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\begin{abstract}
We associate with the Grassmann algebra a topological algebra of distributions, which allows the study of processes analogous to the corresponding free stochastic processes with stationary increments, as well as their derivatives.
\end{abstract}
\maketitle

\noindent MCS classes: 30G35, 15A75

\noindent {\em Key words}: Grassmann algebra, Fock space, stochastic processes
\date{today}
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\tableofcontents

\section{Introduction}
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In the present work, we develop some aspects of the Grassmann algebra counterpart of the noncommutative Fock space \cite{MR1217253} and noncommutative stochastic distributions \cite{MR3231624}. Moreover, we study an analog of stochastic processes with stationary increments (such as the fractional Brownian motion) and their derivatives in the setting of the Grassmann algebra $\Lambda$. The processes introduced here differ from the ones discussed by Rogers in, e.g., \cite{MR925919, rogers1994stochastic, rogers2003supersymmetry}.\smallskip

We recall that $\Lambda$, also called the exterior algebra, is the algebra on a field $\mathcal{K}$ generated by a finite or countable set of elements $i_n$ not belonging to and linearly independent over $\mathcal{K}$, and satisfying
\begin{equation}
i_ni_m+i_mi_n=0, \qquad n,m=1,2,\hdots
\label{anticommutative}
\end{equation}
together with the identity element of $\mathcal{K}$. Usually, the field that is considered is the setting of the complex numbers $\mathbb{C}$. Here, we follow this choice.\smallskip

Due to its role on supersymmetry, an element of $\Lambda$ is often referred to as a supernumber. When the number of generators is finite, say $N$, we use the notation $\Lambda_N$ to evidence it. In our approach, $\Lambda=\cup_{N\in\mathbb{N}}\Lambda_N$. This differs from the way $\Lambda$ is usually treated in the literature, where formal infinite sums are considered. Here, if $z\in\Lambda$, there exists $n(z)$ such that $z\in\Lambda_{n(z)}$. We will consider closures of $\Lambda$ to study cases with an effectively infinite number of generators. The construction of generators in the finite case follows with a matrix representation. In the infinite case, it is less straightforward, but concrete realizations can be given \cite{berezin1979method,rogers1980global}.\smallskip

We note that \eqref{anticommutative} also holds for many types of hypercomplex numbers if $n\neq m$ \cite{gal2002introduction}. The difference here is that it also holds for $n=m$, so that
\[
i_n^2=0,\ldots\quad{\rm for}\quad n=1, \hdots,
\]
and in particular $\Lambda$ (and every $\Lambda_N$) has divisors of $0$.\smallskip

One can say that analysis on this setting started in 1937 with Cartan showing that the Grassmann algebra can represent the exterior algebra if one introduces the idea of derivative (and multiplication) by its generators \cite{cartan1937theorie}. In 1959, Martin considered supernumbers to study ``classical versions'' of physical functions for fermions and obtain their quantization through path integrals \cite{MR0109640,MR0109639}, an idea that would be later used by Schwinger to extend his quantum field theory to fermions \cite{schwinger1966particles}. Moreover, in 1966, Berezin independently started an extensive study of what is now known as supermathematics \cite{berezin1979method, MR914369}.\smallskip

When working with $\Lambda_N$ -- and even with $\Lambda$ to a certain extent -- only algebraic operations are involved, and there is no real problem of convergence since every element in $\Lambda$ generated only by $i_n$'s is nilpotent. On the other hand, $\Lambda$ needs to be completed for various natural problems of analytical character.\smallskip

The purpose of this work is to develop counterparts of classical notions from analysis and stochastic processes theory when taking into consideration the completion of $\Lambda$ to a Hilbert space -- denoted $\overline{\Lambda}^{(2)}$ (see Definition \ref{defdef}) -- and embedding it in a Gel\cprime fand triple
\begin{equation}
\label{gelfand_triple}
\mathfrak S_{1}\subset\overline{\Lambda}^{(2)}\subset\mathfrak S_{-1}.
\end{equation}
The space $\mathfrak S_{-1}$, as we prove, has an algebra structure of the type that was first introduced by Kondratiev \cite{MR1408433} in the setting of Hida's white noise space theory, and studied in a more generalized framework in \cite{MR3404695}. There are a number of parallels (and differences) between the present study and the works \cite{MR2610579,alp}, where the complex numbers were replaced by the commutative algebra of Kondratiev stochastic distributions -- see \cite{MR1408433} for the latter, and \cite{MR1203453,MR2444857} for Hida's white noise space theory and the associated spaces of stochastic distributions. In those cases, the underlying space, i.e., the white noise space, is the commutative Fock space, which is typically associated with bosons. To obtain the so-called full Fock space, one includes the antisymmetric Fock space, which can be associated with fermions. In this context, i.e., when considering the full Fock space, it is also possible to define a noncommutative analog of the Kondratiev space \cite{MR3231624,MR3038506}. Moreover, the same type of tools can be developed in the framework of $Q$-deformed commutation relations \cite{ji2016wick}. Here, we follow a similar approach, envisioning applications on stochastic processes and their derivatives. \smallskip

As in the works \cite{aal3,MR1408433}, and more recently in the quaternionic setting \cite{MR3615375}, a Gel\cprime fand triple together with its algebra structure allows one to consider functions from a compact metric space $E$, say $[0,1]$, into $\overline{\Lambda}^{(2)}$. Such functions may be continuous, but not differentiable, when viewed as an element of $\overline{\Lambda}^{(2)}$. On the other hand, if $f$ is seen as a $\mathfrak S_{-1}$-valued function, one has, under certain hypothesis, differentiability. The differentiability is, {\it a priori}, with respect to the strong topology of $\mathfrak S_{-1}$. However, it, in fact, happens in a Hilbert space, thanks to the assumed compactness of $E$. In particular, one can study stochastic processes and their derivatives in such spaces.\smallskip

The main results of this article are the following: First, in Section \ref{sec2}, after discussing symmetries in $\Lambda$, we endow the latter with a family of norms and obtain new inequalities on these norms -- see Theorem \ref{norm-ineq}. Then, in Section \ref{fock-space}, we introduce the Fock space that can be associated with $\overline{\Lambda}^{(2)}$, making connections with classical aspects of superanalysis, e.g., left derivatives and Berezin integrals. Next, in Section \ref{topological-algebra} we embed the Fock space into Gelfand triples given by \eqref{gelfand_triple} and prove that the product in $\mathfrak S_{-1}$ satisfies V\r{a}ge-like inequalities -- see Theorem \ref{ineq}. Finally, using those Gel\cprime fand triples, we present in Section \ref{stocder} a close counterpart of the free stochastic processes with stationary increments and their derivatives.\smallskip

\section{Symmetries and norms in $\Lambda$}
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\label{sec2}

\subsection{Grassmann algebra and supernumbers}

\begin{definition}
We denote by $\mathfrak I$ the set of $t$-uples $({a_1},\ldots, {a_t})\in\mathbb N^t$, where $t$ runs through $\mathbb N$ and $a_1<{a_2}<\cdots<{a_t}$. For $\alpha=({a_1},\ldots, {a_t})\in\mathfrak I$ we set $i_\alpha=i_{a_1}\cdots i_{a_t}$ and write an element $z\in\Lambda$ as a finite sum
\begin{equation}
\label{Grassmann-decomposition}
z=z_0+\sum_{\alpha\in\mathfrak{I}}z_\alpha i_\alpha,
\end{equation}
where the coefficients $z_0$ and $z_{a_1,\ldots,a_t}$ are complex numbers.
\end{definition}

The term that does not contain any Grassmann generator, $z_0$, is called the {\it body} of the number and is sometimes denoted by $z_B$, while $z_S=z-z_B$ is said to be the {\it soul} of the number \cite{MR1172996}. One can also give a meaning to the sum \eqref{Grassmann-decomposition} when it has an infinite number of terms, as we discuss in Section \ref{comp123}.

Sometimes it is convenient to define $i_0=1$ and ``extend'' the set $\mathfrak{I}$ to accommodate it. We will denote this new set $\mathfrak{I}_0$. Hence, a supernumber can be simply written as
\[
z = \sum_{\alpha\in\mathfrak{I}_0} z_\alpha i_\alpha.
\]

If $z=\sum_{\alpha\in\mathfrak{I}_0}z_\alpha i_\alpha$ and $w=\sum_{\beta\in\mathfrak{I}_0}w_\beta i_\beta$, their product makes sense since the sums are finite, and can be written as
\[
zw = \sum_{\alpha,\beta\in\mathfrak{I}_0}z_\alpha w_\beta i_\alpha i_\beta.
\]
Let $\alpha,\beta\in\mathfrak{I}$.
Note that $i_\alpha i_\beta=0$ when $i_\alpha$ and $i_\beta$ have a common factor $i_u$, with $u\in\mathbb N$. Moreover, when $i_\alpha i_\beta$ does not vanish, it might still not be an element of the set $\{i_\alpha : \alpha\in\mathfrak{I}\}$, since permutations might be necessary to obtain such type of element. However, because permutations only introduce powers of negative one, there exists a uniquely defined $\gamma\in\mathfrak{I}$ such that
\[
i_\alpha i_\beta = (-1)^{\sigma(\alpha,\beta)} i_\gamma,
\]
where $\sigma(\alpha,\beta)$ is the number of permutations necessary to ``build'' $\gamma$ from $\alpha$ and $\beta$. If such a relation holds, we write
\begin{equation}
\alpha\vee\beta=\gamma.
\label{vee}
\end{equation}
So $i_\alpha i_\beta = (-1)^{\sigma(\alpha,\beta)} i_{\alpha\vee\beta}$. To rewrite it in a manner that includes the possibility of $i_\alpha i_\beta=0$, we define $\alpha\vee\beta=\emptyset$ if there is no $\gamma\in\mathfrak{I}_0$ such \eqref{vee} is satisfied. Then,
\[
i_\alpha i_\beta = (-1)^{\sigma(\alpha,\beta)}\sum_{\gamma\in\mathfrak{I}_0} \delta_{\alpha\vee\beta,\gamma} i_\gamma,
\]
where $\delta_{\alpha\vee\beta,\gamma}$ is the Kronecker delta.\smallskip

\begin{remark}
We note that $\mathfrak{I}_0$ defined as above is a monoid, with identity given by $\alpha=0$.
\end{remark}

\begin{remark}
We note that $\mathfrak{I}_0$ is the counterpart of the set of indices $\ell$ considered in the case of infinitely many commuting (resp., noncommuting) variables -- see \eqref{index-commut} and \eqref{index-noncommut}.
\end{remark}

It is important to observe that $\Lambda$ is a $\mathbb{Z}_2$-graded algebra. In fact, the elements that commute with each other are of the form
\begin{equation}
z=z_0+\sum_{\substack{\alpha\in\mathfrak{I} \\ |\alpha| \ {\text even}}}z_\alpha i_\alpha,
\label{even-number}
\end{equation}
where $|\alpha|$ is the number of elements of $\alpha$. Those supernumbers are called the {\it even} supernumbers and their set is denoted by $\Lambda_{even}$. It is easy to verify that they commute with {\it every} element of $\Lambda$ and that, moreover, they form a commutative subalgebra.\smallskip

On the other hand, the elements that anticommute with each other are of the type
\begin{equation}
z=\sum_{\substack{\alpha\in\mathfrak{I} \\ |\alpha| \ {\text odd}}}z_\alpha i_\alpha.
\label{odd-number}
\end{equation}
They are known as {\it odd} supernumbers and do not form a subalgebra. In fact, it is an immediate result that the product of two odd supernumbers is an even supernumber. The set of odd supernumbers is denoted by $\Lambda_{odd}$.\smallskip

The following results are easy, but they are relevant to our discussion.\smallskip

\begin{proposition}
Let $v\in\Lambda_{odd}\subset\Lambda$. Then,
\[
v^2=0.
\]
\label{odd-square}
\end{proposition}

\begin{proof}
The proof follows easily. Let $v=\sum_{\substack{\alpha\in\mathfrak{I} \\ |\alpha| \ {\text odd}}}v_\alpha i_\alpha$
\begin{eqnarray*}
v^2 & = & \frac{1}{2}\sum_{\substack{\alpha,\beta\in\mathfrak{I} \\ |\alpha|,|\beta| \ {\text odd}}} \left(v_\alpha v_\beta i_\alpha i_\beta + v_\beta v_\alpha i_\beta i_\alpha\right) \\
    & = & \frac{1}{2}\sum_{\substack{\alpha,\beta\in\mathfrak{I} \\ |\alpha|,|\beta| \ {\text odd}}}v_\alpha v_\beta \left(i_\alpha i_\beta + i_\beta i_\alpha\right) \\
    & = & 0.
\end{eqnarray*}
\end{proof}

\begin{remark}
Even though Proposition \ref{odd-square} refers to the case where $v$ is an odd supernumber in $\Lambda$, its result is still valid when considering closures of $\Lambda$, i.e., when the set $\mathfrak{I}$ has an infinite number of elements.
\end{remark}

\begin{proposition}
Let $N\in\mathbb{N}$ and consider $N+1$ elements $z_n\in\Lambda_N$ such that ${z_n}_B=0$ for every $n\in{1,\hdots,N+1}$. Then,
\[
\prod_{n=1}^{N+1} z_n = 0.
\]
In particular,
\[
z_S^{N+1} = 0
\]
for every $z=z_B+z_S\in\Lambda_N$.
\end{proposition}

We note the following two corollaries, omitting the proof for the first one.

