, Ch. 23, $\S$ 7,
\cite{KS}, Ch. 2, $\S$ 5, and \cite{St}, Ch.5, $\S$ 3 (the last reference includes results for general Markov
chains, not only for birth-and-death processes).

We are interested in the local large deviation principle (LLDP) for the family of scaled processes
\begin{equation}\label{xit}
\xi_T(t)=\frac{\xi(tT)}{T}, \ 0\leq t\leq 1,
\end{equation}
where $T>0$ is a parameter  (see, \cite{Bor-Mog2}, \cite{Bor-Mog1}).
In a sense, the formulation and analysis of the LLDP should precede the study of
other forms of the large deviation principle.

The validity of our results does not depend on whether or not the process $\xi(\cdot )$ explodes within
a finite time. We focus on the asymptotics of the probability of the event that the trajectories of process
$\xi_T(\,\cdot\, )$ to stay in a neighborhood of a continuous positive function given on the interval $[0,1]$.
It means that we are working on the set of trajectories which do not tend to infinity in the time interval
$[0,T]$. The considered probabilities are positive even if the process $\xi(\cdot)$ explodes (see equation
(\ref{8a}) below).

Let $\mathbb{D}[0,1]$ denote the space of right-continuous functions with left-limit at eact $t\in [0,1]$.
For any  $f,g \in \mathbb{D}[0,1]$, set
$$
\rho(f,g)=\sup\limits_{t\in[0,1]}|f(t)-g(t)|.$$
\def\diy{\displaystyle}

\begin{definition}\label{def1}
The family of random processes $\xi_T(\,\cdot\,)$ satisfies the
LLDP on the set $G\subseteq \mathbb{D}[0,1]$ with a rate functional
$I = I(f)\,:\, \mathbb{D}[0,1] \rightarrow [0,\infty)$ and a normalising function
$T\in (0,\infty )\mapsto \psi(T)>0$ with $\lim\limits_{T\rightarrow\infty}\psi(T) = \infty$
if, for any function $f \in G$, the following equality holds true:
\begin{equation}\label{opr}\begin{array}{l}
\lim\limits_{\varepsilon\rightarrow 0}\limsup\limits_{T\rightarrow \infty}\diy\frac{1}{\psi(T)}
\ln\mathbf{P}(\xi_T(\,\cdot\,)\in U_\varepsilon(f))\\
\qquad =\lim\limits_{\varepsilon\rightarrow 0}\liminf\limits_{T\rightarrow \infty}\diy\frac{1}{\psi(T)}
\ln\mathbf{P}(\xi_T(\,\cdot\,)\in U_\varepsilon(f))=-I(f).\end{array}\end{equation}
Here
\begin{equation}\label{oprU}
U_\varepsilon(f)=\{g\in \mathbb{D}[0,1]: \ \rho(f,g)<\varepsilon\}.\end{equation}
\end{definition}

In the framework of Definition~\ref{def1} there are various cases to consider.
We separate three cases: 1) $l>m$, 2) $l<m$ and $l=m$.

Note that the case $m=1$, $l=0$, follows from \cite{MPY}
(where a two dimensional Markov process is treated). A
similar result is obtained in \cite{Log1}
for solutions of stochastic differential Ito's equations.
The classical case $l=m=0$, $\varphi(T)=T$ follows, for example, from \cite{Bor-Mog}.

In this paper we use the approach developed in \cite{MPY}. We would like to note that the
large deviation principle for the sequence of processes $\xi_T(\,\cdot\,)$ in space
$\mathbb{D}[0,1]$ with Skorohod metric cannot be obtained even for non-exploding processes:
one can show that the corresponding family of
measures is not exponentially dense, except for the case $l=m=0$.

The paper is organized as follows: in $\S$ 2 we introduce our definitions and the system
of notation, as well as the main result (Theorem~\ref{th2.1})
and key lemmas. In $\S$ 3 we prove Theorem~\ref{th2.1} and key lemmas.
In $\S$ 4 (the Appendix) some auxiliary technical assertions are established.

\section{Main results, definitions}

Let $F$ denote the set of functions $f(t) \in \mathbb{C}[0,1]$ such that  $f(0)=0$ and
$f(t)>0$ as $0<t\leq 1$.

\begin{theorem} \label{th2.1}
Let the conditions \eqref{2} and \eqref{1} be fulfilled. Than the random processes sequence
 $\xi_T(\cdot)$ on  $F$ fulfills  the following LLDP:

{\rm{a)}} If  $l>m$ then the normalizing function  $\psi(T)=T^{l+1}$,  and the rate functional has the form
$$
I(f)=P_l\int_0^1f^l(t)dt, \ \ \ f\in F.
$$

{\rm{b)}} If $l=m$ and  $P_l\neq Q_m$ then $\psi(T)=T^{l+1}$ and
$$
I(f)=(\sqrt{P_l}-\sqrt{Q_m})^2\int_0^1f^l(t)dt, \ \ \ f\in F.
$$

{\rm{c)}} If $l<m$ then  $\psi(T)=T^{m+1}$ and
$$
I(f)=Q_m\int_0^1f^m(t)dt, \ \ \ f\in F.
$$
\end{theorem}

The case where $l=m$ and $P_l= Q_m$ needs a different normalization; we do not discuss  it
in this paper.

\bigskip

Consider a space- and time-homogenous Markov process
$\zeta(t)$, $t\in[0,T]$, on the phase space $\mathbb{Z}$, where the jump rate is
equal to $1$, and the jump size is  $\pm 1$, occurring with probability $1/2$.

Denote by $X_T$ the set of all right-continuous step-functions with a finite number of
$\pm 1$-jumps on  $[0,T]$.

\begin{lemma} \label{l2.0}

For any given  $T$, the distribution  ${\mathbf P}^{(\xi)}_T(\,\cdot\,\cap X_T)$
of process  $\xi(\cdot)$ on $X_T$ is absolutely continuous with respect to the distribution
${\mathbf P}^{(\zeta )}_T$
of process $\zeta(\cdot)$ on $X_T$. The corresponding density (the Radon-Nikodym  derivative)
${\mathfrak p}_T(u)=\diy\frac{d{\mathbf P}^{(\xi )}_T}{d{\mathbf P}^{(\zeta )}_T}(u)$, $u\in X_T$,
has the form
\beq
{\mathfrak p}_T(u)=\left\{ \begin{array}{ll}2^{N_T(u)}\left(\prod\limits_{i=1}^{N_T(u)}
e^{-(h(u(t_{i-1}))-1)\tau_{i}}\nu(u(t_{i-1}),u(t_i))\right)\\
\qquad \times e^{-(h(u(t_{N_T(u)})-1))(T-t_{N_T(u)})
},\;\;\qquad\quad\quad \;\; \mbox{ if } \;N_T(u)\geq 1,\\
e^{-(h(0)-1)T}, \qquad\qquad\qquad\qquad\qquad\qquad\qquad  \mbox{ if } \;  N_T(u)= 0.\\
\end{array} \right. \label{8a}\eeq
Here it is supposed that  the function $u(\cdot)$ on  $[0,T]$ has exactly  $N_T(u)$ jumps
at the time points   \
$t_1,t_2,...,t_{N_T(u)}$ \ where  \ $0=t_0<t_1<...<t_{N_T(u)}\leq T$, \
$\tau_i=t_i-t_{i-1}$. Moreover,
$$
\nu (u(t_{i-1}),u (t_i))= \left\{ \begin{array}{ll}\lambda(u (t_{i-1})), &
\mbox{ if }\; u (t_i)-u (t_{i-1})=1;\\
\mu(u (t_{i-1})), & \mbox{ if }\;  u (t_i)-u (t_{i-1})= -1. \end{array} \right.
$$

\end{lemma}

Observe that the probability ${\mathbf P}(\xi(\cdot)\in X_T)$ in Lemma  2.2
is allowed to be less then 1. (Clearly, this probability is positive.) The same density ${\mathfrak p}_T$
was used in  \cite{MPY}.

Let us denote by $N_T(\zeta)$  the random number of jumps in process $\zeta(\cdot)$ on interval
$[0,T]$.

The assertion of Lemma \ref{l2.0} is equivalent to the fact that for any measurable set  $G\subseteq X_T$
\beq \label{3}
\mathbf{P}(\xi(\cdot)\in G)=e^T \mathbf{E}(e^{-A_T(\zeta)}e^{B_T(\zeta)+N_T(\zeta)\ln2};\zeta(\cdot)\in G).
\eeq
We set
\beq  \label{4}\begin{array}{l}
A_T(\zeta)=\int_0^T h(\zeta(t))dt\\
\quad =\left\{ \begin{array}{ll}\sum\limits_{i=1}^{N_T(\zeta)}h(\zeta(t_{i-1}))\tau_{i}+
h(\zeta(t_{N_T(\zeta)}))(T-t_{N_T(\zeta)}),&
\mbox{if }\; N_T(\zeta)\geq1,\\
h(0)T, & \mbox{ if }\;  N_T(\zeta)=0; \end{array} \right.\\
B_T(\zeta)=\left\{ \begin{array}{ll}\sum\limits_{i=1}^{N_T(\zeta)}\ln(\nu(\zeta(t_{i-1}),\zeta(t_i))), &
\mbox{if }\; N_T(\zeta)\geq1,\\
0, & \mbox{ if }\;  N_T(\zeta)=0. \end{array} \right.\end{array}\eeq
The expressions in (\ref{3}) specify, in our context, the statement of the Radon-Nikodym theorem
(see, e.g.,  \cite{D-Sch}, Theorem 2, sec. III, ch. 10). The expressions (\ref{3}) are used for
analysing the asymptotical behaviour of the logarithm of probability
$\mathbf{P}(\xi_T(\cdot)\in U_\varepsilon(f))$, $f\in F$.

Theorem \ref{th2.1} indicates that for $l\neq m$  the main contribution into  the
asymptotics is brought by
 $A_T(\zeta)$, whereas in the case $l=m$ the asymptotics involves both
 $A_T(\zeta)$ and $B_T(\zeta)$.

Consider the family of scaled processes
$$\zeta_T(t)=\frac{\zeta(tT)}{T}, \ \ t\in[0,1].$$
Let $k_+$ and  $k_-$ denote the number of positive and negative jumps in  $\zeta_T(\cdot)$  and
set $L=k_+-k_-$.

For $\zeta_T(\cdot)\in U_\varepsilon(f)$ we have the inequality
\beq
f(1)-\varepsilon \leq \zeta_T(1) \leq f(1)+\varepsilon.  \label{8}
\eeq
The jumps in $\zeta_T(\cdot)$  are  $\pm 1/T$, therefore
 (\ref{8}) yields the inequalities
\beq
(f(1)-\varepsilon)T \leq L \leq (f(1)+\varepsilon)T.  \label{9}
\eeq
With these definitions and observations we can write:
\beq
k_+ + k_- =N_T(\zeta), \ \ \  k_+=\frac{N_T(\zeta)+L}{2}, \ \ \ k_-=\frac{N_T(\zeta)-L}{2}. \label{10}
\eeq
For brevity, we  write  below $\xi_T, \zeta_T$ and $A_T, B_T$ instead of  $\xi_T(\cdot), \zeta_T(\cdot)$
and $A_T(\zeta), B_T(\zeta)$. Also set: $v=\max (l,m)$.

\smallskip

\begin{lemma}\label{l2.3} Let $f\in F$. In case $l\neq m$ we have
$$\lim\limits_{\varepsilon\rightarrow 0} \limsup\limits_{T\rightarrow\infty} \frac{1}{T^{v+1}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\leq0,$$
whereas in case $l=m$
$$\lim\limits_{\varepsilon\rightarrow 0} \limsup\limits_{T\rightarrow\infty} \frac{1}{T^{l+1}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\leq
2\sqrt{P_lQ_m}\int_0^1f^l(s)ds,$$

\end{lemma}

\begin{lemma} \label{l2.4} For $f\in F$, in case $l\neq m$
$$\lim\limits_{\varepsilon\rightarrow 0} \liminf\limits_{T\rightarrow\infty} \frac{1}{T^{v+1}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\geq0,$$
and in case $l=m$
$$\lim\limits_{\varepsilon\rightarrow 0} \liminf\limits_{T\rightarrow\infty} \frac{1}{T^{l+1}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\geq 2\sqrt{P_lQ_m}\int_0^1f^l(s)ds .$$
\end{lemma}

\section{Proofs of Theorem \ref{th2.1} and Lemmas \ref{l2.0}--\ref{l2.4}}

\smallskip
In what follows, $\blacktriangle$ marks the end of a proof.

{\sc Proof of Theorem \ref{th2.1}} We are going to get the LLDP for functions  $f\in F$. First
let us estimate the quantity $A_T$. Fix a value $\varepsilon >0$ until a further notice.

From equation \eqref{4} it follows that
$$
A_T:=\int_0^T h(\zeta(t))dt=T\int_0^1 h(T\zeta_T(s))ds.
$$
If $\zeta_T\in U_\varepsilon(f)$ then
\beq
(f(s)-\varepsilon)\leq \zeta_T(s)\leq (f(s)+\varepsilon). \label{11}
\eeq

Let $\delta\in(0,1)$ be also fixed for the time being and denote
$m_\delta:=\min\limits_{t\in[\delta,1]}f(t)$. Here $m_\delta>0$ for $f\in F$. Therefore,
$k_0=m_\delta-\varepsilon>0$ when $\varepsilon$ is sufficiently small.

Let us estimate  $A_T$ on the set of trajectories where  inequality  (\ref{11}) is valid.
From (\ref{11}) it follows that $T\zeta_T(s)\geq k_0 T$ for $s\in[\delta,1]$. Therefore,
by virtue of condition (\ref{1}), for any $\gamma_0\in (0,1)$ and  $s\in[\delta,1]$,
for $T$ large enough we have the inequalities
\beq\label{12}
1-\gamma_0\leq \frac{h(T\zeta_T(s))}{P_l(T\zeta_T(s))^l}
\leq 1+\gamma_0\ \ \mbox{ in case }\ l>m,\eeq
\beq\label{13}
1-\gamma_0\leq\frac{h(T\zeta_T(s))}{(P_l+Q_m)(T\zeta_T(s))^l}
\leq 1+\gamma_0 \ \   \mbox{in case }\   l=m,\eeq
and
\beq
1-\gamma_0\leq\frac{h(T\zeta_T(s))}{Q_m(T\zeta_T(s))^m}
\leq 1+\gamma_0\ \  \mbox{ in case }\  \ l<m. \label{14}\eeq

Consider the case $l>m$. Owing to (\ref{11}) and (\ref{12}), for $T$ sufficiently large,
we get
\beq\label{15}\begin{array}{l}
T\int_\delta^1(1-\gamma_0)P_l(T(f(s)-\varepsilon))^lds
\ \leq \ A_T\\
\qquad\qquad\leq T\int_0^\delta h(T\zeta_T(s))ds+T\int_\delta^1(1+\gamma_0)P_l(T(f(s)+\varepsilon))^l
ds.\end{array}\eeq

Set $M:=\max(\max\limits_{t\in[0,1]}f(t),1)$. By using (\ref{11}), for $T$ large enough we have that
$$h(T\zeta_T(s))\leq(1+\gamma_0)P_l(T(M+\varepsilon))^l.$$
Consequently, from (\ref{15}) we obtain the inequality
\beq\begin{array}{l}
T^{l+1}P_l\int_\delta^1(1-\gamma_0)(f(s)-\varepsilon)^lds
\ \leq \ A_T \\
\qquad\leq T^{l+1}P_l\delta(1+\gamma_0)(M+\varepsilon)^l+T^{l+1}P_l\int_\delta^1(1+\gamma_0)(f(s)+\varepsilon)^lds.\end{array} \label{16}\eeq
By using the bound (\ref{16}) and equation (\ref{3}), we get the following:
\beq\label{17}\begin{array}{l}
\exp\,[-T^{l+1}P_l\int_\delta^1(1-\gamma_0)(f(s)-\varepsilon)^lds]
e^T \mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\\
\qquad
\geq\mathbf{P}(\xi_T(\cdot)\in U_\varepsilon(f))=e^T \mathbf{E}(e^{-A_T}e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\\
\qquad \geq\exp\,[-T^{l+1}P_l\delta(1+\gamma_0)(M+\varepsilon)^l-T^{l+1}P_l\int_\delta^1(1+\gamma_0)(f(s)+\varepsilon)^lds]\\
\quad\qquad\times e^T \mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f)).\end{array}\eeq
Further, by virtue of (\ref{17})
\beq\begin{array}{l}
-P_l\int_\delta^1(1-\gamma_0)(f(s)-\varepsilon)^lds +
\limsup\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{l+1}}}\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\\
\qquad\geq\limsup\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{l+1}}}\ln\mathbf{P}(\xi_T\in U_\varepsilon(f))\geq\liminf\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{l+1}}}\ln\mathbf{P}(\xi_T\in U_\varepsilon(f))\\
\qquad\geq -P_l\delta(1+\gamma_0)(M+\varepsilon)^l-P_l\int_\delta^1(1+\gamma_0)(f(s)+\varepsilon)^lds\\
\qquad\quad+\liminf\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{l+1}}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f)).
\end{array} \label{18}
\eeq
Next, from (\ref{18}) it follows that
\beq
\begin{array}{l}
-P_l\int_\delta^1(1-\gamma_0)f^l(s)ds+
\lim\limits_{\varepsilon\rightarrow 0}\limsup\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{l+1}}}\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\\
\quad\geq\lim\limits_{\varepsilon\rightarrow 0}\limsup\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{l+1}}}\ln\mathbf{P}(\xi_T(\cdot)\in U_\varepsilon(f))\geq\lim\limits_{\varepsilon\rightarrow 0}\liminf\limits_{T\rightarrow\infty}
{\diy\frac{1}{T^{l+1}}}\ln\mathbf{P}(\xi_T(\cdot)\in U_\varepsilon(f))\\
\quad\geq-P_l\delta(1+\gamma_0)M^l-P_l\int_0^1(1+\gamma_0)f^l(s)ds\\
\qquad\qquad\qquad\qquad\qquad\qquad\quad
+\lim\limits_{\varepsilon\rightarrow 0}\liminf\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{l+1}}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f)).
\end{array} \label{19}
\eeq