\begin{corollary}
Let $z\in\Lambda$ be such that $z_B=0$. Then, there exists $n=n(z)$ such that $z^{n(z)+1}=0$.
\label{nz}
\end{corollary}

\begin{corollary}
Let $z=z_B+z_S\in\Lambda$. Then, $z$ is invertible if and only if $z_B\neq0$.
\end{corollary}

\begin{proof}
On the one hand, assume $z_B\neq0$. Then,
\[
z = z_B \left( 1+ \frac{z_S}{z_B}\right).
\]
According to Corollary \ref{nz}, there exists a $n(z)$ such that $z^{n(z)+1}=0$. Then,
\[
\left(1+z_B^{-1}z_S\right)^{-1} = \sum_{k=0}^{n(z)} \left(-z_B^{-1}z_S\right)^k
\]
and
\[
z^{-1} = z_B^{-1} \sum_{k=0}^{n(z)} \left(-z_B^{-1}z_S\right)^k.
\]

On the other hand, assume $z$ is invertible and let its inverse be $w=w_B+w_S\in\Lambda$. Then, $zw=1$ and, in particular,
\[
z_Bw_B=1\Rightarrow z_B\neq 0.
\]
\end{proof}

\subsection{Symmetries}

We can define a number of conjugations of a supernumber. We start with the one characterized by
\begin{gather*}
i_n^{\dag_1} = -i_n, \quad \forall n \\
\left(z w\right)^{\dag_1} = w^{\dag_1} z^{\dag_1}.
\end{gather*}
Therefore, the conjugation is defined as
\[
z^{\dag_1}=z_0+\sum_{\alpha\in\mathfrak I}z_\alpha i_\alpha^{\dag_1}.
\]

Let
\[
\pi(\alpha) = \frac{|\alpha|(|\alpha|-1)}{2}.
\]
Then, $i_\alpha^{\dag_1} = (-1)^{|\alpha| + \pi(\alpha)} i_\alpha$ and
\[
z^{\dag_1} = z_0+\sum_{\alpha\in\mathfrak I} (-1)^{|\alpha| + \pi(\alpha)} z_\alpha i_\alpha.
\]

In general, we have
\begin{align*}
\left(z+w\right)^{\dag_1} &= z^{\dag_1} + w^{\dag_1},\\
\left(z^{\dag_1}\right)^{\dag_1} &= z. \\
\end{align*}

The next conjugation is given by the conjugation of the complex coefficients. The Grassmann generators are invariant under it, i.e.,
\[
i_\alpha^{\dag_2} = i_\alpha.
\]
Therefore,
\[
z^{\dag_2}=\overline{z_0}+\sum_{\alpha\in\mathfrak I}\overline{z_\alpha} i_\alpha,
\]
where the overline represents the usual conjugation of a complex number. In general, it holds
\begin{align*}
\left(z+w\right)^{\dag_2} &= \left(z\right)^{\dag_2} + \left(w\right)^{\dag_2}\\
\left(z^{\dag_2}\right)^{\dag_2} &= z \\
\label{z1z2}
\left(z w\right)^{\dag_2} &= z^{\dag_2} w^{\dag_2}.
\end{align*}

The next conjugation is motivated by the fact already mentioned that $\Lambda$ is a $\mathbb{Z}_2$-graded algebra, i.e., the fact that an arbitrary supernumber $z\in\Lambda$ can be written as $z=u+v$, where $u\in\Lambda_{even}$ and $v\in\Lambda_{odd}$. We, then, define the conjugation $\dag_3$ in a way that $u^{\dag_3}=u$ and $v^{\dag_3} = -v$. Hence, it holds in general
\[
z^{\dag_3} = z_0 + \sum_{\alpha\in\mathfrak I}(-1)^{|\alpha|} z_\alpha i_\alpha.
\]

Observe that
\begin{align*}
\left(z+w\right)^{\dag_3} &= \left(z\right)^{\dag_3} + \left(w\right)^{\dag_3}\\
\left(z^{\dag_3}\right)^{\dag_3} &= z \\
\left(z w\right)^{\dag_3} &= \left(z\right)^{\dag_3} \left(w\right)^{\dag_3}.
\end{align*}

We note that the aforedefined conjugations commute with each other:
\begin{eqnarray*}
\left(z^{\dag_1}\right)^{\dag_2} & = & \left(z^{\dag_2}\right)^{\dag_1}, \\
\left(z^{\dag_2}\right)^{\dag_3} & = & \left(z^{\dag_3}\right)^{\dag_2}, \\
\left(z^{\dag_3}\right)^{\dag_1} & = & \left(z^{\dag_1}\right)^{\dag_3}.
\end{eqnarray*}

Therefore, four more conjugations can be defined:
\begin{align*}
z^{\dag_4} &\equiv \left(z^{\dag_1}\right)^{\dag_2} = \overline{z_0} + \sum_{\alpha\in\mathfrak I} (-1)^{|\alpha|+\pi(\alpha)} \overline{z_\alpha} i_\alpha, \\
z^{\dag_5} &\equiv \left(z^{\dag_2}\right)^{\dag_3} = \overline{z_0}+\sum_{\alpha\in\mathfrak I} (-1)^{|\alpha|} \overline{z_\alpha} i_\alpha, \\
z^{\dag_6} &\equiv \left(z^{\dag_3}\right)^{\dag_1} = z_0 + \sum_{\alpha\in\mathfrak I}(-1)^{\pi(\alpha)} z_\alpha i_\alpha, \\
z^{\dag_7} &\equiv \left(\left(z^{\dag_1}\right)^{\dag_2}\right)^{\dag_3} = \overline{z_0}+\sum_{\alpha\in\mathfrak I} (-1)^{\pi(\alpha)} \overline{z_\alpha} i_\alpha.
\end{align*}

We call special attention to $\dag_7$ because this is the conjugation that normally appears in the literature \cite{MR1172996}. Observe that it is very similar to $\dag_2$: it can be characterized as the complex conjugation of the coefficients and
\[
i_n^{\dag_7} = i_n.
\]
However, in general $i_\alpha^{\dag_7}\neq i_\alpha$. The reason for it is that given two supernumbers $z$ and $w$,
\[
\left(zw\right)^{\dag_7} = \left(w\right)^{\dag_7} \left(z\right)^{\dag_7}.
\]

Motivated by this conjugation, a {\it real supernumber} $z_R$ is defined as a supernumber with the property $\left(z_R\right)^{\dag_7}=z_R$. Analogously, a {\it imaginary supernumber} $z_C$ is a supernumber such that $\left(z_C\right)^{\dag_7}=-z_C$. With that, it is easy to see that a supernumber $z$ can be written as $z=z_R+z_C$.

\begin{remark}
The conjugations $\dag_i$ together with the identity $I$ form a commutative group with the composition law. In fact, it is the elementary abelian group $E_8$.

\end{remark}

\begin{remark}
It is clear that $zz^{\dag_i}$ (and, in particular, the modulus $\left|z\right|_i^2=zz^{\dag_i}$ induced by $\dag_i$), with $i\in\{1,2,\hdots,7\}$, is not a real number in general. To see it, first observe that $\dag_4$, $\dag_5$, $\dag_6$, and $\dag_7$ are just compositions of conjugations. Then, let $i\in\mathbb{C}$ be the complex unit and consider the examples $z=i_1 i_2 i_3 - i_4$, $w=ii_1+i_3$, and $r=1+i_1 i_2 + i_3$, which give
\[
zz^{\dag_1}=2i_1 i_2 i_3 i_4 \quad ww^{\dag_2}=2i i_1 i_2 \quad rr^{\dag_3}=1 + 2 i_1 i_2.
\]
\end{remark}

\begin{remark}
The choices of $z$ and $w$ made in the previous remark also show that
\[
zz^{\dag_1} \neq z^{\dag_1}z \quad ww^{\dag_2} \neq w^{\dag_2} w.
\]
\end{remark}

\begin{remark}
In spite of the previous remark, $zz^{\dag_3}=z^{\dag_3}z$ for every $z\in\Lambda$. In fact, $z=u+v$ and, using Proposition \ref{odd-square},
\[
zz^{\dag_3} = \left(u+v\right)\left(u-v\right)=u^2=\left(u-v\right)\left(u+v\right)=z^{\dag_3}z.
\]
\end{remark}

\begin{remark}
It is clear that the conjugations defined here are continuous in $\Lambda_N$ with respect to its natural topology induced by $\mathbb{C}^{2^N}$.
\end{remark}

\subsection{Norms and completions}
\label{comp123}

Because the conjugations defined above induce moduli that in general are not real, we introduce now the $p$-norm of a supernumber.

\begin{definition}
Let $p\geq 1$ be a real number. The $p$-norm of a supernumber $z\in\Lambda$ is defined as
\begin{equation}
\Vert z \Vert_p = \left(\sum_{\alpha\in\mathfrak{I}_0} |z_\alpha|^p \right)^{1/p},
\label{pnorm}
\end{equation}
where $|\cdot|$ is the usual modulus of a complex number.
\end{definition}

The above definition makes sense for any real $p\geq1$. However, for the purposes of this paper, we consider only $p\in\mathbb{N}$.

\begin{theorem}
Let $z,w\in\Lambda$. If $p=1$,
\begin{equation}
\Vert zw \Vert_1 \leq \Vert z \Vert_1 \Vert w \Vert_1.
\label{1norm-ineq}
\end{equation}
If $p> 1$,
\begin{equation}
\Vert zw \Vert_p^p \leq \Vert z \Vert_1^p \Vert w \Vert_{2^{p-1}} \prod_{k=1}^{p-1} \Vert w\Vert_{2^k}
\label{pnorm-ineq}
\end{equation}
and
\begin{equation}
\Vert zw \Vert_p^p \leq \Vert w \Vert_1^p \Vert z \Vert_{2^{p-1}} \prod_{k=1}^{p-1} \Vert z\Vert_{2^k}.
\label{pnorm-ineq2}
\end{equation}
\label{norm-ineq}
\end{theorem}

\begin{proof}
The proof of \eqref{1norm-ineq} is straightforward and is, then, omitted.

To start the proof of \eqref{pnorm-ineq}, we note that
\begin{eqnarray*}
\Vert zw \Vert_p^p & = & \left\Vert\sum_{\gamma\in\mathfrak{I}_0} \sum_{\alpha\vee\beta=\gamma} (-1)^{\sigma(\alpha,\beta)} z_\alpha w_\beta i_\gamma \right\Vert_p^p \\
     & = & \sum_{\gamma\in\mathfrak{I}_0} \left|\sum_{\alpha\vee\beta=\gamma} (-1)^{\sigma(\alpha,\beta)} z_\alpha w_\beta\right|^p \\
     & \leq & \sum_{\gamma\in\mathfrak{I}_0} \sum_{k=1}^p \sum_{\alpha_k\vee\beta_k=\gamma} \left|z_{\alpha_1}\right| \cdots \left|z_{\alpha_p}\right| \left|w_{\beta_1}\right| \cdots \left|w_{\beta_p}\right| \\
     & \leq & \sum_{\alpha_1,\cdots,\alpha_p\in\mathfrak{I}_0} \left|z_{\alpha_1}\right| \cdots \left|z_{\alpha_p}\right| \sum_{k=1}^p \sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta_k\in\mathfrak{I}_0 \\ \alpha_k\vee\beta_k=\gamma}} \left|w_{\beta_1}\right| \cdots \left|w_{\beta_p}\right|.
\end{eqnarray*}
Now, using the Cauchy-Schwarz inequality,
\begin{eqnarray*}
\sum_{k=1}^p \sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta_k\in\mathfrak{I}_0 \\ \alpha_k\vee\beta_k=\gamma}} \left|w_{\beta_1}\right| \cdots \left|w_{\beta_p}\right| & \leq & \left(\sum_{k=1}^{p-1} \sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta_k\in\mathfrak{I}_0 \\ \alpha_k\vee\beta_k=\gamma}} \left(\left|w_{\beta_1}\right| \cdots \left|w_{\beta_{p-1}}\right|\right)^2\right)^{1/2} \left(\sum_{\substack{\gamma\in\mathfrak{I}_0 \\ \alpha_p\vee\beta_p}}|w_{\beta_p}|^2\right)^{1/2} \\
    & \leq & \left(\sum_{k=1}^{p-2} \sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta_k\in\mathfrak{I}_0 \\ \alpha_k\vee\beta_k=\gamma}} \left(\left|w_{\beta_1}\right|
\cdots \hspace{-2mm}
\left|w_{\beta_{p-2}}\right|\right)^4\right)^{1/4} \times \\
     &&\hspace{5mm}\times\left(\sum_{\substack{\gamma\in\mathfrak{I}_0 \\ \alpha_{p-1}\vee\beta_{p-1}}} |w_{\beta_{p-1}}|^4\right)^{1/4} \left(\sum_{\beta_p\in\mathfrak{I}_0}|w_{\beta_p}|^2\right)^{1/2} \\
     & \leq & \left(\sum_{k=1}^{p-2} \sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta_k\in\mathfrak{I}_0 \\ \alpha_k\vee\beta_k=\gamma}} \left(\left|w_{\beta_1}\right| \cdots \left|w_{\beta_{p-2}}\right|\right)^4\right)^{1/4} \hspace{-2mm}\left(\sum_{\beta_{p-1}} |w_{\beta_{p-1}}|^4\right)^{1/4}\hspace{-3mm} \Vert w \Vert_2 \\
     & \leq & \left(\sum_{k=1}^{p-n} \sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta_k\in\mathfrak{I}_0 \\ \alpha_k\vee\beta_k=\gamma}} \left(\left|w_{\beta_1}\right| \cdots \left|w_{\beta_{p-n}}\right|\right)^{2^n}\right)^{1/2^n} \prod_{k=1}^{n} \Vert w \Vert_{2^k} \\
     & \leq & \left(\sum_{k=1}^2 \sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta_k\in\mathfrak{I}_0 \\ \alpha_k\vee\beta_k=\gamma}} \left(\left|w_{\beta_1}\right| \left|w_{\beta_2}\right|\right)^{2^{p-2}}\right)^{1/2^{p-2}} \prod_{k=1}^{p-2} \Vert w \Vert_{2^k} \\
     & \leq & \Vert w\Vert_{2^{p-1}} \prod_{k=1}^{p-1} \Vert w \Vert_{2^k}.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\Vert zw \Vert_p^p & \leq & \sum_{\alpha_1,\cdots,\alpha_p\in\mathfrak{I}_0} \left|z_{\alpha_1}\right| \cdots \left|z_{\alpha_p}\right| \Vert w\Vert_{2^{p-1}} \prod_{k=1}^{p-1} \Vert w \Vert_{2^k} \\
     & \leq & \left(\sum_{\alpha\in\mathfrak{I}_0}|z_\alpha|\right)^p \Vert w\Vert_{2^{p-1}} \prod_{k=1}^{p-1} \Vert w \Vert_{2^k} \\
     & \leq & \Vert z\Vert_1^p \Vert w\Vert_{2^{p-1}} \prod_{k=1}^{p-1} \Vert w \Vert_{2^k},
\end{eqnarray*}
proving \eqref{pnorm-ineq}. The proof of \eqref{pnorm-ineq2} follows in a similar manner.
\end{proof}

\begin{remark}
The inequalities presented in Theorem \ref{norm-ineq} are homogeneous.
\end{remark}

\begin{remark}
The inequalities \eqref{pnorm-ineq} and \eqref{pnorm-ineq2} are not enough to state that, in the closure $\overline{\Lambda}^{(p)}$ of $\Lambda$ with respect to the $p$-norm, the product of elements is a law of composition, i.e., if $z,w\in\overline{\Lambda}^{(p)}$, the aforementioned inequalities do not guarantee that the product converges in $\overline{\Lambda}^{(p)}$. In this paper, we are particular interested in $\overline{\Lambda}^{(2)}$. However, we were unable to show whether the product is or is not a law of composition in this space. Although the solution to this question is important and can reveal interesting aspects of $\overline{\Lambda}^{(2)}$, the main results we present are independent of it.
\end{remark}

\begin{remark}
In \cite{MR1099318}, a Fr\'echet structure modeled on a therein defined sequence space is given to $\Lambda$ by endowing it with seminorms.
\end{remark}

Up to now, besides the last remarks, we only considered algebraic properties of $\Lambda$. To study convergence problems, it is necessary to complete this set. The completion with respect to the $1$-norm, i.e., the space $\overline{\Lambda}^{(1)}$ has a Banach algebra structure and is already known in the literature. It was introduced by Rogers \cite{rogers1980global, zbMATH00861741}. In the next section, we study $\overline{\Lambda}^{(2)}$.

\section{The Fock space}
\setcounter{equation}{0}
\label{fock-space}

The classical Fock space associated with $\ell_2$, i.e., the reproducing kernel
Hilbert space introduced by Bargmann with reproducing kernel \cite{bargmann}
\begin{equation}
\label{barg}
e^{\langle z,w\rangle_{\ell_2}}=\sum_{\alpha\in\ell}\frac{z^\alpha \overline{w}^{\alpha}}{\alpha!}
\end{equation}
corresponds to function theory in a (countably) infinite number of commuting
complex variables. In \eqref{barg} we have used the several complex variables notation and set $z=(z_1,z_2,\ldots)\in\ell_2(\mathbb N)$, $\ell$ to be the family of sequences
\begin{equation}
\alpha=(\alpha_1,\alpha_2,\ldots), \qquad \alpha_j\in\mathbb{N}_0
\label{index-commut}
\end{equation}
where at most a finite number of $\alpha_j$ are different from $0$,
\[
z^\alpha=z_1^{\alpha_1}z_2^{\alpha_2}\cdots,
\]
and $\alpha!=\alpha_1!\alpha_2!\cdots$.

In the noncommutative setting, the commuting variables give place to noncommuting ones and, then, one needs a different set of indexes $\widetilde{\ell}$. We consider $\alpha\in\widetilde{\ell}$ given by
\begin{equation}
\alpha = ((\alpha_1,n_1),(\alpha_2,n_2),\hdots,(\alpha_m,n_m)),
\label{index-noncommut}
\end{equation}
where $n_m\in\mathbb{N}$ and $\alpha_u\neq\alpha_{u+1}$, for $u=1,\hdots,m-1$. Hence, the new kernel is written as
\[
\sum_{\alpha\in\widetilde{\ell}}z^\alpha \overline{w^\alpha},
\]
where here $\overline{w^\alpha}=\cdots \overline{w_2}^{n_2} \overline{w_1}^{n_1}$.

Note that the left concatenation gives $\widetilde{\ell}$ a monoid structure. Furthermore, it defines a partial order as follows: For $\alpha, \beta\in\widetilde{\ell}$ we say that $\beta\le \alpha$ if there is $\gamma\in\widetilde{\ell}$ such that $\alpha=\beta\gamma$.\smallskip

The parallel with the structure of the Grassmann algebra is clear. Motivated by it, we give an inner product to $\overline{\Lambda}^{(2)}$ envisioning the construction of the counterpart of the Fock space here.

\begin{definition}
The inner product $\langle \cdot,\cdot \rangle$ between two supernumbers $z,w\in\Lambda$ is defined as
\[
\langle z,w\rangle = \sum_{\alpha\in\mathfrak{I}_0} z_\alpha \overline{w_\alpha}.
\]
\end{definition}

\begin{remark}
Let $z$ and $w$ be two supernumbers. Then,
\[
\langle w,z\rangle = \langle z^{\dag_2},w^{\dag_2}\rangle = \langle z^{\dag_4},w^{\dag_4}\rangle = \langle z^{\dag_5},w^{\dag_5}\rangle = \langle z^{\dag_7},w^{\dag_7}\rangle.
\]
\end{remark}

\begin{proposition}
$\Lambda$ endowed with the aforementioned defined inner product and the $2$-norm is a Hilbert space.
\end{proposition}

\begin{proof}
It suffices to observe that the $2$-norm is induced by the above defined inner product. In fact,
\[
\Vert z\Vert_2 \equiv \langle z, z\rangle^{1/2}.
\]
\end{proof}

\begin{definition}
\label{defdef}
By analogy with the noncommutative setting, as discussed in the beginning of this section (and also noting the definition in \cite{MR1217253}), $\overline{\Lambda}^{(2)}$ is called the Fock space.
\end{definition}

In general, we consider functions $f:\mathbb{\mathcal{I}}\rightarrow\overline{\Lambda}^{(2)}$, where $\mathcal{I}$ is the domain of $f$, usually $\mathbb{C}$ or $\mathbb{R}$. Whenever we write $f\in\overline{\Lambda}^{(2)}$, we mean that $f(x)\in\overline{\Lambda}^{(2)}$ for every $x\in\mathcal{I}$.\smallskip

As in the classical examples of Fock space, it is possible to define the left multiplication operation $M_f$ in $\overline{\Lambda}^{(2)}$. But there is a caveat: as already discussed, the multiplication might not be a law of composition in $\overline{\Lambda}^{(2)}$. If that is the case, $M_f$ is unbounded for an arbitrary $f\in\overline{\Lambda}^{(2)}$. One can, then, use two different approaches: study such operators in a space of stochastic distributions $\mathfrak{S}_{-1}$, as we discuss in the next section, or restrict $M_f$ to $f\in\overline{\Lambda}^{(1)}\subset\overline{\Lambda}^{(2)}$ since Theorem \ref{norm-ineq} assures $M_f$ is bounded in this case, i.e., if $f\in\overline{\Lambda}^{(1)}\subset\overline{\Lambda}^{(2)}$ and $g\in\overline{\Lambda}^{(2)}$, we have
\[
M_f g = fg \in\overline{\Lambda}^{(2)}.
\]

In this section, we consider only $M_f$ with $f\in\overline{\Lambda}^{(1)}$. When we get back to it in Section \ref{stocder}, we consider the general case where $f\in\overline{\Lambda}^{(2)}$ since the product is a law of composition in the space of stochastic distributions, which is introduced in the next section.

Note that, if $f=\sum_{\alpha\in\mathfrak{I}_0}f_\alpha i_\alpha$,
\begin{equation}
M_f = \sum_{\alpha\in\mathfrak{I}_0}f_\alpha M_{i_\alpha}.
\end{equation}
Therefore, we can focus our analysis on multiplication by elements of $\mathfrak{I}_0$. Furthermore, we note that the focus can be on the set $\mathfrak{I}$, since multiplication by the generator of the body, i.e., $M_1$ is just the identity operator.

In the case of elements associated with $\mathfrak{I}_0$, we already know that
\[
i_\alpha i_\beta = (-1)^{\sigma(\alpha,\beta)} \sum_{\eta\in\mathfrak{I}} \delta_{\alpha\vee\beta,\eta} i_\eta.
\]
It is also straightforward from the definition of the inner product that
\[
\langle i_\alpha, i_\beta\rangle = \delta_{\alpha,\beta}.
\]
Then,
\begin{equation}
\langle M_{i_\alpha} i_\beta, i_\gamma \rangle = (-1)^{\sigma(\alpha,\beta)} \sum_{\eta\in\mathfrak{I}} \delta_{\alpha\vee\beta,\eta} \langle i_\eta, i_\gamma \rangle = (-1)^{\sigma(\alpha,\beta)} \delta_{\alpha\vee\beta,\gamma}.
\end{equation}

We can also look for an expression for the adjoint $M^*_{i_\alpha}$ of $M_{i_\alpha}$:
\begin{equation}
\langle M^*_{i_\alpha} i_\beta, i_\gamma\rangle = \langle i_\beta, M_{i_\alpha}i_\gamma\rangle = (-1)^{\sigma(\alpha,\gamma)} \delta_{\beta,\alpha\vee\gamma}.
\end{equation}
By analogy with the classical cases, we say $M^*_{i_\alpha}$ is the left derivative with respect to $i_\alpha$.

Observing that
\[
M_{i_\alpha} = M_{i_{a_1}} M_{i_{a_2}} \hdots M_{i_{a_t}}
\]
and, as a consequence,
\begin{equation}
M_{i_\alpha}^* = M_{i_{a_t}}^* M_{i_{a_{t-1}}}^* \hdots M_{i_{a_1}}^*,
\label{left-deriv}
\end{equation}
we can pay close attention to the left derivative with respect to single generators. Then,
\[
M^*_{i_n} i_\alpha = \left\{
  \begin{array}{l l}
    0, & \text{if } a_k\neq n, \forall a_k\in\{1,\cdots,|\alpha|\} \\
    (-1)^{k-1} i_{a_1}i_{a_2}\cdots i_{a_{k-1}} i_{a_{k+1}}\cdots i_{a_{|\alpha|}}, & \text{if } \exists k; a_k=n
  \end{array}
\right.
\]
Therefore, the left derivative constructed here corresponds to the one that is traditionally defined in superanalysis and its applications \cite{berezin1979method, MR914369, MR1172996, zbMATH00861741}.

\begin{remark}
Expression \eqref{left-deriv} clearly shows that the operator conjugation $*$ is an extension of the conjugation of supernumbers $\dag_7$.
\end{remark}

\begin{remark}
The Berezin integral is a concept widely used in superanalysis and supersymmetry \cite{berezin1979method, MR914369, MR1172996, zbMATH00861741}. It coincides with the left derivative and, then, can be also defined in terms of $M^*_{i_n}$:
\[
\int di_{n} f \equiv M_{i_n}f.
\]
More generally, if $|\alpha|<\infty$,
\[
\int di_{\alpha} f \equiv \int di_{a_{|\alpha|}} \cdots\int i_{a_1} f = M^*_{i_{a_{|\alpha|}}}\cdots M^*_{i_{a_1}} f.
\]
Moreover, if $f$ is generated by $i_{a_1},\hdots,i_{a_N}$ and $i_\alpha=i_{a_1}\hdots i_{a_N}$, the Berezin integral
\[
\int di_{\alpha} f = f_{1,2,\hdots,N}
\]
reduces to
\[
\int di_{\alpha} f = \langle M_f 1, i_{\alpha}\rangle,
\]
which has some resemblance to a residue.
\end{remark}

It is also worth considering the self-adjoint operator
\begin{equation}
T_f = M_f + M_f^*.
\label{operator}
\end{equation}

\begin{proposition}
Let $f,g\in\overline{\Lambda}^{(2)}$ with $f_B=g_B=0$, then the following holds:
\[
\langle T_f 1, T_g 1\rangle = \langle f, g\rangle.
\]
\label{ipr}
\end{proposition}

\begin{proof}
First, note that $M_f^* 1 = f_B$ and $M_g^* 1 = g_B$. Then, if $f_B=g_B=0$,
\[
\langle T_f 1, T_g 1\rangle = \langle f + f_B, g + g_B\rangle = \langle f, g \rangle.
\]
\end{proof}

Such an operator $T_f$ is used later in this paper when we define stochastic processes associated with the Fock space.

\section{A topological algebra associated with $\Lambda$}
\setcounter{equation}{0}
\label{topological-algebra}

Let $\mathscr S$ denote the space of Schwartz functions (the space of test functions), and let  $\mathscr S^\prime$  be its dual (the space of tempered distributions). The Gel\cprime fand triple $(\mathscr S,\mathbf L_2(\mathbb R,dx),\mathscr S^\prime)$ plays an important role in classical analysis, see for instance \cite{MR0209834}. Gel\cprime fand triples are also defined in Hida's white noise space theory \cite{MR1408433,MR1387829} and in its noncommutative counterpart \cite{MR3231624,MR3038506}, and are used to solve stochastic differential equations and model stochastic processes and their derivatives. In this section we define Gel\cprime fand triples in the Grassmann setting, with the aim of solving similar problems.\\

One of the reasons commonly used for introducing a Gel\cprime fand triple is the fact that in a Hilbert space, one can define products different from the inner product between two elements and, often, those products are not a law of composition on such a space, i.e., the result of the product of two elements does not necessarily belong to the Hilbert space. For instance, in the white noise space, the Wick product is not a law of composition. Then, one embeds the white noise space into a space of stochastic distributions to make it a law of composition. Various choices are possible to do so.