Note that the inequatity  (\ref{19}) is valid for all $\gamma_0,\delta>0$. Letting
 $\gamma_0\rightarrow 0$ and $\delta\rightarrow 0$ we get that
\beq\begin{array}{l} -P_l\int_0^1f^l(s)ds+
\lim\limits_{\varepsilon\rightarrow 0}\limsup\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{l+1}}}\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\\
\quad\geq\lim\limits_{\varepsilon\rightarrow 0}\limsup\limits_{T\rightarrow\infty}\frac{1}{T^{l+1}}\ln\mathbf{P}(\xi_T(\cdot)\in U_\varepsilon(f))\geq\lim\limits_{\varepsilon\rightarrow 0}\liminf\limits_{T\rightarrow\infty}
{\diy\frac{1}{T^{l+1}}}\ln\mathbf{P}(\xi_T(\cdot)\in U_\varepsilon(f))\\
\quad\geq-P_l\int_0^1f^l(s)ds+\lim\limits_{\varepsilon\rightarrow 0}\liminf\limits_{T\rightarrow\infty}
{\diy\frac{1}{T^{l+1}}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f)).
\end{array} \label{20}
\eeq

In a similar way, by using (\ref{13}) and  (\ref{14}) we obtain inequalities
for the case $ l = m $:
\beq \begin{array}{l}
-(P_l+Q_m)\int_0^1f^l(s)ds+
\lim\limits_{\varepsilon\rightarrow 0}\limsup\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{l+1}}}\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\\
\quad\geq\lim\limits_{\varepsilon\rightarrow 0}\limsup\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{l+1}}}\ln\mathbf{P}(\xi_T\in U_\varepsilon(f))\geq\lim\limits_{\varepsilon\rightarrow 0}\liminf\limits_{T\rightarrow\infty}
{\diy\frac{1}{T^{l+1}}}\ln\mathbf{P}(\xi_T\in U_\varepsilon(f))\\
\quad\geq-(P_l+Q_m)\int_0^1f^l(s)ds+\lim\limits_{\varepsilon\rightarrow 0}\liminf\limits_{T\rightarrow\infty}
{\diy\frac{1}{T^{l+1}}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f)),\end{array} \label{21}\eeq
and for the case $l<m$:
\beq\begin{array}{l}
-Q_m\int_0^1f^m(s)ds+
\lim\limits_{\varepsilon\rightarrow 0}\limsup\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{m+1}}}\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\\
\quad\geq\lim\limits_{\varepsilon\rightarrow 0}\limsup\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{m+1}}}
\ln\mathbf{P}(\xi_T(\cdot)\in U_\varepsilon(f))\geq\lim\limits_{\varepsilon\rightarrow 0}\liminf\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{m+1}}}\ln\mathbf{P}(\xi_T(\cdot)\in U_\varepsilon(f))\\
\quad\geq-Q_m\int_0^1f^m(s)ds+\lim\limits_{\varepsilon\rightarrow 0}\liminf\limits_{T\rightarrow\infty}
{\diy\frac{1}{T^{m+1}}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f)).
\end{array} \label{22}\eeq

Observe that in the course if deducing the estimates  (\ref{20}), (\ref{21}) and (\ref{22})
the limit  $T\to \infty$ precedes the limit $\varepsilon \to 0$.

Applying Lemmas \ref{l2.3} and \ref{l2.4} to (\ref{20})--(\ref{22})
completes the proof of the LLDP for the functions  $\in F$. \quad $\blacktriangle$

\begin{remark} \label{r3.1} The above argument allows us to extend the assertion of
Theorem {\rm\ref{th2.1}} to the set of functions $f\in{\rm C}[0,1]$ with $f(0)=0$,
$f(t)\geq 0$ for $0<t\leq 1$ and $f(t)=0$ at finitely many points in $[0,1]$.
\end{remark}

\begin{remark} \label{r3.2} For the Yule process (a process of pure birth  with
$l>0$, $P_l>0$ and $\mu(x)\equiv 0$, see e.g.  \cite{Fell}), the rate functional has the form
$$I(f)=P_l\int_0^1f^l(t)dt, \ \ \ f\in F_M,$$
where $F_M$ is a set of not-decreasing continuous functions $f(t)$ on  $[0,1]$ with $f(0)=0$.
\end{remark}

\medskip

{\sc Proof of Lemma \ref{l2.0}.} Let $N_T(\xi)$ be the number of jumps in process $\xi$
in the time interval  $[0,T]$. In the course of the proof we work on  the event that the
trajectoriy of $\xi$ belongs to $X_T$, i.e., that  $N_T<\infty$. This event has a positive probability.

As was mentioned earlier, the statement of the lemma means that for any measurable set
$G\subseteq X_T$ the equality  \reff{3} is valid.
Denote by  $X_T^{(n)}$ a set of functions $u\in X_T$ with $N_T(u)=n,\ n=0,1,\dots$.
Consider one-to-one mapping
\beq\label{mf}
 u\in X_T^{(n)} \mapsto
 (t_1,\ldots,t_n ; \Delta_1,\ldots ,\Delta_n)\in {\mathfrak X}^{(n)}_T=[0,T]^n_<\times \{+1,-1\}^n,\ n=1,2\ldots .
 \eeq
Here $t_1,\ldots,t_n$ is a sequence of jump times for function $u$ in $[0,T]$,
$\Delta_i$ is a size of jump $u(t_i)-u(t_{i-1})$\ (with $\Delta_1=u(t_1)$). Next $[0,T]^{n}_<$
stands for an $n$-dimensional simplex  $\{(t_1,\ldots,t_n)\ : 0<t_1<\ldots t_n\leq T\}$.

The probabilities $\mathbf{P}(\xi(\cdot)\in G)$ and $\mathbf{P}(\zeta(\cdot)\in G)$ are determiined by

a) the respective densities $\mathfrak{f}_{\xi}$ and $\mathfrak{f}_{\zeta}$ relative to
the summation measure
$\sum\limits_{n\geq 1}\prod\limits_{j=1}^n{\rm d}t_j$
on  ${\mathfrak X}_T:=\bigcup_{n\geq 1}{\mathfrak X}^{(n)}_T$ (here $t_0=0$ as $j=1$), and

b) the probabilities
$\mathbf{P}(\xi(t)=0, 0\leq t\leq T)=e^{-\lambda(0)T}$ ,  $\mathbf{P}(\zeta(t)=0, 0\leq t\leq T)=e^{-T}$.

The densities  $\mathfrak{f}_{\xi}$  and $\mathfrak{f}_{\zeta}$  are of form
\beq\label{fxi}
  \mathfrak{f}_{\xi}(t_1,\ldots,t_n;\Delta_1,\ldots,\Delta_n)
 =\left(\prod_{i=1}^n\nu(x_{i-1},x_i)e^{-h(x_{i-1})\tau_i}\right)e^{-h(x_n)(T-t_n)},\eeq
and
\beq\label{fz}
 \mathfrak{f}_{\zeta}(t_1,\ldots,t_n;\Delta_1,\ldots,\Delta_n)=
2^{-n} \left(\prod_{i=1}^ne^{-\tau_i}\right)e^{-(T-t_n)},
\eeq
where $x_0=0,\ x_i=\sum_{j=1}^i\Delta_j$,
$i=1,\ldots,n$.

Each factor $\nu(x_{i-1},x_i)e^{-h(x_{i-1})\tau_i}$ in (\ref{fxi}) gives the probability density $h(x_{i-1})e^{-h(x_{i-1})\tau_i}$ for the  time the process
 $\xi$ spent at state  $x_{i-1}$ multiplied  by the probability $\nu(x_{i-1},x_i)/h(x_{i-1})$
of a jump from $x_{i-1}$ to $x_i$. The factor  $e^{-h(x_n)(T-t_n)}$
is the probability to stay at  $x_n$ until time $T$. A similar meaning is attributed to the factors
$\frac{1}{2}e^{-\tau_i}$ and $e^{-(T-tto stay a_n)}$. The products of terms in (\ref{fxi}) and (\ref{fz})
reflect the Markovian character of both processes.

The Radon--Nikodym derivative ${\mathfrak p}_T=
{\rm d}{\mathbf P}^{(\xi)}_T(\,\cdot\,\cap X_T)\big/{\rm d}{\mathbf P}^{(\zeta )}_T$ in \eqref{8a}
is a ratio $\mathfrak{f}_\xi /\mathfrak{f}_\zeta $
because the mapping $X^{(n)}_T\to{\mathfrak X}^{(n)}_T$ is one-to-one. The
Radon-Nikodym theorem can be applied here as both densities
$\mathfrak{f}_{\xi}$ and $\mathfrak{f}_{\zeta}$ are positive on  ${\mathfrak X}^{(n)}_T$
$\sum\limits_{n\geq 1}\prod\limits_{j=1}^n{\rm d}t_j$
and measure  ${\mathfrak X}_T$  is finite (for formulation and proof of Radon-Nicodim Theorem
see, e.g., \cite{D-Sch}, Chapter III, Section 10, Theorem 2, or \cite{Ru}, Theorem 6.10). \quad
$\blacktriangle$

\medskip

{\sc Proof of Lemma  \ref{l2.3}.} First, we upper-bound the expected value
$\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))$.