In our case, as already discussed, it is not yet clear if the product is a law of composition in $\overline{\Lambda}^{(2)}$. So one could question if it is necessary to make use of an analogous embedding. However, making the product a law of composition is not the only reason to introduce the space of stochastic distributions. In fact, when considering stochastic processes, such spaces are necessary for the study of their derivatives. Because of it, we introduce in this section the analogous of the space of stochastic distributions. Before doing so, we review a few facts from the classical case as well as from the theory of perfect spaces and strong algebras. We refer the reader to \cite{MR0435834, GS2_english} for more information on these spaces.\smallskip

Our starting point is a decreasing family of Hilbert spaces $(\mathcal H_p,\|\cdot\|_{\mathcal{H}_p})_{p\in\mathbb Z}$, with increasing norms. The intersection $\mathcal F=\cap_{p=0}^\infty \mathcal H_p$ is a Fr\'echet space, which we assume perfect, meaning that compactness is equivalent to being bounded and compact. This will happen in particular when for every $p$ there exists $q>p$ such that the injection map from ${\mathcal H_q}$ into ${\mathcal H_p}$ is compact. An important instance is when the injection is nuclear. We identify ${\mathcal H}_p^\prime$ with ${\mathcal H}_{-p}$. Our main interest is the dual ${\mathcal F}^\prime
=\cup_{p=0}^\infty{\mathcal H}_{-p}$, which together with $\mathcal F$ and $\mathcal H_0$, for a Gel\cprime fand triple.\\

We endow the dual $\mathcal F^\prime$ with the strong topology, defined in terms of the bounded sets of $\mathcal F$. The space $\mathcal F^\prime$ is then locally convex, and the strong topology coincides with the inductive limit topology. See \cite[Section 3]{MR3029153} for a discussion.\\

Therefore, the analysis is, {\sl a priori}, done in a larger space of distributions, which is a (non-metrizable) inductive limit of Hilbert spaces (distributions here being understood as continuous functionals on the space $\mathcal F^\prime$). However, is in fact done locally in a Hilbert space. There are two reasons why this happens. The first reason is that the space of distributions is the dual of a Fr\'echet nuclear space (dual of a perfect space would suffice). The second reason is the algebra structure of $\mathfrak{S}_{-1}$, and V\r{a}ge inequality \eqref{vage_ineq}. We now state two of the main results related to such spaces.
They are used in the proofs of Theorems \ref{stochasticintegral} and \ref{thm55}.

\begin{proposition}
\label{compact}
A set is (weakly or strongly) compact in $\mathcal F^\prime$ if and only if it is compact in one of the spaces ${\mathcal H}_{-p}$ in the corresponding norm.
\end{proposition}

\begin{proposition}
Assume $\mathcal F^\prime$ perfect. Then, weak and strong convergence of sequences are equivalent, and a sequence converges (weakly or strongly) if and only if it converges in one of the spaces ${\mathcal H}_{-p}$ in the corresponding norm.
\label{convergence}
\end{proposition}

A topological algebra is assumed to be separately continuous in each variable. It is not a trivial fact that a strong algebra is in fact jointly continuous in the two variables see \cite[IV.26, Theorem 2]{Bourbaki81EVT} and
the discussion in \cite[pp. 215-216]{MR3404695}.\\

In the case we are interested, we define
\[
\mathcal{H}_{-p}(c_\alpha) = \left\{f = \sum_{\alpha\in\mathfrak{I}_0} f_\alpha i_\alpha \in\Lambda^{(2)} \suchthat \sum_{\alpha\in\mathfrak{I}_0} |f_\alpha|^2 c_\alpha^{-2p}<\infty \right\},
\]
where $p\in\mathbb{Z}$ and the coefficients $c_\alpha$'s form a sequence of positive real numbers such that
\begin{equation}
c_\alpha c_\beta \leq c_\gamma
\label{cond1}
\end{equation}
if $\alpha\vee\beta=\gamma$ and
\begin{equation}
\sum_{\alpha\in\mathfrak{I}_0} c_\alpha^{-2d} < \infty,
\label{cond2}
\end{equation}
where $d$ is a positive integer. Observe that
\[
\mathcal{H}_{-q}(c_\alpha) \subseteq \mathcal{H}_{-p}(c_\alpha)
\]
if $p\geq q$.

Henceforth, we denote $\mathcal{H}_{-p}(c_\alpha)$ simply by $\mathcal{H}_{-p}$.

\begin{definition}
The norm $\left\Vert f\right\Vert_{\mathcal{H}_{-p}}$ of $f\in\mathcal{H}_{-p}$ is defined as
\[
\left\Vert f\right\Vert_{\mathcal{H}_{-p}} \equiv \sum_{\alpha\in\mathfrak{I}_0} |f_\alpha|^2 c_\alpha^{-2p}.
\]
\end{definition}

\begin{proposition}
If $c_{\alpha\vee\beta}=c_\alpha c_\beta$, then $c_0=1$.
\label{c0}
\end{proposition}

\begin{proof}
On the one hand, if $c_0>1$, the inequality \eqref{cond1} does not hold in general since
\[
c_0 c_\alpha > c_\alpha = c_{0\vee\alpha}.
\]
On the other hand, if $c_0<1$, the condition \eqref{cond2} is not satisfied. In fact, $c_0^{-2d}<c_\alpha^{-2d}$ for every $\alpha\in\mathfrak{I}$ and, then, the sum in \eqref{cond2} diverges.
\end{proof}

\begin{proposition}
Let $f\in\mathcal{H}_{-p}$ with $c_\alpha>1$ if $\alpha\neq0$ and $c_0=1$. Then,
\[
\lim_{p\rightarrow\infty}\left\Vert f\right\Vert_{\mathcal{H}_{-p}} = |f_0|^2.
\]
\label{limit-norm}
\end{proposition}

\begin{proof}
Because $\lim_{p\rightarrow\infty} c_\alpha^{-2p}=0$ for every $\alpha\neq 0$,
\begin{eqnarray*}
\lim_{p\rightarrow\infty}\left\Vert f\right\Vert_{\mathcal{H}_{-p}} & = & \lim_{p\rightarrow\infty}\sum_{\alpha\in\mathfrak{I}_0} |f_\alpha|^2 c_\alpha^{-2p} \\
     & = & \sum_{\alpha\in\mathfrak{I}_0} |f_\alpha|^2 \lim_{p\rightarrow\infty} c_\alpha^{-2p} \\
     & = & |f_0|^2.
\end{eqnarray*}
\end{proof}

The following theorem introduces a V\r{a}ge-like inequality, which is the analogous of a result due to V\r{a}ge \cite{vage96} and allows the analysis of stochastic processes to be done locally in a Hilbert space.

\begin{theorem}
If $f\in\mathcal{H}_{-q}$ and $g\in\mathcal{H}_{-p}$, with $p> q$, then
\begin{equation}
\label{vage_ineq}
\left\Vert fg \right\Vert_{\mathcal{H}_{-p}} \leq C_{p-q} \left\Vert f\right\Vert_{\mathcal{H}_{-q}} \left\Vert g\right\Vert_{\mathcal{H}_{-p}},
\end{equation}
where $C_{p-q}$ is a positive constant.
\label{ineq}
\end{theorem}

\begin{proof}
Let $f\in\mathcal{H}_{-q}$ and $g\in\mathcal{H}_{-p}$. Hence, using the Cauchy-Schwarz inequality,
\begin{eqnarray*}
\left\Vert fg \right\Vert_{\mathcal{H}_{-p}}^2 & = & \sum_{\gamma\in\mathfrak{I}_0} |(fg)_\gamma|^2 c_\gamma^{-2p} \\
     & = & \sum_{\gamma\in\mathfrak{I}_0} \left|\sum_{\alpha\vee\beta=\gamma}(-1)^{\sigma(\alpha,\beta)}f_\alpha g_\beta\right|^2 c_\gamma^{-2p} \\
     & \leq & \sum_{\gamma\in\mathfrak{I}_0} \left(\sum_{\substack{\alpha\vee\beta=\gamma \\ \alpha'\vee\beta'=\gamma}}|f_\alpha| |g_\beta| |f_{\alpha'}| |g_{\beta'}| \right) c_\gamma^{-2p} \\
     & \leq & \sum_{\gamma\in\mathfrak{I}_0} \left(\sum_{\substack{\alpha\vee\beta=\gamma \\ \alpha'\vee\beta'=\gamma}}|f_\alpha| c_\alpha^{-p} \ |g_\beta| c_\beta^{-p} \ |f_{\alpha'}| c_{\alpha'}^{-p} \ |g_{\beta'}| c_{\beta'}^{-p} \right) \\
     & \leq & \sum_{\alpha,\alpha'\in\mathfrak{I}_0} |f_\alpha| c_\alpha^{-p} \ |f_{\alpha'}| c_{\alpha'}^{-p} \left( \sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta,\beta' \\ \alpha\vee\beta=\gamma \\ \alpha'\vee\beta'=\gamma}} |g_\beta| c_\beta^{-p} \ |g_{\beta'}| c_{\beta'}^{-p} \right) \\
     & \leq & \sum_{\alpha,\alpha'\in\mathfrak{I}_0} |f_\alpha| c_\alpha^{-p} \ |f_{\alpha'}| c_{\alpha'}^{-p} \left( \sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta \\ \alpha\vee\beta=\gamma}} |g_\beta|^2 c_\beta^{-2p}\right)^{1/2} \left(\sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta' \\ \alpha'\vee\beta'=\gamma}} |g_{\beta'}|^2 c_{\beta'}^{-2p} \right)^{1/2} \\
     & \leq & \sum_{\alpha,\alpha'\in\mathfrak{I}_0} |f_\alpha| c_\alpha^{-p} \ |f_{\alpha'}| c_{\alpha'}^{-p} \left( \sum_{\beta\in\mathfrak{I}_0} |g_\beta|^2 c_\beta^{-2p}\right)^{1/2} \left(\sum_{\beta'\in\mathfrak{I}_0} |g_{\beta'}|^2 c_{\beta'}^{-2p} \right)^{1/2} \\
     & \leq & \left( \sum_{\alpha\in\mathfrak{I}_0} |f_\alpha| c_\alpha^{-p} \right)^2 \left\Vert g\right\Vert_{\mathcal{H}_{-p}}^2 \\
     & \leq & \left( \sum_{\alpha\in\mathfrak{I}_0} |f_\alpha| c_\alpha^{-q} \ c_\alpha^{q-p} \right)^2 \left\Vert g\right\Vert_{\mathcal{H}_{-p}}^2 \\
     & \leq & \left(\sum_{\alpha\in\mathfrak{I}_0} c_\alpha^{-2(p-q)} \right) \left\Vert f\right\Vert_{\mathcal{H}_{-q}}^2 \left\Vert g\right\Vert_{\mathcal{H}_{-p}}^2.
\end{eqnarray*}

Now, it remains to be shown that there exist $c_\alpha$'s such that $\sum_{\alpha\in\mathfrak{I}_0} c_\alpha^{-2(p-q)}<\infty$. We recall that $\alpha\in\mathfrak{I}$ means that $\alpha=(a_1,\hdots,a_n)$ for some integer $n\geq 1$, where $a_1,\hdots,a_n$ are positive integers such that $a_1< a_2 < \cdots < a_n$. The ``extension'' $\mathfrak{I}_0$ adds one more element element to $\mathfrak{I}$, namely $\alpha=0$. With that in mind, one possible family of weights $c_\alpha$ that satisfy \eqref{cond1} and \eqref{cond2} is given by
\[
c_\alpha = e^{\sum_{k=1}^n \varphi(a_k)}
\]
for every $\alpha\in\mathfrak{I}_0$, where $\varphi$ is a monotonically increasing real power series -- or at least the values $\varphi(n)$, for non-negative integers $n$, form a monotonically increasing sequence -- with $\varphi(0)=0$. We also require the coefficients of the power series to be bigger than $\ln 2^{1/2(p-q)}$.