Given $a>1$ we write:
\beq\label{27}
\begin{array}{l}
\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f)):= E_{1}+E_{2},\\
 E_1:=\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f);N_T(\zeta)\leq T^a),\\
  E_2:=\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f);N_T(\zeta)> T^a).
\end{array}\eeq
Let us bound $E_1$ from above. If $\zeta_T\in U_\varepsilon(f)$ and $N_T(\zeta)\leq T^a$ then,
by virtue of (\ref{1}), it follows that for  any $\gamma_1>0$ and $T$ large enough,
$$\begin{array}{l}B_T=\sum\limits_{i=1}^{N_T(\zeta)}\ln(\nu(\zeta(t_{i-1}),\zeta(t_{i}))\\
\quad\quad\leq T^a(\ln(P_lT^l(M+\varepsilon)^l(1+\gamma_1))+\ln(Q_mT^m
(M+\varepsilon)^m(1+\gamma_1))).\end{array}$$
Here, as before, $M=\max\left[\max\limits_{t\in[0,1]}f(t),1\right]$.

Set $k_1=P_lQ_m(M+\varepsilon )^{l+m}(1+\gamma_1)^2$. Then the following inequality
is fulfilled:
\beq
E_1\leq \exp\{(T^a+1)\ln(k_1T^{l+m})\}.\label{E1}\eeq
\smallskip

Next, we establish an upper bound for  $E_2$.

Fix $\varepsilon$ and $\delta\in (0,1)$ until the completion of the argument.
Denote $M_\delta:=\max\limits_{s\in[0,\delta]}f(s)$. Given  $u\in X_T$, set
$$\widetilde{\nu}(u(t_{i-1}),u(t_i))=\begin{cases}
P_l(u(t_{i-1}))^l,&\hbox{if }\; u(t_i)-u(t_{i-1})=1, \ t_i\geq\delta T,\\
Q_m(u(t_{i-1}))^m,&\hbox{if }\; u(t_i)-u(t_{i-1})=-1, \ t_i\geq\delta T,\\
P_l(T(M_\delta+\varepsilon))^l,&\hbox{if }\;u(t_i)-u(t_{i-1})=1, \ t_i<\delta T,\\
Q_m(T(M_\delta+\varepsilon))^m,&\hbox{if }\ u(t_i)-u(t_{i-1})=-1, \ t_i<\delta T.
\end{cases}$$
As earlier, $t_i$ are the times of jumps in $u$.

If $\zeta_T\in U_\varepsilon(f)$ then, by (\ref{1}) and the form of function
$\widetilde{\nu}(u(t_{i-1}),u(t_i))$, for $T$ sufficiently large  and
$t_{i-1}<\delta T$  we have an inequality
\beq
\widetilde{\nu}(\zeta(t_{i-1}),\zeta(t_i))\geq\nu(\zeta(t_{i-1}),\zeta(t_i)). \label{art4}
\eeq
Next, if $\zeta_T\in U_\varepsilon(f)$ and $\varepsilon$ is sufficiently small then
for $s>\delta$ we have $\zeta_T(s)>\min\limits_{s\in[\delta,1]}f(s)-\varepsilon>0$.
Thus, for $t_{i-1}\geq\delta T$ the condition (\ref{1}) implies that for any $\gamma_2\in (0,1)$
and $T$ large enough
\beq
(1-\gamma_2)\leq\frac{\nu(\zeta(t_{i-1}),\zeta(t_i))}{\widetilde{\nu}(\zeta(t_{i-1}),\zeta(t_i))}\leq(1+\gamma_2).        \label{29}
\eeq
Owing to inequalities (\ref{art4}), (\ref{29}) for any  $\gamma_2\in (0,1)$ and $T$ sufficiently large
we have that
$$\begin{array}{l}
\prod\limits_{i=1}^{N_T(\zeta)}\nu(\zeta(t_{i-1}),\zeta(t_{i}))
\mathbf{1}(\zeta_T\in U_\varepsilon(f),N_T(\zeta)> T^a)\\
\qquad\leq (1+\gamma_2)^{N_T(\zeta)}\prod\limits_{i=1}^{N_T(\zeta)}\widetilde{\nu}(\zeta(t_{i-1}),\zeta(t_{i}))
\mathbf{1}(\zeta_T\in U_\varepsilon(f),N_T(\zeta)> T^a).\end{array}$$

Next, set
$$\widetilde{f}_\delta(s)=\begin{cases}
M_\delta,&\hbox{if }\; t\in[0,\delta),\\
f(s),&\text{if }\; t\in[\delta,1].\end{cases}$$

From the form of  $\widetilde{\nu}(\zeta(t_{i-1}),\zeta(t_i))$ it follows that for
$\zeta_T\in U_{\varepsilon}(f)$ one of inequalities holds true, depending upon the
sign of $\zeta (t_i)-\zeta (t_{i-1})$: either
\beq
\widetilde{\nu}(\zeta(t_{i-1}),\zeta(t_i))\leq P_l(T(\widetilde{f}_\delta(t_{i-1}/T)+\varepsilon))^l,
\label{art1}\eeq
or
\beq
\widetilde{\nu}(\zeta(t_{i-1}),\zeta(t_i))\leq Q_m(T(\widetilde{f}_\delta(t_{i-1}/T)+\varepsilon))^m.
\label{art2}\eeq

If  $\zeta_T\in U_\varepsilon(f)$ then, by virtue of (\ref{10}), process
$\zeta_T$ has \ $\diy\frac{N_T(\zeta)+L}{2}$ \ positive jumps and \
$\diy\frac{N_T(\zeta)-L}{2}$ \ negative jumps. Hence, from (\ref{art1}), (\ref{art2}) we obtain
that
\beq\begin{array}{l}\prod\limits_{i=1}^{N_T(\zeta)}\nu(\zeta(t_{i-1}),\zeta(t_{i}))
\mathbf{1}(\zeta_T\in U_\varepsilon(f),N_T(\zeta)> T^a) \\
\quad \leq\begin{cases}
                              (1+\gamma_2)^{N_T(\zeta)}T^{vL/2}P_l^{\frac{N_T(\zeta)+L}{2}}
Q_m^{\frac{N_T(\zeta)-L}{2}}\prod\limits_{i=1}^{N_T(\zeta)}T^{\frac{l+m}{2}}(M+\varepsilon)^v,
&\hbox{if }\ l\neq m,\\
(1+\gamma_2)^{N_T(\zeta)}P_l^{\frac{N_T(\zeta)+L}{2}}
Q_m^{\frac{N_T(\zeta)-L}{2}}\prod\limits_{i=1}^{N_T(\zeta)}
T^l(\widetilde{f}_\delta(t_{i-1}/T)+\varepsilon)^l,&\text{if }\; l=m.\end{cases}\end{array}
\label{new1}\eeq

Set
$$k_2(T):= \min \left(1,(P_l/Q_m)^{(f(1)-\varepsilon)T/2}\right),\
k_3(T):= \max \left(1,(P_l/Q_m)^{(f(1)+\varepsilon)T/2}\right).$$
Then from (\ref{9}) it follows that
\beq
k_2(T)\leq\biggl(\frac{P_l}{Q_m}\biggl)^{L/2}\leq  k_3(T). \label{31}\eeq

In addition, set $k_4=
\left(\frac{\widetilde{f}_\delta(0)+\varepsilon}{\widetilde{f}_\delta(t_{N_T(\zeta)}/T)+\varepsilon}\right)^l$.
Owing to inequalities \reff{new1}, (\ref{31}),
for $T$ sufficiently large
$$E_2\leq \begin{cases}
k_3(T)T^{\frac{v(M+\varepsilon)}{2}}\mathbf{E}\prod\limits_{i=1}^{N_T(\zeta)}
                              2P_l^{\frac{1}{2}}
Q_m^{\frac{1}{2}}(1+\gamma_2)T^{(l+m)/2}(M+\varepsilon)^v,&\hbox{if }\; l\neq m,\\
k_3(T)k_4\mathbf{E}\prod\limits_{i=1}^{N_T(\zeta)}2P_l^{\frac{1}{2}}
Q_m^{\frac{1}{2}}(1+\gamma_2)
T^l(\widetilde{f}_\delta(t_i/T)+\varepsilon)^l,&\text{if }\; l=m.
\end{cases}$$

Following Remark  \ref{r4.2} from Appendix, we get an exponential bound for  $E_2$:
$$E_2\leq
\begin{cases} k_3(T)e^{-T}T^{\frac{v(M+\varepsilon)}{2}}\exp\left\{2P_l^{\frac{1}{2}}
Q_m^{\frac{1}{2}}(1+\gamma_2)T^{(l+m)/2+1}(M+\varepsilon)^v\right\},&\hbox{if }\; l\neq m,\\
k_3(T)k_4e^{-T}\exp\left\{2P_l^{\frac{1}{2}}
Q_m^{\frac{1}{2}}(1+\gamma_2)T^{l+1}\int_0^1(\widetilde{f}_\delta(s)+\varepsilon)^lds\right\},
&\hbox{if }\ l=m. \end{cases}$$