Then, it is straightforward that
\[
c_\alpha c_\beta = c_\gamma
\]
if $\alpha\vee\beta=\gamma$ and $c_0=1$, as required by Proposition \ref{c0}. Moreover, if $d=p-q>0$
\[
\sum_{\alpha\in\mathfrak{I}_0} c_\alpha^{2(q-p)} = 1+  \sum_{n=1}^\infty \sum_{\substack{\alpha\in\mathfrak{I}; \\ |\alpha|=n}} e^{-2d\sum_{k=1}^n \varphi(a_k)}.
\]
Note that there exists $\xi>\ln 2^{1/2d}$ such that
\[
\sum_{\substack{\alpha\in\mathfrak{I}; \\ |\alpha|=1}} e^{-2d\varphi(a_1)} = \sum_{a_1=1}^\infty e^{-2d\varphi(a_1)} \leq \sum_{a_1=1}^\infty e^{-2d\xi a_1} = \frac{1}{e^{2d\xi}-1}<1.
\]
Also,
\[
\sum_{\substack{\alpha\in\mathfrak{I}; \\ |\alpha|=2}} e^{-2d (\varphi(a_1)+\varphi(a_2))} \leq \left(\sum_{a_1=1}^\infty e^{-2d\varphi(a_1)}\right)\left(\sum_{a_2=1}^\infty e^{-2d\varphi(a_2)}\right) \leq \left(\frac{1}{e^{2d\xi}-1}\right)^2
\]
and, in general,
\[
\sum_{\substack{\alpha\in\mathfrak{I}; \\ |\alpha|=n}} e^{-2d (\varphi(a_1)+\cdots+\varphi(a_n))} \leq \left(\sum_{a_1=1}^\infty e^{-2d\varphi(a_1)}\right)\cdots\left(\sum_{a_n=1}^\infty e^{-2d\varphi(a_n)}\right) \leq \left(\frac{1}{e^{2d\xi}-1}\right)^n.
\]
Hence,
\[
\sum_{\alpha\in\mathfrak{I}_0} c_\alpha^{-2d} \leq 1+  \sum_{n=1}^\infty \left(\frac{1}{e^{2d\xi}-1}\right)^n = 1+\frac{1}{e^{2d\xi}-2}.
\]
\end{proof}

We present the following corollary, whose proof is analogous to the one just presented for the previous theorem.

\begin{corollary}
If $f\in\mathcal{H}_{-p}$ and $g\in\mathcal{H}_{-q}$, with $p> q$, then
\begin{equation}
\left\Vert fg \right\Vert_{\mathcal{H}_{-p}} \leq C_{p-q} \left\Vert f\right\Vert_{\mathcal{H}_{-p}} \left\Vert g\right\Vert_{\mathcal{H}_{-q}},
\end{equation}
where $C_{p-q}$ is a positive constant.
\label{ineq2}
\end{corollary}

\begin{definition}
We define the space
\[\mathfrak{S}_1 = \cap_{p\in\mathbb{Z}} \mathcal{H}_p\]
and its topological dual
\[\mathfrak{S}_{-1}=\cup_{p\in\mathbb{Z}}, \mathcal{H}_{-p},\]
which are respectively the analogous of the space of test functions and the space of tempered distributions in the classical cases.
\end{definition}

\begin{corollary}
The space $\mathfrak{S}_{-1}$ endowed with the product is a strong algebra.
\label{strong}
\end{corollary}

\begin{proof}[Outline of the proof]
We first endow $\mathfrak{S}_{-1}$ with the inductive topology. Theorem \ref{ineq} implies that the product is separately continuous in each $\mathcal{H}_{-p}$, which is equivalent to continuity in the inductive topology. Furthermore, $\mathfrak{S}_{-1}$ inherits the associativity of the product in $\Lambda$. We have, then, a Banach algebra structure. Thus, $\mathfrak{S}_{-1}$ can be seen as the inductive limit of Banach spaces, which makes it a strong algebra. See \cite{MR3404695} for more details.
\end{proof}

\begin{remark}
Note that the inductive topology is equivalent to the strong topology.
\end{remark}

\begin{remark}
The product would also be associative in $\overline{\Lambda}^{(2)}$ if it was a law of composition there.
\end{remark}

\begin{corollary}
Let $n\in\mathbb{N}$ and $f\in\mathcal{H}_{-p}\subseteq\mathcal{H}_{-p-2}$. Then,
\[
\left\Vert f^n\right\Vert_{\mathcal{H}_{-p-2}} \leq C_2^{n-1} \left\Vert f\right\Vert_{\mathcal{H}_{-p}}^n.
\]
\label{f-conv}
\end{corollary}

\begin{proof}
First, note that $\left\Vert f\right\Vert_{\mathcal{H}_{-p-2}} \leq \left\Vert f\right\Vert_{\mathcal{H}_{-p}}$ for every $f\in\mathcal{H}_{-p}\subseteq\mathcal{H}_{-p-2}$. Hence, using Theorem \ref{ineq},
\begin{eqnarray*}
\left\Vert f^n\right\Vert_{\mathcal{H}_{-p-2}} & \leq & C_2 \left\Vert f\right\Vert_{\mathcal{H}_{-p}} \left\Vert f^{n-1}\right\Vert_{\mathcal{H}_{-p-2}} \\
     & \leq & C_2^2 \left\Vert f\right\Vert_{\mathcal{H}_{-p}}^2 \left\Vert f^{n-2}\right\Vert_{\mathcal{H}_{-p-2}} \\
     & \leq & C_2^{n-1} \left\Vert f\right\Vert_{\mathcal{H}_{-p}}^n.
\end{eqnarray*}
\end{proof}

\begin{corollary}
Let
\begin{equation}
F(\lambda)=\sum_{n\in\mathbb{N}_0} \alpha_n \lambda^n
\label{exp-ps}
\end{equation}
be an absolutely convergent power series in the open disk with radius $R$, where $\alpha_n,\lambda\in\mathbb{C}$ and $\mathbb{N}_0=\mathbb{N}\cup\{0\}$. If $f\in\mathcal{H}_{-p}$, then $F(f)$ converges in $\mathcal{H}_{-p-2}$ if
\begin{equation}
\left\Vert f\right\Vert_{\mathcal{H}_{-p}} < \frac{R}{C_2}.
\label{cond}
\end{equation}
\label{ps}
\end{corollary}

\begin{proof}
By assumption, if $|\lambda|<R$, the power series \eqref{exp-ps} converges absolutely, i.e.,
\[
\sum_{n\in\mathbb{N}_0} \left|\alpha_n \lambda^n\right|=\sum_{n\in\mathbb{N}_0} \left|\alpha_n\right| \left|\lambda^n\right|<\infty.
\]
Using Corollary \ref{f-conv}, we can study the absolutely convergence of $F(f)$ in $\mathcal{H}_{-p}$:
\begin{eqnarray*}
\sum_{n\in\mathbb{N}_0} \left\Vert\alpha_n f^n\right\Vert_{\mathcal{H}_{-p-2}} & = & \sum_{n\in\mathbb{N}_0} |\alpha_n|^2 \left\Vert f^n\right\Vert_{\mathcal{H}_{-p-2}} \\
     & \leq & \alpha_0+C_2^{-1} \sum_{n\in\mathbb{N}} |\alpha_n|^2 \left(C_2 \left\Vert f\right\Vert_{\mathcal{H}_{-p}}\right)^n.
\end{eqnarray*}
Then, $F(f)$ converges absolutely in $\mathcal{H}_{-p-2}$ if
\[
C_2 \left\Vert f\right\Vert_{\mathcal{H}_{-p}} < R \Rightarrow \left\Vert f\right\Vert_{\mathcal{H}_{-p}} < \frac{R}{C_2}.
\]
\end{proof}

\begin{corollary}
Let $F(\lambda)$ be a power series as in Corollary \ref{ps}. Then, $F(f)$ converges in $\mathfrak{S}_{-1}$ for $f\in\mathfrak{S}_{-1}$ if the body of $f$ satisfies \eqref{cond}.
\label{body-red}
\end{corollary}

\begin{proof}
If $f\in\mathfrak{S}_{-1}$, there exists an integer $q_0$ such that $f\in\mathcal{H}_{-q}$ for every $q\geq q_0$. By Theorem \ref{ps}, for $F(f)$ to converge, it is necessary that $\left\Vert f\right\Vert_{\mathcal{H}_{-q}} < R/C_2$, what does not hold in general. However, because of Proposition \ref{limit-norm}, we can reduce this condition to
\[
\left| f_0\right|^2 < \frac{R}{C_2},
\]
proving the corollary.
\end{proof}

\begin{corollary}
Let $f\in\mathfrak{S}_{-1}$. Then, $f$ is invertible if and only if its body $f_0$ satisfies $f_0\neq 0$.
\end{corollary}

\begin{proof}
On the one hand, if $g$ is the inverse of $f$ and its body is given by $g_0$, then
\[
fg=1 \Rightarrow f_0 g_0 = 1 \Rightarrow f_0\neq 0.
\]
On the other hand, if $f_0\neq 0$, consider without loss of generality $f_0=1$.  Then, Corollary \ref{body-red} implies that
\[
F(f) = \sum_{n\in\mathbb{N}_0} (1-f)^n
\]
converges if the body of $1-f$ is smaller than $C_2^{-1}$. However, $(1-f)_B=0$. Therefore, $g=F(f)\in\mathfrak{S}_{-1}$ and $g$ is the inverse of $f$.
\end{proof}

\section{Stochastic processes and their derivatives}
\setcounter{equation}{0}
\label{stocder}

In the literature, and in particular by Rogers (see, e.g., \cite{MR925919, rogers1994stochastic, rogers2003supersymmetry}), a class of $\Lambda$-valued functions of a real variable is considered and defined as stochastic processes. Here, we look at $\overline{\Lambda}^{(2)}$-valued functions, and present a different approach to stochastic processes in the Grassmannian setting. Our aim is to obtain a close counterpart of the noncommutative white noise space theory \cite{MR3231624,MR3038506}.\smallskip

Let us start reviewing this framework -- we refer the reader to \cite{MR3231624,MR1217253} for more details. First, we recall that the study of Gaussian stochastic processes can be made through the analysis of positive-definite kernels since there is a one-to-one correspondence between the two notions \cite{MR58:31324b,montreal}. In fact, the kernel coincides with the covariance of the stochastic process.

In the framework we are basing our model, as can be seen in \cite[Section 3]{aal2}, the processes are associated with positive-definite kernels of the form
\begin{equation}
K_\sigma(t,s) = \int_\mathbb{R} \frac{(e^{iut}-1)(e^{-ius}-1)}{u^2} d\sigma(u),
\label{kernel}
\end{equation}
where $\sigma$ is absolutely increasing continuous with respect to the Lebesgue measure, $d\sigma(u)=m(u)du$, such that the Stieltjes integral
\begin{equation}
\label{qwertyu}
\int_\mathbb{R}\frac{m(u)du}{u^2+1}<\infty.
\end{equation}
The reason for such choice is the fact that integrals of the form \eqref{kernel} correspond to correlation functions of zero-mean Gaussian processes with stationary increments -- see \cite{MR0012176,MR0004644}. An important example of such processes is the fractional Brownian motion, for which $d\sigma(u)=|u|^{1-2H}du$, with $H\in(1,2)$. In this case, the correlation function $K_\sigma(t,s)$ becomes
\begin{equation}
\label{h2h}
K(t,s)=\gamma_H\left(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\right)
\end{equation}
where $\gamma_H$ depends only on $H$ and, if $H\neq 1/2$, is equal to
\[
\gamma_H = \frac{\cos(\pi H)\Gamma(2-2H)}{(1-2H)H},
\]
where $\Gamma$ is the Euler's Gamma function. Moreover, $\gamma_{1/2}= \pi $ by continuity.\smallskip

We, then, define an operator $S_m$ in $\mathbf{L}_2(\mathbb{R})$ such that
\begin{equation}
\widehat{S_mf}(u) = \sqrt{m(u)}\widehat{f}(u),
\label{Sm-def}
\end{equation}
where $\widehat{f}$ is the Fourier transform of $f$. Note that $S_m$ is, in general, unbounded. Its domain is
\[
\text{dom}\,S_m = \left\{f\in\mathbf{L}_2(\mathbb{R})\suchthat \int_\mathbb{R} m(u) |\widehat{f}(u)|^2 du<\infty\right\},
\]
which contains $\mathbf{1}_{[0,t]}$. Defining
\[
f_m(t) = S_m\mathbf{1}_{[0,t]}
\]
and using Plancherel's equality,
\begin{eqnarray*}
\left\langle f_m(t),f_m(s)\right\rangle_{\mathbf{L}_2(\mathbb{R})} & = & \frac{1}{2\pi} \left\langle \widehat{f}_m(t),\widehat{f}_m(s)\right\rangle_{\mathbf{L}_2(\mathbb{R})} \\
     & = & \frac{1}{2\pi} \left\langle \sqrt{m(u)}\widehat{\mathbf{1}}_{[0,t]},\sqrt{m(u)}\widehat{\mathbf{1}}_{[0,s]}\right\rangle_{\mathbf{L}_2(\mathbb{R})} \\
     & = & \frac{1}{2\pi} \left\langle m(u)\frac{e^{-iut}-1}{u},\frac{e^{-ius}-1}{u}\right\rangle_{\mathbf{L}_2(\mathbb{R})} \\
     & = & \frac{1}{2\pi} \int_\mathbb{R} \frac{(e^{iut}-1)(e^{-ius}-1)}{u^2} m(u) du.
\end{eqnarray*}

In order to obtain the stochastic processes we are interested in, we construct a random variable associated with the functions $f_m(t)$. This is done with the introduction of the creation operator $\ell_h$, with $h\in\mathbf{L}_2(\mathbb{R})$, defined by
\[
\ell_h(f) = h\otimes f, \qquad f\in\Gamma(\mathbf{L}_2(\mathbb{R})),
\]
where $\Gamma(\mathbf{L}_2(\mathbb{R}))$ denotes the full Fock space associated with $\mathbf{L}_2(\mathbb{R})$. Finally, letting $T_h = \ell_h + \ell_h^*$, we define a random variable $X_m(t)$ as
\begin{equation}
X_m(t) \equiv T_{f_m(t)}.
\label{random-var-classical}
\end{equation}

Observe that the expected value of a random variable $X_m(t)$ can be defined by
\[
E(X_m(t)) = \left\langle \Omega,T_{f_m(t)}(\Omega)\right\rangle_\Gamma,
\]
where $\Omega$ is the vacuum state of $\Gamma$. Moreover, as expected:
\begin{equation}
E(X_m(t)X_m(s)) = \left\langle T_{f_m(t)}(\Omega),T_{f_m(s)}(\Omega)\right\rangle_\Gamma = \left\langle f_m(t),f_m(s) \right\rangle_{\mathbf{L}_2(\mathbb{R})}=K_\sigma(t,s),
\label{free-expected}
\end{equation}
where $K_\sigma(t,s)$ is given by \eqref{kernel} with $d\sigma(u)=m(u)du$, as already discussed.

The stochastic free processes associated with those variables have the concept of freeness -- in opposition to independence -- associated with them. Moreover, instead of Gaussian distributions, they have semi-circle distributions. Again, we refer the reader to \cite{MR3615375,MR1217253} for an extended discussion on this topic.