Then, for $T$ sufficiently large, selecting  $a<\frac{l+m}{2}+1$ we obtain from the bound
(\ref{E1}) that
$$\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))=
E_1+E_2\leq 2E_2.$$
Finally, by taking into account that the value $\ln \Big(k_3(T)T^{\frac{v(M+\varepsilon )}{2}}\Big)$
is of order $T\ln T$, while $\ln\big(k_3(T)k_4\big)$ is of order $T$, we conclude:
for any  $\gamma_2\in (0,1)$ and $\delta\in(0,1)$, the following bounds hold true:
$$\begin{array}{l}
\lim\limits_{\varepsilon\rightarrow 0} \limsup\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{v+1}}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\\
\qquad\qquad\qquad \leq \begin{cases}0,&\hbox{if }\; l\neq m,\\
2\sqrt{P_lQ_m}(1+\gamma_2)\int_0^1\widetilde{f}_\delta^l(s)ds,&\hbox{if }\ l=m.
\end{cases}\end{array}$$
Taking the limit as $\gamma_2 ,\delta\rightarrow 0$ completes the proof. \quad $\blacktriangle$

\medskip

{\sc Proof of Lemma \ref{l2.4}.} Let us now lower-bound the value  ${E}_2$ from (\ref{27}).
As before, we fix a sufficiently small $\varepsilon$ until the end of the argument. Everywhere
below, $[\,\cdot\,]$ stands for the integer part.

Introduce the event $D:=\left\{\max\limits_{1\leq k\leq N_T(\zeta)+1}\tau_k\leq T^{1-\beta}\right\}$,
where $1<\beta<a$ and $\tau_{N_T(\zeta)+1}:=T-t_{N_T(\zeta)}$. Also consider the event
$C_{\varepsilon}:=\left\{\inf\limits_{t\in [t_{[\varepsilon T/4]},T]}\zeta(t)>\varepsilon/16\right\}$,
where $t_{[\varepsilon T/4]}$ is the time of the  $[\varepsilon T/4]$-th jump in
 $\zeta$.

Obviously
$$\begin{array}{l}
{E}_2 \;=\; 2^{N_T(\zeta)}\mathbf{E}\prod\limits_{i=1}^{N_T(\zeta)}\nu(\zeta(t_{i-1}),\zeta(t_{i}))
\mathbf{1}(\zeta_T\in U_\varepsilon(f),N_T(\zeta)> T^a)\\
\qquad \geq \mathbf{E}2^{N_T(\zeta)}\prod\limits_{i=1}^{N_T(\zeta)}\nu(\zeta(t_{i-1}),\zeta(t_{i}))
\mathbf{1}(D,C_{\varepsilon},\zeta_T\in U_\varepsilon^+(f),N_T(\zeta)> T^a),\end{array}$$
where $U_\varepsilon^+(f):=\left\{g: \min\limits_{t\in[0,1]}g(t)\geq 0\right\}\cap U_\varepsilon(f)$.

Let $\delta=\min\{s:\min\limits_{t\in[s,1]}f(t)\geq 2\varepsilon\}$ and
denote $r(\delta):=\min\{i:t_{i}\geq T\delta\}$.

Suppose that $\zeta_T\in U_\varepsilon^+(f)$ and $r(\delta)+1\leq i\leq N_T(\zeta)$.
By condition (\ref{1}),
depending upon the sign of $\zeta(t_i)-\zeta(t_{i-1})$, we  have ether
\beq
\nu(\zeta(t_{i-1}),\zeta(t_i))\geq(1-\gamma_3)P_l(T(f(t_{i-1}/T)-\varepsilon))^l \label{art5}
\eeq
or
\beq
\nu(\zeta(t_{i-1}),\zeta(t_i))\geq(1-\gamma_3)Q_m(T(f(t_{i-1}/T)-\varepsilon))^m. \label{art6}
\eeq

If the event $C_\varepsilon$ has occurred, and $[\varepsilon T/4]\leq i\leq r(\delta)$, then,
owing to condition (\ref{1}), for any $\gamma_3\in (0,1)$  and a sufficiently large $T$
the following inequality holds true:
\beq
\nu(\zeta(t_{i-1}),\zeta(t_i))\geq(1-\gamma_3)(T\varepsilon/16)^w, \label{art7}
\eeq
where $w:=\min(l,m)$.

For $\zeta_T\in U_\varepsilon^+(f)$ and $1 \leq i\leq [\varepsilon T/4]$ we have
\beq
\nu(\zeta(t_{i-1}),\zeta(t_i))\geq k_5:=\min\left[\inf\limits_{x\in \mathbb{Z}_+}\lambda(x),\inf\limits_{x\in \mathbb{N}}\mu(x)\right]. \label{art8}
\eeq

Let us introduce the function
$$\widehat{f}_\varepsilon(s)=\begin{cases}
\diy\frac{\varepsilon}{16\max(1,P_l,Q_m)},&\hbox{if }\ s\in[0,\delta),\\
f(s)-\varepsilon,&\hbox{if }\ s\in[\delta,1]. \end{cases}$$

Using (\ref{10}), (\ref{art5}), (\ref{art6}), (\ref{art7}), (\ref{art8}), we get the bound
$$\begin{array}{l}
E_2\geq k_6(T)\mathbf{E}\Bigg[ P_l^{\frac{N_T(\zeta)+L}{2}}
Q_m^{\frac{N_T(\zeta)-L}{2}}(1-\gamma_3)^{N_T(\zeta)}2^{N_T(\zeta)}\\
\qquad \times\prod\limits_{i=[\varepsilon T/4]+1}^{N_T(\zeta)}(T\widehat{f}(t_{i-1}/T))^w
\mathbf{1}(D,C_\varepsilon,\zeta_T\in U_\varepsilon^+(f),N_T(\zeta)> T^a)\Bigg],
\end{array}$$
where $k_6(T):=\left(\diy\frac{k_5}{\max(P_l,Q_m)}\right)^{[\varepsilon T/4]}$.

From inequalities (\ref{9}), (\ref{31}) we obtain
$$\begin{array}{l}
E_2\geq k_7(T)\mathbf{E}\bigg[P_l^{\frac{N_T(\zeta)}{2}}
Q_m^{\frac{N_T(\zeta)}{2}}(1-\gamma_3)^{N_T(\zeta)}2^{N_T(\zeta)}\\
\qquad \times\prod\limits_{i=1}^{N_T(\zeta)}(T\widehat{f}(t_{i}/T))^w
\mathbf{1}(D,C_\varepsilon,\zeta_T\in U_\varepsilon^+(f),N_T(\zeta)> T^a)\bigg],\end{array}$$
where  $k_7(T):=\diy\frac{k_6(T)k_2(T)}{M^wT^{w[\varepsilon T/4]}}$.

From Lemma \ref{l4.4} of Appendix it follows that for any
$\gamma_4\in (0,1)$ and $T$ sufficiently large  the following holds true
\beq
{E}_2 \geq k_7(T)\sum\limits_{n=[T^a]+1}^{\infty}2^n(1-\gamma_4)^{n}P_l^{\frac{n}{2}}
Q_m^{\frac{n}{2}}\mathbf{E}
\prod\limits_{i=1}^{n}
T^w(\widehat{f}(t_i/T))^w
\mathbf{1}(D,N_T(\zeta)=n).  \label{35}
\eeq
Here  $\gamma_4$ is expressed via $\gamma_3$ and $\theta$ whereas $\theta\in (0,1)$
is introduced in Lemmas  \ref{l4.3}, \ref{l4.4} from Appendix.

To estimate the product from (\ref{35}), we use Lemma \ref{l4.1}.  Taking into account that
$n>T^a$, we get that  for $T$ large enough,
\beq\begin{array}{l}\mathbf{E}\left[\prod\limits_{i=1}^{n}T^w(\widehat{f} (t_i/T))^w
\mathbf{1}(D,N_T(\zeta)=n)\right]
=\mathbf{E}\left[\prod\limits_{i=1}^{n}T^w(\widehat{f}(t_i/T))^w\mathbf{1}(N_T(\zeta)=n)\right]\\
\qquad\qquad\qquad
-\mathbf{E}\left[\prod\limits_{i=1}^{n}T^w(\widehat{f}(t_i/T))^w\mathbf{1}(\overline{D},N_T(\zeta)=n)\right]\\
\qquad \diy\geq
\frac{\left(T^w\int_0^T(\widehat{f}(t_i/T))^w dt\right)^n}{n!}e^{-T}
-2T^\beta\frac{\left(T^w\int_0^T(\widehat{f}(t_i/T))^w dt-T^{w+1}\alpha_{1/T^\beta}\right)^n}{n!}e^{-T}. \end{array}\label{32}\eeq
Here $\alpha_{1/T^\beta}=\diy\frac{1}{2T^\beta}\inf\limits_{s\in[0,1]}\;\left(\widehat{f}(s)\right)^w=
\frac{1}{2T^\beta}\left(\frac{\varepsilon}{16\max(1,P_l,Q_m)}\right)^w$ (compare to equation
(\ref{47}) in Appendix) .