In the case of the stochastic processes we desire to define in the setting of the Grassmann numbers, we replace the operator $T_{f}$ in the expression \eqref{random-var-classical} by the operator we defined in expression \eqref{operator} with
\begin{eqnarray*}
f_m(t) & = & \sum_{n\in\mathbb{N}} \left\langle S_m \mathbf{1}_{[0,t]}, \xi_n\right\rangle_{\mathbf{L}_2(\mathbb{R})} i_n \\
     & = & \sum_{n\in\mathbb{N}} \left\langle \mathbf{1}_{[0,t]}, S_m\xi_n\right\rangle_{\mathbf{L}_2(\mathbb{R})} i_n \\
     & = & \sum_{n\in\mathbb{N}} \left(\int_0^t (S_m\xi_n)(u) du\right)i_n,
\end{eqnarray*}
where $\xi_n$ denotes the Hermite functions.

Hence,
\[
X_m(t) = \sum_{n\in\mathbb{N}} \left(\int_0^t (S_m\xi_n)(u) du\right)T_{i_n}.
\]

We can also define the expected value function $E$ in a similar manner as defined in the ``classical'' case:
\[
E(X_m(t)) = \langle 1, X_m(t) 1\rangle_{\overline{\Lambda}^{(2)}},
\]
where $1$ is the vacuum state in the Fock space introduced in Section \ref{fock-space}. Note that $E(X_m(t))=0$.

Using \eqref{ipr}, we observe that the covariance, which gives the kernel $K(t,s)$, satisfies:
\[
E(X_m(t)X_m(s)) = K(t,s) = \langle X_m(t)1, X_m(s)1\rangle_{\overline{\Lambda}^{(2)}} = \langle f_m(t), f_m(s)\rangle_{\overline{\Lambda}^{(2)}}.
\]
Hence,
\[
K(t,s) = \sum_{n\in\mathbb{N}} \left(\int_0^t(S_m\xi_n)(u)du\right)\left(\int_0^s (S_m\xi_n)(u') du'\right),
\]
which is equivalent to the kernel \eqref{kernel}.

Now, we gather a few results concerning bounds for the operator $T_f$, where $f\in\mathcal{H}_{-p}$ for some $p\in\mathbb{N}$. They are relevant in the proof of Theorem \ref{thm55}.

\begin{proposition}
For every $f\in\mathfrak{S}_{-1}$ with $\left\Vert f\right\Vert_{\mathcal{H}_{-p}}<\infty$, the operator $M_f$ is bounded from $\mathcal{H}_p$ into $\mathcal{H}_{-p}$.
\label{Mf-result}
\end{proposition}

\begin{proof}
Let $g\in\mathcal{H}_p$. Then, using Corollary \ref{ineq2}, we have
\[
\left\Vert M_f g\right\Vert_{\mathcal{H}_{-p}} = \left\Vert f g\right\Vert_{\mathcal{H}_{-p}} \leq C_{2p} \left\Vert f \right\Vert_{\mathcal{H}_{-p}} \left\Vert g\right\Vert_{\mathcal{H}_{p}}.
\]
\end{proof}

For the next result, and henceforth, we consider spaces $\mathcal{H}_{-p}$ where the coefficients $c_\alpha$ of the norm are of the type introduced in the proof of Theorem \ref{ineq}. In particular, if $\alpha\vee\beta=\gamma$, $c_\alpha c_\beta = c_\gamma$.

\begin{proposition}
For every $f\in\mathcal{H}_{-q}$, the operator $M^*_f$ is bounded from $\mathcal{H}_p$ into $\mathcal{H}_{-p}$, where $q<p$.
\end{proposition}

\begin{proof}
Let $g\in\mathcal{H}_{p}$. Then, we have
\begin{eqnarray*}
\left\Vert M^*_f g \right\Vert_{\mathcal{H}_{-p}}^2 \leq \left\Vert M^*_f g \right\Vert_{\mathcal{H}_{p}}^2 & = & \sum_{\gamma\in\mathfrak{I}_0} \left| \sum_{\alpha\vee\gamma=\beta} (-1)^{\sigma(\alpha,\gamma)}f_\alpha g_\beta\right|^2 c_\gamma^{2p} \\
     & \leq & \sum_{\gamma\in\mathfrak{I}_0} \left( \sum_{\alpha\vee\gamma=\beta} |f_\alpha| |g_\beta| \right)^2 c_\gamma^{2p} \\
     & \leq & \sum_{\alpha,\alpha'\in\mathfrak{I}_0} |f_\alpha| c_\alpha^{-p} |f_\alpha'| c_{\alpha'}^{-p} \sum_{\substack{\gamma\in\mathfrak{I}_0; \exists \beta,\beta' \\ \alpha\vee\gamma=\beta \\ \alpha'\vee\gamma=\beta'}} |g_\beta| c_\beta^{p} |g_\beta'| c_{\beta'}^{p} \\
     & \leq & \sum_{\alpha,\alpha'\in\mathfrak{I}_0} |f_\alpha| c_\alpha^{-p} |f_\alpha'| c_{\alpha'}^{-p} \left\Vert g\right\Vert_{\mathcal{H}_p}^2 \\
     & \leq & \left(\sum_{\alpha\in\mathfrak{I}_0} |f_\alpha| c_\alpha^{-q} c_\alpha^{-(p-q)} \right)^2 \left\Vert g\right\Vert_{\mathcal{H}_p}^2 \\
     & \leq & C_{p-q}^2 \left\Vert f\right\Vert_{\mathcal{H}_{-q}}^2 \left\Vert g\right\Vert_{\mathcal{H}_p}^2.
\end{eqnarray*}
\end{proof}

\begin{corollary}
For every $f\in\mathfrak{S}_{-1}$, the operator $T_f=M_f+M^*_f$ is bounded from $\mathfrak{S}_1$ into $\mathfrak{S}_{-1}$ and there exist $p>q$ such that
\[
\Vert T_f g\Vert_{-p} \leq 2C_1 \Vert f\Vert_{-q} \Vert g\Vert_p.
\]
\label{ineq-operator}
\end{corollary}

\begin{proof}
This is a direct consequence of the last two propositions. However, since Proposition \ref{Mf-result} is, in some sense, stronger than what we need, we note that if $f\in\mathcal{H}_{-q}$ and $g_\in\mathcal{H}_p$, the following holds from the last proposition and Theorem \ref{ineq}:
\begin{eqnarray*}
\Vert T_f g\Vert_{-p} & \leq & \Vert M_f g\Vert_{-p} + \Vert M^*_f g\Vert_{-p}\\
     & \leq & C_{p-q} \Vert f\Vert_{-q} \Vert g\Vert_{-p} + C_{p-q} \Vert f\Vert_{-q} \Vert g\Vert_p \\
     & \leq & 2C_{p-q} \Vert f\Vert_{-q} \Vert g\Vert_p \\
     & \leq & 2C_1 \Vert f\Vert_{-q} \Vert g\Vert_p.
\end{eqnarray*}
\end{proof}

To compute $dX_m(t)/dt$, we will make use of the spaces $\mathfrak{S}_1$ and $\mathfrak{S}_{-1}$ and assume certain growth conditions for $m$. First, we present the following
proposition, the proof for which we refer the reader to \cite[Proposition 3.7 and Lemma 3.8]{aal3}.

\begin{proposition}
Let $m$ satisfy
\begin{equation}
m(u) \leq \left\{
\begin{array}{l l}
K |u|^{-b} & |u|\leq 1, \\
K |u|^{2N} & |u|> 1,
\end{array}
\right.
\label{m-form}
\end{equation}
where $b<2$, $N\in\mathbb{N}_0$, and $K$ is a positive real constant. Then,
\begin{equation}
\left|S_m\xi_n(t)\right| \leq D_1 n^{\frac{N+1}{2}} + D_2,
\label{S_m-ineq1}
\end{equation}
and
\begin{equation}
\left|S_m\xi_n(t)-S_m\xi_n(s)\right| \leq \left|t-s\right| \left(D_3 n^{\frac{N+2}{2}} + D_4\right),
\label{S_m-ineq2}
\end{equation}
where $D_1$, $D_2$, $D_3$, and $D_4$ are non-negative functions independent of $n$.
\end{proposition}

\begin{theorem}
\label{thm55}
Let $m$ be a positive measurable function, satisfying \eqref{m-form} and \eqref{qwertyu} (the latter for $d\sigma(t)=m(t)dt$).
Then, for every $g\in\mathfrak{S}_1$ the function $t\mapsto X_m(t)f$ is strongly continuous in $\mathfrak{S}_{-1}$ and there exists a continuous operator $W_m(t)$ from $\mathfrak{S}_1$ into $\mathfrak{S}_{-1}$ such that
\begin{equation}
\frac{d}{dt}X_m(t)g = W_m(t)g.
\label{deriv-proc}
\end{equation}
Finally the function $t\mapsto W_m(t)g$ is continuous from $[a,b]$ into $\mathfrak{S}_{-1}$.
\end{theorem}

\begin{proof}
Since $f_m(t) = \sum_{n\in\mathbb{N}} f^m_n(t) i_n$, with $f^m_n = \int_0^t S_m\xi_n(u) du$, we use \eqref{S_m-ineq1} to obtain
\[
\left\Vert\frac{d}{dt} f_m(t) \right\Vert_{\mathcal{H}_{-p}}^2 = \sum_{n\in\mathbb{N}} \left|S_m\xi_n(t)\right|^2 c_n^{-2p} \leq \sum_{n\in\mathbb{N}} (D_1 n^{\frac{N+1}{2}}+D_2)^2 c_n^{-2p}.
\]
Then, for every suitable choice of coefficients $c_n$, there exists a positive integer $p_0$ such that, for every $p\geq p_0$, $df_m(t)/dt\in\mathcal{H}_{-p}$. Moreover, for $s=t+h$, where $h\neq0$ is a real number, there exists $p_1$ such that,
\[
K_p=\sum_{n\in\mathbb{N}}(D_3 n^{\frac{N+2}{2}}+D_4)^2 c_n^{-2p}<\infty, \qquad \forall p\geq p_1
\]
and, with \eqref{S_m-ineq2}, the $\mathcal{H}_{-p}$ norm of the difference of derivatives satisfies
\[
\sum_{n\in\mathbb{N}} \left|S_m\xi_n(s)-S_m\xi_n(t)\right|^2 c_n^{-2p} \leq  K_{p_1}|h|^2.
\]
Therefore, $W_m=T_{df_m/dt}$ is a continuous operator from $\mathfrak{S}_1$ into $\mathfrak{S}_{-1}$.

Finally, to see that \eqref{deriv-proc} holds, observe that for a real number $h\neq 0$ and $g\in\mathcal{H}_p$
\[
\left(\frac{X_m(t+h)-X_m(t)}{h}-W_m(t)\right)g = \sum_{n\in\mathbb{N}} \frac{\int_t^{t+h}(S_m\xi(u)-S_m\xi(t))du}{h} M_{i_n}g = X_{\Delta(t,h)}g,
\]
with
\[
\Delta(t,h) = \sum_{n\in\mathbb{N}} \frac{\int_t^{t+h}(S_m\xi(u)-S_m\xi(t))du}{h} i_n.
\]
Then, there exists $p>q\geq p_1$ such that, using Corollary \ref{ineq-operator},
\[
\left\Vert X_{\Delta(t,h)}g\right\Vert_{\mathcal{H}_{-p}} \leq 2C_1 \left\Vert\Delta(t,h)\right\Vert_{\mathcal{H}_{-q}} \left\Vert g\right\Vert_{\mathcal{H}_p} \leq \left(2C_1 K_{p_1}\left\Vert g\right\Vert_{\mathcal{H}_p}\right) |h|^2.
\]

Therefore, $W_m(t)=T_{df_m/dt}(t)\equiv dX_m(t)/dt$ is a continuous operator from $\mathfrak{S}_1$ into $\mathfrak{S}_{-1}$. Finally, using Proposition \ref{convergence}, one sees that the function $t\mapsto W_m(t)g$ is continuous.
\end{proof}

Once the treatment for derivatives is formalized, the next natural step would be the development of the counterpart of Ito/Malliavin stochastic calculus. The first step in this direction is the introduction of stochastic integrals. This is done in the following theorem, while stochastic calculus itself will be developed elsewhere.

\begin{theorem}
\label{stochasticintegral}
Let $t\mapsto Y(t)$, with $t\in[a,b]$, be a continuous $\mathfrak{S}_{-1}$-valued function in the strong topology of $\mathfrak{S}_{-1}$. If $g\in\mathfrak{S}_1$, there exists a positive integer $p$, which depends on $g$, such that the Pettis integral
\begin{equation}
\int_a^b Y(t) W_m(t)g \ dt
\label{integral}
\end{equation}
can be computed as a limit of Riemann sums and converges in $\mathcal{H}_{-p}$.
\end{theorem}

\begin{proof}[Outline of the proof]
As in \cite{aal3} the function $t\mapsto W_m(t)g$ is continuous from $[a,b]$ to $\mathfrak{S}_{-1}$. Since the product is jointly continuous in $\mathfrak{S}_{-1}$, the map $t\mapsto Y(t)W_m(t)g$ is continuous and its image is therefore compact in $\mathfrak{S}_{-1}$. By Proposition \ref{compact}, there exists $p$ such that the image of $t\mapsto Y(t)W_m(t)g$ is in $\mathcal{H}_{-p}$; this function is still continuous with respect to the topology of $\mathcal{H}_{-p}$, as is seen using Theorem \ref{convergence}. We then compute \eqref{integral} in $\mathcal{H}_{-p}$ using Riemann sums.
\end{proof}

As we already discussed, the random variables introduced here have the same expected value and the same covariance of the variables associated with free stochastic processes. In spite of it, the fact that the product in the integral \eqref{integral} is, ultimately, the product of $i_\alpha$'s, what contrasts with the ``classical'' case, which is a product of Hermite functions, shows that the processes induced by them are very different.

\section{Final remarks}
\setcounter{equation}{0}
\label{final-rmk}

In this paper, we have studied some aspects of the closure of the Grassmann algebra with respect to the $2$-norm, which we called the Fock space, as well as its embedding in Gel\cprime fand triples. We also introduced a new type of stochastic processes and an approach to study their derivatives. We present now research questions that emerge from the ideas discussed here. \smallskip

An important matter that is still not resolved is whether the product is a law of composition in the Fock space. We know that even if there is no problem of convergence for the product, the evaluation of stochastic integrals could be problematic if we had not introduced the space of stochastic distributions. However, addressing this question is crucial and either answer will, surely, lead to new results on the structure of the Fock space introduced here. \smallskip

Another direction that can be investigated is related to Wiener algebras. In \cite{MR3404695}, a special type of Wiener algebra which can be associated with any strong algebra was introduced in a generic framework. An open question is regarding whether the counterpart of such a Wiener algebra can reveal important aspects of the strong algebra we introduced, uncovering new aspects of it. \smallskip

The stochastic processes we used as a basis have the concept of freeness -- in opposition to independence -- associated with their random variables, and the distribution associated with them are semi-circles -- not Gaussians. A possible research direction is, then, the investigation of the questions: What is the independence-like concept associated with the random variables defined here? What are the distributions associated with the stochastic processes generated by them? Furthermore, as already mentioned at the end of the previous section, there is a possibility to develop the counterpart of Ito and Malliavin stochastic calculus in the present setting. One could also look for the physical processes modeled by them. \smallskip

Furthermore, another interesting direction is to look into generalizations of the processes defined here. We considered in Section \ref{stocder} the special cases where $d\sigma(t)=m(t)dt$. For a general $\sigma$, one cannot introduce the operator $S_m$ defined by \eqref{Sm-def}. However, it is possible to prove that there exists a continuous positive operator $A$ from the Schwartz space $\mathcal S$ into its dual $\mathcal S^\prime$ such that
\begin{equation}
\int_\mathbb{R} |\widehat{f}(u)|^2 d\sigma(u)=\langle Af,f\rangle_{\mathcal S^\prime,\mathcal S}.
\end{equation}
The operator $A$ can be factorized via a Hilbert space since $\mathcal S$ is nuclear -- see \cite{MR647140} for factorization theorems. An explicit construction of $A$ in the form $A=Q_\sigma^* Q_\sigma$, where $Q_\sigma$ is continuous from $\mathcal S$ into $\mathbf L_2(\mathbb R)$, is given in \cite{MR2793121}. \smallskip

Finally, a problem we intend to address soon in a different paper concerns the construction of the counterpart of linear systems and rational functions in the setting of the supernumbers. For the theory of linear systems in the setting of Hida's white noise theory and its noncommutative counterpart, see \cite{MR0255260, MR0452844, SSR}, and \cite{MR2610579, alp, MR3038506} for more recent works.

\section*{Acknowledgement}
Daniel Alpay thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research. Ismael L. Paiva acknowledges financial support from the Science without Borders program (CNPq/Brazil). Daniele C. Struppa thanks the Donald Bren Distinguished Chair in Mathematics, which supported this research.

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TITLE: Norm of Fredholm operator in $L^1$
QUESTION [2 upvotes]: Let $T:L^1([0,1])\rightarrow L^1([0,1])$ be the Fredholm integral operator given by
$$
Tf(x)=\int_0^1 k(x,y)f(y)\, dy
$$
where $k \in C([0,1]^2)$ is called the kernel of $T$. My problem is to find $\|T\|$ in terms of it's kernel.
It's easy to show that it is bounded, but I'm failing to prove that the bound it is the norm (using sign functions). I know a similar result for $T$ acting on $C([a,b])$, a bound for $T$ acting in $L^2$, (and, more generally, from $L^p$ to $L^q$), but here I need help in this case.

REPLY [0 votes]: Here is my solution. The idea is to not to find a function satisfying $||T|| = \frac{||Tf||_{L^1[0,1]}}{||f||_{L^1[0,1]}}$, but to find a sequence of functions such that $||T||=\lim\limits_{n \to \infty} \frac{||Tf_n||_{L^1[0,1]}}{||f_n||_{L^1[0,1]}}$.
Using Fubini's theorem and Hölder's inequality it's easy to see:
$$||Tf||_{L^1[0,1]} = \int_0^1|Tf(y)|dy = \int_0^1\left|\int_0^1k(x,y)f(x)dx\right|dy \le \int_0^1\int_0^1|k(x,y)||f(x)|dxdy = \int_0^1 dx|f(x)|\int_0^1 dy|k(x,y)| \le ||f||_{L^1[0,1]} \left|\left|\int_0^1 |k(x,y)|dy\right|\right|_{C[0,1]} $$
So the guess is that $||T|| = \left|\left|\int_0^1 |k(x,y)|dy\right|\right|_{C[0,1]}$.
$\DeclareMathOperator*{\argmax}{arg\,max}$
Consider $f_n(x)=I_n(x)$ $-$ indicator of $U_{\frac{1}{2n}}(x_0)\cap[0,1]$, where $x_0 = \argmax_\limits{x \in [0,1]}\int_0^1 |k(x,y)|dy$.
Using integral mean value theorem:
$$||T||=\lim\limits_{n \to \infty} \frac{||Tf_n||_{L^1[0,1]}}{||f_n||_{L^1[0,1]}} = \lim\limits_{n \to \infty} \frac{\int_0^1\left|\int_{x_0-\frac{1}{2n}}^{x_0+\frac{1}{2n}}k(x,y)dx\right|dy}{\frac{1}{n}} = \lim\limits_{n \to \infty}\frac{\int_0^1|\frac{1}{n}k(\xi_n,y)|dy}{\frac{1}{n}} = \int_0^1|k(x_0,y)|dy = \left|\left|\int_0^1 |k(x,y)|dy\right|\right|_{C[0,1]} $$


TITLE: How to find the general solution set to a constrained system of linear equations
QUESTION [0 upvotes]: Consider the following general system of linear equations
$$
\begin{pmatrix} a & -b\\ -c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} v \\ w \end{pmatrix}
$$
where $a, b, c, d \in \mathbb{N}^+$, $x, y, v, w \in \mathbb{R}^+$ so that the following constraints are also satisfied: $0 \leq v, w <e$, $0 < x,y < e$ where $e$ is the exponential constant.
I am trying to understand what the general solution set to such a system would look like.  That is, for all $a, b, c, d \in \mathbb{N}^+$ and $ v, w \in \mathbb{R}^+$, what is the corresponding set of solutions $(x, y)^T$?
For example, I am unsure if every $x, y$ pair satisfying $0 < x,y < e$ is a solution to one of these general systems satisfying the constraints, or if only a subset of such $x, y$ values create the solution set to such an arbitrary system.  If this is indeed the case, what would that (solution) subset be?

REPLY [1 votes]: According to your last paragraph it seems you are interested in the problem:
Given $0 < x, y, < \mathrm{e}$ find $a, b, c, d \in ℕ+$ and $\mathrm{e} > v, w \geq 0$ such that the above system of equations is true. Especially you want to know whether there is a solution for arbitrary $0 < x, y < \mathrm{e}$.
And the answer is that there is.
The problem can be reformulated in a following way: for given $x, y$ find $a, b, c, d \in ℕ+$:
$$
\mathrm{e} > ax - by \geq 0\\
\mathrm{e} > -cx + dy \geq 0.
$$
Let's take the first inequality. We can transform it -- add a term $by$ first and then divide by $bx$ (recall that $x, b > 0$ thus the inequalities don't switch the direction) to
$$
\frac{y}{x} + \frac{\rm e}{bx} > \frac{a}{b} \geq \frac{y}{x}. \tag{1}\label{1}
$$
Let's continue with the left hand side inequality.  If we managed to find
$a/b$ such that
$$
\frac{y}{x} + \inf_{0 < x < \mathrm{e}} \frac{\rm e}{bx} > \frac{a}{b}\tag{2}\label{2}
$$
then the LHS inequality of \eqref{1} would be fulfilled for free.
Well, let's try it. The infimum equals to $\frac{\rm e}{b{\rm e}}$. Cancel those $\rm e$ in it and replace the left inequality of \eqref{1} by \eqref{2}
$$
 \frac{y}{x} + \frac{1}{b} > \frac{a}{b} \geq \frac{y}{x}.
$$
Now take the positive real number $y/x$ and round it upwards to the first decimal place.  The error is clearly less than 1/10 and since it is rounded upwards it is at least 1/10.
In other words:
$$
\frac{y}{x} + \frac{1}{10} > \frac{a}{10} \geq \frac{y}{x},
$$ where $a/10$ is the rounded value.  Clearly, $a \geq 1$ as the value is at least $1/10$.  Thus, you have found $a, b \in ℕ+$ and it suffices to set $v = ax - by$.
The second inequality can be solved in the same way.


TITLE: Finding particular solution for PDE
QUESTION [0 upvotes]: $$(p^2+q^2)x=pz, p=\frac{\partial z}{\partial x}, q=\frac{\partial z}{\partial y}$$
For the above PDE, I need to find the solution that passes through $x=0, z^2=4y$
I used Charpit's method to get complete solution as $$z^2=a^2x^2+(ay+b)^2$$
To find the particular solution, I substitute $x=0, z^2=4y$ here to get $$a^2y^2+(2ab-4)y+b^2=0.$$
But I do not know how to proceed from here. Can somebody help?

REPLY [0 votes]: I want to start off by saying that I don't understand how and why this method works, I am picking it from a MD Raisinghania, a standard book for PDE. If anybody can explain the intuition, that would be very helpful.
The complete solution is $f:z^2=a^2x^2+(ay+b)^2$
Substituting $x=0, z^2=4y$ into $f$, we get
$$g:a^2y^2+(2ab-4)y+b^2=0$$
$g$ is an equation with $y$ as parameter. So we eliminate $y$ from $g=0$ and $\frac{\partial g}{\partial y}=0$ (like finding an envelop).
This gives us $ab=1$.
Substituting $b=1/a$ in complete solution $f$, we get
$f_2:z^2-a^2x^2-(ay+\frac 1 a)^2=0$
We now find envelop of $f_2=0$ with $a$ as parameter, ie eliminate $a$ from $f_2=0$ and $\frac{\partial f_2}{\partial y}=0$. This gives us, $$(z^2-2y)^2=4(x^2+y^2)$$
This is the required solution that passes through given curve.


TITLE: Integral paradox: Deterministic integral interpreted as limiting case of stochastic integral
QUESTION [4 upvotes]: The value of a stochastic integral, in this case integrating a Wiener process with respect to itself $$\int_0^T W(t)\;dW(t)$$ is dependent on the chosen position of the endpoint of the subintervals.
For example the Ito integral (leftpoint chosen) gives: $$\int_0^T W(t)\;dW(t)=\frac{W(T)^2}{2}-\frac{T}{2}$$
The extra term $-T/2$ is the so called Ito correction term.
My question concerns this very term: If $W(t)$ wasn't stochastic but deterministic the classical calculus would be used and this extra term wouldn't be there. You can e.g. show via simulation that it disappears when you set $\sigma=0$ and thereby render Brownian motion deterministic.
My question
When the deterministic integral is interpreted as the limiting case ($\sigma \rightarrow 0$) of the stochastic integral where the extra term $-T/2$ vanishes when $\sigma$ reaches $0$ (and thereby becomes deterministic), how can it be that the extra term $-T/2$ is only dependent on $T$ and not on $\sigma$?

REPLY [1 votes]: Let $B(t)$ be the standard Wiener process, so that $\operatorname{var}(B(t))=t$.  Let $W(t) = \sigma B(t)$, so that $\operatorname{var}(W(t))=\sigma^2 t$.  If
$$
\int_0^T B(t)\;dB(t) = \frac{B(T)^2}{2} - \frac T2,
$$
then
$$
\begin{align}
\int_0^T W(t) \; dW(t) & = \sigma^2\int_0^T B(t)\;dB(t) \\  \\
& = \sigma^2\left( \frac{B(T)^2}{2} - \frac T2 \right) = \frac{\sigma^2 B(T)^2}{2} - \frac{\sigma^2 T}{2} \\  \\
&  = \frac{W(T)^2}{2} - \frac{\sigma^2 T}{2}.
\end{align}
$$
Now let $\sigma^2\to0$.


TITLE: Resonance of a damped harmonic oscillator under forced oscillations
QUESTION [1 upvotes]: Suppose we have a damped harmonic oscillator and we also apply an external force such that our system oscillates in steady state. If the frequency of my force matches the natural frequency of my oscillator, I will have resonance. However, I've also read the description of resonance as the unidirectional flow of energy from the external force to the oscillator. Can anyone explain what this exactly means and how?
Also, I've read other posts here, where they use the Fourier transform of force while deriving $x$. I have no idea why Fourier transform is used here, and how does it make the calculations any simpler. Please illuminate me on this!!

REPLY [0 votes]: the unidirectional flow of energy from the external force to the oscillator

means that the external force is giving the oscillator energy to balance the energy lost in the dampening. By knowing the general solution
$$ x(t) = A_1 e^{-\gamma t} \cos(\omega_1 t + \phi_1) +  A_2 \cos(\omega_2 t + \phi_2) $$
In stationary condition it becomes ($t \rightarrow \infty $)
$$ x(t) = A_2 \cos(\omega t + \phi) $$
we can now differentiate and get
$$ \dot x (t) -\omega A_2 \sin(\omega t + \phi_2)$$
Energy is the sum of kinetic and potential energy:
$$ E(t) \frac{1}{2} m A_2^2 \omega_0^2 \cos^2(\omega t + \phi_2) + \frac{1}{2}mA_2^2 \omega^2 sin^2(\omega t + \phi_2)$$
In resonance conditions ($\omega = \omega_0$) we get
$$ E (t) = \frac{1}{2} m A_2 \omega_0^2 $$
Energy does not depend on time; this means that all the energy lost because of the dampening is immediately brought back by the external force. I guess this is what your text book means with 'unidirectional flow of energy'.
Hope this helps.