Let us now estimate the last summand in the right side of (\ref{32}).
Denote $k_8:=\sup\limits_{s\in[0,1]}(\widehat{f}(s))^w$. As $a>\beta$, for a sufficiently large
$T$ the following inequalities hold true:
$$\begin{array}{l}
2T^\beta\left(T^w\int_0^T(\widehat{f}({t}/{T}))^w dt-T^{w+1}\alpha_{1/T^\beta}\right)^n\\
\qquad\leq
2T^\beta\left(T^w\int_0^T(\widehat{f}({t}/{T}))^w dt\right)^n
\left(1-\diy\frac{\varepsilon^w}{2k_8T^\beta (16\max(1,P_l,Q_m))^w}\right)^n\\
\qquad \leq 2T^\beta\left(T^w\int_0^T(\widehat{f}({t}/{T}))^w dt\right)^n
\left(1-\diy\frac{\varepsilon^w}{2k_8T^\beta (16\max(1,P_l,Q_m))^w}\right)^{T^a}\\
\qquad  \leq\left(T^w\int_0^T(\widehat{f}({t}/{T}))^w dt\right)^n
\exp\left(\beta\ln(2T)-\diy\frac{\varepsilon^w}{2k_8(16\max(1,P_l,Q_m))^w}T^{a-\beta}\right)\\
\qquad\leq{\diy\frac{1}{2}}\left(T^w\int_0^T\widehat{f}^w({t}/{T}) dt\right)^n.\end{array}$$
Consequently, from (\ref{32}) it follows that
$$\mathbf{E}\left[\prod\limits_{i=1}^{n}T^w(\widehat{f} (t_i/T))^w
\mathbf{1}(D,N_T(\zeta)=n)\right]\geq\frac{1}{2}
\frac{\left(T^w\int_0^T(\widehat{f}({t}/{T}))^w dt\right)^n}{n!}e^{-T}.$$
By virtue of (\ref{35}), for $T$ sufficiently large,
$$
{E}_2 \geq \frac{k_7(T)}{2}\sum\limits_{n=[T^a]+1}^{\infty}2^n(1-\gamma_4)^{n}\sqrt{P_lQ_m}
\frac{\left(T^w\int_0^T(\widehat{f}({t}/{T}))^w dt\right)^n}{n!}e^{-T}.
$$
From this it follows that, selecting $a<w+1$, for $T$ large enough we obtain the inequalities
\beq \label{33} \begin{array}{l}
E_2 \geq{\diy\frac{k_7(T)e^{-T}}{2}}\exp\left(2(1-\gamma_4)\sqrt{P_l Q_m}
T^{w+1}\int_0^1\widehat{f}(s)^w ds\right)\\
\qquad\qquad
-{\diy\frac{k_7(T)e^{-T}}{2}}\exp\left(a\ln(T)+(w+2)T^a\ln(T)\right)\\
\geq{\diy\frac{k_7(T)e^{-T}}{4}}\exp\left(2(1-\gamma_4)\sqrt{P_l Q_m}
T^{w+1}\int_0^1\widehat{f}(s)^w ds\right).\end{array}\eeq

By virtue of (\ref{33}) and the fact that  $\ln k_7$ is a quantity of order $T\ln T$,
we now obtain that
$$\begin{array}{l} \liminf\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{v+1}}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\\
\qquad\qquad\qquad \geq \begin{cases} 0,&\text{if }\; l\neq m,\\
2(1-\gamma_4)\sqrt{P_l Q_m}\int_0^1\widehat{f}(s)^l ds,&\hbox{if }\ l=m.\end{cases}
\end{array}$$
Furthermore, taking into account the definition of function $\widehat{f}(s)$, we obtain that
$$\begin{array}{l}
\lim\limits_{\varepsilon\rightarrow 0}\liminf\limits_{T\rightarrow\infty}{\diy\frac{1}{T^{v+1}}}
\ln\mathbf{E}(e^{B_T+N_T(\zeta)\ln2};\zeta_T\in U_\varepsilon(f))\\
\qquad\qquad\qquad \geq\begin{cases} 0,&\text{if }\; l\neq m,\\
2(1-\gamma_4)\sqrt{P_lQ_m}\int_0^1f^l(s)ds,&\text{if }\; l=m.\end{cases}
\end{array}$$

Taking the limit as $\delta \to 0$ and $\gamma_4\rightarrow 0$  completes the proof of the
lemma. \quad $\blacktriangle$

\section  {Appendix}

\ \ \ \ \ \ In this section we prove the auxiliary assertions used in earlier arguments.

Let $X_T^{(n)}$ stand for the event that process $\zeta$ has exactly $n$ jumps on the
interval $[0,T]$.

\begin{lemma} \label{l4.1} Let  $g(t)$ be a non-negative bounded Borel function and $n\geq 1$.
Then
\beq
\mathbf{E}\left[\prod\limits_{i=1}^n g(t_i)\mathbf{1}(X_T^{(n)})\right]=
\frac{\left(\int_0^Tg(s)ds\right)^n}{n!}e^{-T}, \label{46}
\eeq
and
\beq
\mathbf{E}\left[\prod\limits_{i=1}^ng(t_i)\mathbf{1}(X_T^{(n)})
\mathbf{1}\big(\max\limits_{1\leq k\leq n+1}\tau_k>T\Delta\big)\right]\leq
\frac{2}{\Delta}\frac{\left(\int_0^Tg(s)ds-T\alpha_\Delta\right)^n}{n!}e^{-T}. \label{47}
\eeq
Here  $\Delta>0$ is a constant and $\alpha_\Delta:=\diy\frac{\Delta}{2} \inf\limits_{t\in[0,T]}\;g(t)$.
Further, $t_1,\dots,t_n$ are jump times on $[0,T]$ in process $\zeta$ and $\tau_{n+1}:=T-t_n$.
\end{lemma}

{\sc Proof.} First, we prove \eqref{46}. To this end, write:
$$\mathbf{E}\left(\prod\limits_{i=1}^ng(t_i)\biggl|X_T^{(n)}\right)=
\mathbf{E}\left(\prod\limits_{i=1}^ng(t_i)\biggl|\eta(T)=n\right),$$
where $\eta$ is a Poisson process with mean $\mathbf{E}\eta(t)=t$.

From \cite{KT}, Theorem 2.3, p. 126, it follows that
$$\begin{array}{l}
\mathbf{E}\left(\prod\limits_{i=1}^ng(t_i)\biggl|\eta(T)=n\right)\\
\quad
={\diy\frac{n!}{T^n}}\int_0^T\left(\int_{s_1}^T...\left(\int_{s_{n-1}}^T\prod_{i=1}^n g(s_i)ds_n\right)...ds_2\right)ds_1=
{\diy\frac{1}{T^n}}\left(\int_0^Tg(s)ds\right)^n.\end{array}$$
Therefore,
$$\mathbf{E}\left[\prod\limits_{i=1}^ng(t_i)\mathbf{1}(X_T^{(n)})\right]=
\frac{1}{T^n}\left(\int_0^Tg(s)ds\right)^n\mathbf{P}(\eta(T)=n)=\frac{1}{n!}\left(\int_0^Tg(s)ds\right)^n e^{-T}.
$$

Next, we turn to the proof of \eqref{47}. Here
$$\begin{array}{l}
\mathbf{E}\left[\prod\limits_{i=1}^ng(t_i)\mathbf{1}(X_T^{(n)})
\mathbf{1}\big(\max\limits_{1\leq k\leq n+1}\tau_k>T\Delta\big)\right]\\
\quad \leq\sum\limits_{r=1}^{[2/\Delta]}\mathbf{E}\left[\prod\limits_{i=1}^ng(t_i)\mathbf{1}(\eta(T)=n)
\mathbf{1}\biggl(\eta\biggl(\frac{rT\Delta}{2}\biggl)-\eta\biggl(\frac{(r-1)T\Delta}{2}\biggl)=0\biggl)\right]:=
\sum\limits_{r=1}^{[2/\Delta]}D_r.\end{array} $$
Using the fact that  $\max\limits_{1\leq k\leq n+1}\tau_k>T\Delta$, we get that
there exists an $r$ with $1\leq r \leq [\frac{2}{\Delta}]$ and with no jumps on interval
$\left[\frac{(r-1)T\Delta}{2},\frac{rT\Delta}{2}\right]$.

Write
$$\begin{array}{l}
D_1=\mathbf{E}\left[\prod\limits_{i=1}^ng(t_i)\mathbf{1}(\eta(T)=n)
\mathbf{1}\biggl(\eta\biggl(\frac{T\Delta}{2}\biggl)=0\biggl)\right]\\
\qquad
=\mathbf{E}\left[\prod\limits_{i=1}^ng(t_i)\mathbf{1}\biggl(\eta(T)-\eta\biggl(\frac{T\Delta}{2}\biggl)=n\biggl)
\mathbf{1}\biggl(\eta\biggl(\frac{T\Delta}{2}\biggl)=0\biggl)\right].\end{array}$$
By using the independence of increments in, and the homogeneity of, the Poisson process and
formula (\ref{46}) we obtain
$$D_1=\frac{\left(\int_{\frac{T\Delta}{2}}^Tg(s)ds\right)^n}{n!}e^{-T(1-\Delta/2)}
\mathbf{P}\biggl(\eta\biggl(\frac{T\Delta}{2}\biggl)=0\biggl)=
\frac{\left(\int_{\frac{T\Delta}{2}}^Tg(s)ds\right)^n}{n!}e^{-T}.
$$
Similarly  for any  $1\leq r \leq [\frac{2}{\Delta}]$ one obtains that
$$
D_r=\frac{\biggl(\int\limits_{[0,T]\setminus B_{r,\Delta}}g(s)ds\biggl)^n}{n!}e^{-T},
$$
where $B_{r,\Delta}=\left[\diy\frac{(r-1)T\Delta}{2},\frac{rT\Delta}{2}\right]$.

In view of the relations
$$\min\limits_{1\leq r \leq \left[\frac{2}{\Delta}\right]}\int\limits_{\left[\frac{(r-1)T\Delta}{2},
\frac{rT\Delta}{2}\right]}g(s)ds
\geq T\frac{\Delta}{2}\inf\limits_{s\in[0,T]}g(s)=T\alpha_\Delta ,$$
we get that
$$
\mathbf{E}\left[\prod\limits_{i=1}^ng(t_i)\mathbf{1}(X_T^{(n)})
\mathbf{1}\big(\max\limits_{1\leq k\leq n+1}\tau_k>T\Delta\big)\right]\leq
\frac{2}{\Delta}\frac{\left(\int_0^Tg(s)ds-T\alpha_\Delta\right)^n}{n!}e^{-T}.\blacktriangle
$$

\begin{remark} \label{r4.2} Lemma {\rm\ref{l4.1}} implies that
$$
\mathbf{E}\prod\limits_{i=1}^{\eta(T)}g(t_i)\mathbf{1}(\eta(T)\geq 1)=
e^{-T}\left(\exp\left\{\int_0^Tg(s)ds\right\}-1\right).
$$
\end{remark}

\begin{lemma} \label{l4.3} Consider a sequence  $b_1, b_2, ..., b_n$, where each $b_i$ equals
$-1$ or $1$. Define by  $c_d$ the number of sequences with  following property:
$$
\left|\sum\limits_{k=1}^r b_k\right|\leq d, \ \forall \ 1\leq r \leq n.
$$
Take $d= [T\Delta]$ and $n= O(T^\beta)$ where $T\rightarrow\infty$ while $\Delta>0$, $\beta>1$
are fixed. Then for any $\theta\in (0,1)$ and  sufficiently large $T$ we hjave the bound
$$c_d\geq (1-\theta)^{n+1} 2^n.$$
\end{lemma}

{\sc Proof.} It is clear that if a sequence $b_{2(p-1)d+1},...,b_{2pd}$, with $1 \leq p \leq\diy\frac{n}{2d}$,
has an equal number of  $1$ and $-1$, and in the sequence $b_{2d[\frac{n}{2d}]+1},...,b_n$ the
difference  between the numbers of  $1$ and  $-1$ in the absolute value is at most 1 then required
property is fulfilled.
The number of  such sequences is not less then $\left(C_{2d}^d\right)^{[\frac{n}{2d}]}$.

Using Stirling's formula gives that
$$
\left(C_{2d}^d\right)^{[\frac{n}{2d}]}\sim \left(\frac{\sqrt{2}(2d)^{2d}}{\sqrt{\pi d} d^{2d}}\right)^{[\frac{n}{2d}]}=
\left(\frac{\sqrt{2} 2^{2d}}{\sqrt{\pi d}}\right)^{[\frac{n}{2d}]}\geq 2^{n-2d}(\pi d)^{-\frac{n}{4d}}.
$$
Thus, owing to the fact that $-2d\ln\,2-\diy\frac{nrd}{4d}=o(n)$, we obtain that,
for any $\theta\in (0,1)$ and $T$ sufficiently large,
$$c_d\geq (1-\theta) 2^n \exp\left(-2d \ln2 - \frac{n\ln\pi d}{4d}\right)
\geq (1-\theta)^{n+1} 2^n.\quad \blacktriangle $$

\begin{lemma} \label{l4.4} Take $\beta>1$ and $n\geq T^\beta$ and let $g(\cdot)$ be a non-negative
bounded Borel function. For any  $\theta>0$ and all $T$ sufficiently large the following estimate holds
true
$$\begin{array}{l}
\mathbf{E}\left[g(t_1,...,t_n)\mathbf{1}\left(\max\limits_{1\leq k\leq n+1}\tau_k\leq T^{1-\beta}\right)
\mathbf{1}(X_T^{(n)})\mathbf{1}(\zeta_T\in U_\varepsilon^+(f))\mathbf{1}(C_\varepsilon)\right]\\
\qquad\qquad\qquad \geq(1-\theta)^{2n}\mathbf{E}\left[g(t_1,...,t_n)
\mathbf{1}\left(\max\limits_{1\leq k\leq n+1}\tau_k\leq T^{1-\beta}\right)
\mathbf{1}(X_T^{(n)})\right].\end{array}$$
Here $U_\varepsilon^+(f):=\left\{g: \min\limits_{t\in[0,1]}g(t)\geq 0\right\}\cap U_\varepsilon(f)$,
$C_\varepsilon:=\left\{\inf\limits_{t\in [t_{[\varepsilon T/4]},T]}\zeta(t)>\varepsilon/16\right\}$
and $t_{[\varepsilon T/4]}$ is the point of the $[\varepsilon T/4]$-th jump
in process  $\zeta$.
\end{lemma}

{\sc Proof} As  $f$  is  uniformly continuous on
$[0,1]$, then for $\delta>0$  sufficiently small we have the inequality
$$\sup\limits_{s,t:|s-t|\leq\delta}|f(s)-f(t)|<\frac{\varepsilon}{4}.$$

Fix $\delta$ with $1/\delta\in{\mathbb N}$ and let  $1 \leq r \leq 1/\delta$.

Denote by $B_{m_r,\delta r}$ the event where process $\zeta$ has exactly $m_r$ jumps on
the interval \\
$[T\delta (r-1),T\delta r]$.

Then we can write
$$\begin{array}{l}
\mathbf{E}\left[g(t_1,...,t_n)\mathbf{1}\left(\max\limits_{1\leq k\leq n+1}\tau_k\leq T^{1-\beta}\right)
\mathbf{1}(X_T^{(n)})\mathbf{1}(\zeta_T\in U_\varepsilon^+(f))\mathbf{1}(C_\varepsilon)\right]\\
\;\; =\sum\limits_{m_1,...,m_{1/\delta}}\mathbf{E}\left[g(t_1,...,t_n)
\mathbf{1}\left(\max\limits_{1\leq k\leq n+1}\tau_k\leq T^{1-\beta}\right)
\mathbf{1}(X_T^{(n)})\mathbf{1}(\zeta_T\in U_\varepsilon^+(f))\mathbf{1}(C_\varepsilon)
\prod\limits_{r=1}^{1/\delta}\mathbf{1}(B_{m_r,\delta r})\right].\end{array}$$
The summation here is over all collections with
$\min\limits_{r}m_r\geq \delta T^\beta$, $\sum\limits_r m_r=n$.

Take a collection $m_1,\dots,m_r$ satisfying the above condition.

Consider a piece of a trajectory of $\zeta$ on the interval $[0,\delta T]$.
Denote by $t_{1,1},...,t_{m_1,1}$ the jump points of $\zeta$ lying in this interval.
Suppose that the jumps satisfy the following conditions:

1) The jumps at times $t_{1,1},...,t_{d_{\varepsilon ,1},1}$ are positive where
$d_{\varepsilon ,1}:=[T\varepsilon /4]$.

2) The jumps at times $t_{d_{\varepsilon ,1}+1,1},... ,t_{m_1,1}$ are such that
for any integer $k\in\big[d_{\varepsilon ,1}+1,m_1]$ we have the inequality
$$\left|\sum\limits_{l=d_{\varepsilon ,1}+1}^k \zeta(t_{l,1})\right|\leq
\left[\frac{T\varepsilon}{8}\right].$$

Then, for $T$ large enough, the trajectory $\zeta_T(t)$ has the following properties.

1) The trajectory is non-negative and lies in an $\varepsilon$-neighborhood of function $f$
for $t\in [0,\delta ]$.

2) $\zeta_T(t)\geq\varepsilon/16$ for $t>t_{d_{\varepsilon,1},1}/T$.

3) $|\zeta_T(\delta)-f(\delta)|\leq 3\varepsilon/8$.

Now consider a piece of a trajectory of $\zeta$ defined on the interval $[\delta T,2\delta T]$
and having the property $|\zeta(\delta T)-Tf(\delta )|\leq 3\varepsilon T/8$.
Denote by $t_{1,2},...,t_{m_2,2}$ the jump points of $\zeta$ in this interval.

Let these jumps satisfy the conditions:

1) At times $t_{1,2},...,t_{|d_{\varepsilon,2}|,2}$ the jumps are positive or negative in accordance
with the sign of the value
$d_{\varepsilon,2}:=[T(\max(\varepsilon/4,f(2\delta))-\zeta_T(\delta))]$.

2) At times $t_{|d_{\varepsilon,2}|+1,2},...,t_{m_2,2}$ are such that for any integer
$k\in [|d_{\varepsilon,2}|+1,m_2]$ the following inequality holds true:
$$
\biggl|\sum\limits_{l=|d_{\varepsilon,2}|+1}^k \zeta(t_{l,2})\biggl|\leq \bigg[\frac{T\varepsilon}{8}\bigg].
$$
Then, again for $T$ large enough, the trajectory $\zeta_T(t)$ has the following properties.

1) The trajectory is non-negative and lies in a $\varepsilon$-neighborhood of  $f$ for
$t\in [\delta,2\delta]$.

2) $\zeta_T(t)\geq\varepsilon/16$ as $t\in [\delta,2\delta]$.

3) $|\zeta_T(2\delta)-f(2\delta)|\leq 3\varepsilon/8$.

Further pieces of the trajectory are dealt with by induction.

Let us count the trajectories whose jumps satisfy the above properties.

As $\max\limits_{1\leq k \leq n+1}\tau_k\leq T^{1-\beta}$, we have that for any $r$
with $1 \leq r \leq 1/\delta$ the interval $[T\delta (r-1),T\delta r]$ contains at least
$[\delta T^\beta]$ jumps of process $\zeta$, where $\beta>1$. Using Lemma  \ref{l4.3} yields
that, when  $T$ is sufficiently large, we will have on $[T\delta (r-1),T\delta r]$
a number of pieces of the trajectory with the aforementioned properties which is not
less than
$$(1-\theta)^{m_r+1-|d_{\varepsilon,r}|}
2^{m_r-|d_{\varepsilon,r}|}>(1-\theta)^{2m_r}2^{m_r}.$$

Consequently, the number of trajectories that fulfill the above properties for all $r$
is not less then
\beq
\prod\limits_r(1-\theta)^{2m_r}2^{m_r}=(1-\theta)^{2n}2^n. \label{a.1}
\eeq

Next, the jump directions in $\zeta$ are mutually independent and do not depend either on
the number of jumps within the interval or on the jump times. Hence, we can use equality
\eqref{a.1} and get that
$$\begin{array}{l}
\sum\limits_{m_1,...,m_{1/\delta}}\mathbf{E}\left[g(t_1,...,t_n)\mathbf{1}(\max\limits_{1\leq k\leq n+1}\tau_k\leq T^{1-\beta})\mathbf{1}(X_T^{(n)})\mathbf{1}(\zeta_T\in U_\varepsilon^+(f))\mathbf{1}(C_\varepsilon)
\prod\limits_{r=1}^{1/\delta}\mathbf{1}(B_{m_r,\delta r})\right]\\
\qquad \geq\sum\limits_{m_1,...,m_{1/\delta}}{\diy\frac{(1-\theta)^{2n}2^n}{2^n}}\mathbf{E}
\left[g(t_1,...,t_n)\mathbf{1}(\max\limits_{1\leq k\leq n+1}\tau_k\leq T^{1-\beta})
\prod\limits_{r=1}^{1/\delta}\mathbf{1}(B_{m_r,\delta r})\right]\\
\qquad =(1-\theta)^{2n}\mathbf{E}\Big[g(t_1,...,t_n)
\mathbf{1}(\max\limits_{1\leq k\leq n+1}\tau_k\leq T^{1-\beta})
\mathbf{1}(X_T^{(n)})\Big].\quad\blacktriangle\end{array}$$
\bigskip

\section*{Acknowledgments}

The authors are grateful to A.A. Borovkov, B.M. Gurevich,
A.M. Mogulsky and E.A. Pechersky for their interest and useful comments.
E.A. Pechersky should be credited for an initial statement of the probem.
We also thank the anonymous referees for critical remarks and suggestions.

AVL thanks FAPESP for the financial support via Grant 2017/20482.
YMS thanks Math Department, Penn State University, for hospitality and support
and StJohn's College, Cambridge, for support.
AAY thanks CNPq and  FAPESP for the financial support via
Grants 301050/2016-3 and 2017/10555-0, respectively.
\bigskip


TITLE: An inequality of integral on [0,1] relative to Fourier transform.
QUESTION [0 upvotes]: For $\alpha > 0$, I want to prove that there exists a positive number $C$ s.t. the following inequality holds.
$$ \int_0^1 (1-t)^{\alpha} cos(2 \pi xt) dt \leq \frac{C}{(1+|x|)^{min \{ 1+\alpha , 2 \} } }$$
I can't find any relation between Riemann-Lebesgue theorem for fourier transform on [0,1] and this inequality, but there is a hint that using the proof of it.
Thank you for your answers.

REPLY [1 votes]: This is a very similar argument to Van der Corput's Lemma (which is really just integration by parts to force the decay). I'll do a sketch, omitting some computations.
First, notice that it's enough to consider $x>0$ since $\cos$ is even. Moreover, since $|\cos(y)|\leq 1$ and $\int_0^1(1-t)^\alpha\, dt= c_\alpha<\infty$, it's enough to prove the bound for $x>1$.
First, do a change of variables (and use that $\cos$ is even again) to write
$$
f(x):=\int_0^1 (1-t)^\alpha \cos(2\pi xt)\, dt = \int_0^1 t^\alpha \cos(2\pi x(t-1))\, dt.
$$
Now integrate by parts, after multiplying and dividing by $2\pi x$, to get
$$
f(x)= \dfrac{\alpha}{2\pi x}\int_0^1 t^{\alpha-1} \sin(2\pi x(t-1))\, dt.
$$
Now if $\alpha=1$ we can evaluate this directly and get the right bound; on the other hand if $\alpha>1$, then we can integrate by parts again in the same way to get the bound by $x^{-2}$. Therefore it remains the case $0<\alpha<1$, and recall that we're using $x>1$. For this we break up the integral above
$$
f(x)= \dfrac{\alpha}{2\pi x} \int_0^{1/x} t^{\alpha-1} \sin(2\pi x(t-1))\, dt + \dfrac{\alpha}{2\pi x}\int_{1/x}^1 t^{\alpha-1} \sin(2\pi x(t-1))\, dt =I+II.
$$
To handle $I$ we just use the trivial bound for $\sin$,
$$
|I|\leq \dfrac{\alpha}{2\pi x}\int_{0}^{1/x} t^{\alpha-1}\, dt  \leq \dfrac{1}{2\pi x^{1+\alpha}}.
$$
To deal with with $II$, we integrate by parts as in the very first step to get a similar bound as in $I$.


TITLE: Degrees of freedom of rolling coin
QUESTION [1 upvotes]: I gave a mental ability test yesterday in which this question was asked.

A rolling coin on a flat surface has how many degrees of freedom?

I read about degrees of freedom but I haven't had much interaction with mechanics after high school(4 years ago), can someone please explain on how to solve this question?

REPLY [3 votes]: There is one condition, namely that the height coordinate $z$ of the lowest point of the coin is $z_{\rm lowest}=0$. This reduces the $3+3=6$ degrees of freedom of the coin by one (because there was one condition) to five. Note that those $3+3$ I started with are 3 degrees of freedom from a position of the object's center-of-mass; and 3 degrees of freedom parameterizing a rotation in $SO(3)$ determining the orientation (i.e. two angular coordinates to determine the axis; plus one angular coordinate to determine the right rotation around the axis).
Equivalently, one may parameterize these degrees of freedom in many ways. For example, there are 2 horizontal coordinates of the center-of-mass of the coin. And one may also determine the longitude and latitude (2) of the direction of the axis of the coin (directed from tails to heads) plus the right rotation angle $\phi$ around this axis. Again, $2+3=5$. Everything else (namely the right amount of "vertical translation") is determined by saying that the coin sits on the table at a given moment. (I've changed the counting from the first version of the answer, the newer one seems more logical to me.)
Note that these are five x-like degrees of freedom. Each of them has a velocity or a momentum so the "phase space" has $2 \times  5 = 10$ dimensions.
As the Ghost of Perdition mentioned in the comments, one reduces the degrees of freedom if the coin is preventing from slipping. For a general change of the 5 x-like coordinates above, the coin was slipping by $\Delta x$ and $\Delta y$ in the two horizontal directions. Setting them to zero removes 2 x-like coordinates and also their velocities. So one ends up with 3 degrees of freedom - and the phase space is locally 6-dimensional.
However, there's no natural way to extend this 6-dimensional space to all times because the precise location and orientation that is allowed at a given moment depends on the whole history. If one wants to include all possible positions and orientations of the coin into which the coin can get by a history, he still needs the 10-dimensional phase space. It's only true that for the dynamics around some point in time, the no-slipping constraint will effectively reduce the 10-dimensional phase space to a 6-dimensional one.
In other words, we need a 5-dimensional configuration space with 2 (no-slipping) constraints or Lagrange multipliers that could be used to reduce 5 to 3 if these constraints were simple and "integrable". However, they are not, so the reduction or the "simplification" of the problem (elimination of the constraints along with some coordinates) cannot be done.