REPLY [0 votes]: If the external force frequency is different from the resonant frequency, the force does positive work for part of each oscillation cycle and negative work for the other part.
To see what this means physically, suppose the force frequency applied to a spring and mass system is very low. For half the cycle, the force does work compressing the spring. For the other half, the spring does work as it is released.
At resonance, the force is 90 degrees out of phase with the displacement, and the force does positive work for the complete cycle.
I have no idea what you mean by "using fourier transform of force while deriving $x$." For example do you really mean "derive" or are you making the common mistake of writing "derive" when you mean "differentiate"? (And what does $x$ represent anyway?)


text:
\section{Primitive of Exponential of a x by Hyperbolic Cosine of b x}
Tags: Primitives involving Exponential Function, Primitives involving Hyperbolic Cosine Function

\begin{theorem}
:$\displaystyle \int e^{a x} \cosh b x \rd x = \frac {e^{a x} \paren {a \cosh b x + b \sinh b x} } {a^2 - b^2} + C$
\end{theorem}

\begin{proof}
{{begin-eqn}}
{{eqn | l = \int e^{a x} \cosh b x \rd x
      | r = \int e^{a x} \paren {\frac {e^{b x} + e^{-b x} } 2} \rd x
      | c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 2 \int e^{a x} \paren {e^{b x} + e^{-b x} } \rd x
      | c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = \frac 1 2 \int \paren {e^{a x} e^{b x} + e^{a x} e^{-b x} } \rd x
      | c =
}}
{{eqn | r = \frac 1 2 \int \paren {e^{a x + b x} + e^{a x - b x} } \rd x
      | c = [[Exponent Combination Laws/Product of Powers|Exponent Combination Laws: Product of Powers]]
}}
{{eqn | r = \frac 1 2 \int e^{a x + b x} \rd x + \frac 1 2 \int e^{a x - b x} \rd x
      | c = [[Linear Combination of Integrals]]
}}
{{eqn | r = \frac 1 2 \int e^{\paren {a + b} x} \rd x + \frac 1 2 \int e^{\paren {a - b} x} \rd x
      | c =
}}
{{eqn | r = \frac 1 2 \frac {e^{\paren {a + b} x} } {a + b} + \frac 1 2 \frac {e^{\paren {a - b} x} } {a - b} + C
      | c = [[Primitive of Exponential of a x|Primitive of $e^{a x}$]]
}}
{{eqn | r = \frac 1 2 \frac {e^{a x + b x} } {a + b} + \frac 1 2 \frac {e^{a x - b x} } {a - b} + C
      | c =
}}
{{eqn | r = \frac 1 2 \frac {e^{a x} e^{b x} } {a + b} + \frac 1 2 \frac {e^{a x} e^{-b x} } {a - b} + C
      | c = [[Exponent Combination Laws/Product of Powers|Exponent Combination Laws: Product of Powers]]
}}
{{eqn | r = \frac 1 2 \frac {e^{a x} e^{b x} \paren {a - b} } {\paren {a + b} \paren {a - b} } + \frac 1 2 \frac {e^{a x} e^{-b x} \paren {a + b} } {\paren {a - b} \paren {a + b} } + C
      | c =
}}
{{eqn | r = \frac {e^{a x} e^{b x} \paren {a - b} + e^{a x} e^{-b x} \paren {a + b} } {2 \paren {a + b} \paren {a - b} } + C
      | c =
}}
{{eqn | r = \frac {e^{a x} e^{b x} \paren {a - b} + e^{a x} e^{-b x} \paren {a + b} } {2 \paren {a^2 - b^2} } + C
      | c = [[Difference of Two Squares]]
}}
{{eqn | r = \frac {a e^{a x} e^{b x} - b e^{a x} e^{b x} + a e^{a x} e^{-b x} + b e^{a x} e^{-b x} } {2 \paren {a^2 - b^2} } + C
      | c =
}}
{{eqn | r = \frac {e^{a x} } {\paren {a^2 - b^2} } \paren {\frac {a e^{b x} - b e^{b x} + a e^{-b x} + b e^{-b x} } 2} + C
      | c =
}}
{{eqn | r = \frac {e^{a x} } {\paren {a^2 - b^2} } \paren {a \frac {e^{b x} + e^{-b x} } 2 + b \frac {e^{b x} - e^{-b x} } 2} + C
      | c =
}}
{{eqn | r = \frac {e^{a x} } {\paren {a^2 - b^2} } \paren {a \frac {e^b x + e^{-b} x} 2 + b \sinh b x} + C
      | c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{a x} } {\paren {a^2 - b^2} } \paren {a \cosh b x + b \sinh b x } + C
      | c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{a x} \paren {a \cosh b x + b \sinh b x} } {a^2 - b^2} + C
      | c =
}}
{{end-eqn}}
{{qed}}
\end{proof}

config: wiki


TITLE: Proof explanation: Hahn-Banach Theorem in Stein's functional analysis
QUESTION [2 upvotes]: Theorem: Let $V$ be a linear space and $V_{0}$ is its subspace. Also let $p$ be a finite convex functional in $V$ and $l_{0}$ is linear functional in $V_{0}$.
  While $l_{0}(v)\le p(v),v\in V_{0}$
  Then exists linear functional $l$ defined in $V$, which is extension of functional $l_{0}$ and
  $l(v)\le p(v),v\in V$

Stein's proof in functional analysis page$_{21}$ :

......
  Well-order all vectors in $V$ that do not belongs to $V_0$ , and denote this ordering by $\lt$ . Among these vectors we call a vector $v$ "extendable"if the linear functional $l_0$ has an extension of the kind desired to the subspace spanned by $V_0 , v$ and all vectors $\lt v$ . What we want to prove is in effect that all vectors not in $V_0$ are extendable .
  ......

Since we can not apply well-ordering principle to find the largest element in $V$ . If we have proved all vectors not in $V_0$ is extendable , then how to show $V$ is extendable ?

REPLY [0 votes]: I think your concerns about the proof are correct. Let me try to offer a repaired version of the proof.
Well-order $V \setminus V_0$. Let $v_1$ be the least element of $V \setminus V_0$. Let $S_{\leq v}$ denote the span of $V_0 \cup \{ w: w \leq v \}$.
Let $S_{< v}$ denote the span of $V_0 \cup \{ w: w < v \}$.
We use (transfinite) recursion to define, for each $v \in V \setminus V_0$, a linear functional $\ell_v$ on $S_{\leq v}$ that is dominated by $p$, that extends $\ell_0$, and that extends $\ell_u$ for each $u < v$. It is then immediate that defining $\ell$ on $V$ by $\ell(v) = \ell_v(v)$ gives the desired functional (to verify linearity use that $\ell_v$ is extended by $\ell_w$ whenever $v<w$).
For the basis step, define a linear functional $\ell_{v_1}$ on $S_{\leq v_1}$ that is dominated by $p$ and extends $\ell_0$ in the usual way. For the inductive step, suppose that for each $u < v$ we have defined a linear functional $\ell_u$ on $S_{\leq w}$ that is $p$-dominated and extends $\ell_0$. Suppose also that $\ell_t = \ell_u$ for each $t \leq u < v$. Define $\ell'_v$ on the span of $S_{< v}$ as follows. For each $u$ in $S_{<v}$, choose $u'<v$ such that $u \in S_{\leq u'}$ and define $\ell'_v(u) = \ell_{u'}(u)$.
It is clear that $\ell'_v$ is dominated by $p$ and that $\ell'_v$ extends $\ell_u$ for each $u < v$. Using this last fact, it is easy to check that $\ell'_v$ is linear. Now extend $\ell'_v$ to a linear functional $\ell_v$ on $S_{\leq v}$ in the usual way. This completes the recursive definition.


TITLE: Describing two-step iteration in terms of complete Boolean algebras.
QUESTION [1 upvotes]: Suppose $B$ is a complete Boolean and let $\dot{C}\in V^B$ be such that $$\|\dot{C}\text{is a complete Boolean algebra}\|_B=1.$$
Let us consider all $\dot{c}\in V^B$ such that $\|\dot{c}\in\dot{C}\|=1$ and the equivalence relation $c_1\equiv c_2$ if and only if $\|\dot{c_1}=\dot{c_2}\|=1$. Let $D$ be the set of equivalence classes of this relation. We can make $D$ an equivalence relation as follows: if $\dot{c_1}$ and $\dot{c_2}$ are in $D$, there is a unique $\dot{c}\in D$ such that $\|\dot{c}=\dot{c_1}+_{\dot{C}}\dot{c_2}\|=1$, then we let $\dot{c}=\dot{c_1}+_D\dot{c_2}$, we define similarly $\cdot_D$ and $-_D$.
Furthermore, $D$ is a complete Boolean algebra: If $X\subseteq D$, let $\dot{X}\in V^B$ be given by $dom(\dot{X})=X$ and $\dot{X}(\dot{c})=1$, then by the fullness of $V^B$ there is some $\dot{c}\in D$ such that $\|\dot{c}=\sum_{\dot{C}}\dot{X}\|=1$, then $\dot{c}=\sum_D X$.
In the proof of lemma 16.3 of Jech's Set Theory, the author states the following fact:

Given any $b\in B$, there is a unique $\dot{c}\in D$ such that $\|\dot{c}=1_{\dot{C}}\|=b$ and $\|\dot{c}=0_{\dot{C}}\|=-b$.

Does anyone know why this is true?
Thanks

REPLY [1 votes]: This is simply definition by cases of names.
$\{b,-b\}$ is a maximal antichain (or a partition of $1$ in terms of Boolean algebras). So we can define a name whose values are determined by that antichain.


TITLE: NP-complete decision problems
QUESTION [0 upvotes]: Are the following statements true or false? Why? What could I use to prove these statements?

A decision problem is NP-complete if and only if there is a polynomial-time
reduction from SAT to it.

A decision problem is NP-hard if and only if there is a polynomial-time reduction from it to SAT.

REPLY [2 votes]: Note that SAT is an NP-hard problem, meaning for any problem X in NP, there is a polynomial-time reduction from X to SAT. If there is a p-time reduction from SAT to Y, then by transitivity, any problem in NP has a p-time reduction to Y, hence Y must be NP-hard. But we do not know if it is in NP or not, hence we cannot deduce that it is NP-complete. It could be a problem that is much higher in the hierarchy. So the first item is false. On the other hand, if there is a p-time reduction from X to SAT, then X must be in NP, since SAT itself is in NP. But is it NP-hard? Not necessarily, X could be in P, for example. So the second item is also false.


TITLE: markov chain computation
QUESTION [1 upvotes]: I consider a 2 state Markov chain: $X = \{1,2\}$, transitions are
$M(i,j)$ and the matrix has a unique stationary distribution $\pi$:
$$    \pi(1) = \frac{M(2,1)}{2-M(1,1) - M(2,2)}
    \\
    \pi(2) = \frac{M(1,2)}{2-M(1,1) - M(2,2)}.
$$
I try to compute
$$
\sum_{i,j\in X} \pi(i) (M^k(i,j) -\pi(j)) \mu_i \mu_j
$$
where $\mu_1, \mu_2$ are reals. This is supposed to be
$$
\pi(1) \pi(2) (\mu_1 - \mu_2)^2 (1-M(1,2) - M(2,1))^k
$$but why?

REPLY [0 votes]: Calculating the powers of $M$ (see here: How to compute the powers of $2\times2$ Markov matrices)
shows that the matrix whose $(i,j)^{\rm th}$ entry is $\pi(i)(M^k(i,j)-\pi(j))$
can be written  $$A:=\underbrace{\pi(1)\pi(2)(1-M(1,2)-M(2,1))^k}_{\rm scalar}\pmatrix{1&-1\\-1&1}.$$
The expression you want is
\begin{eqnarray*} \mu^T A \mu &=&\pi(1)\pi(2)(1-M(1,2)-M(2,1))^k \pmatrix{\mu_1&\mu_2}\pmatrix{1&-1\\-1&1}\pmatrix{\mu_1\\ \mu_2}\\
&=&\pi(1)\pi(2)(1-M(1,2)-M(2,1))^k (\mu_1-\mu_2)^2.
\end{eqnarray*}


text:
\section{Coercivity and continuity analysis \label{sec:stability} }

As a first result, we establish coercivity and continuity properties of the bilinear form $a_h$. To this end, we make the following assumption, all the subsequent analysis will rely on.
\begin{assumption}\label{ass:d-smallness}
There exist constants $c_d > 0$ and $\zeta >0$ such that
\begin{equation}\label{eq:d-smallness}
\Vert \bs{d}(\tx) \Vert \leq c_d \, h_T^{1+\zeta} \,, \qquad \forall \tx \in T \cap \tG\,, \quad \forall T \in \tT \,.
\end{equation}
\end{assumption}
\noindent This condition requires the distance $\Vert \bs{d} \Vert$ between $\tG$ and $\Gamma$ to go to zero slightly faster that the local meshsize. It can be realized in practice by slightly shifting the nodes on $\tG$ towards $\Gamma$ while refining the grid.

We introduce the following mesh parameters
\begin{equation}\label{eq:mesh-param}
h_\Gamma := \max_{T\in \tT : T {\color{violet}\cap} \tG \not = \emptyset} h_T\,, \qquad h_\Omega := \max_{T\in \tT} h_T \,.
\end{equation}

The following well-known scaled trace inequalities in shape-regular triangulations will be used in different forms throughout the paper.
\begin{property}
There exists a constant $C_I >0$ independent of the meshsize such that for any $T \in \tT$ and any edge/face $E \subset \partial T$ it holds
\begin{equation}\label{eq:scaled-1}
\Vert h_T^{1/2} \nabla w \Vert_{0,E}^2 \leq C_I \, \Vert \nabla w \Vert_{0,T}^2 \,, \qquad \forall w \in \mathbb{P}_1(T)\,.
\end{equation}
This immediately gives
\begin{equation}\label{eq:scaled-2}
\Vert h^{1/2} \nabla w \Vert_{0,\tG}^2 \leq C_I \, \Vert \nabla w \Vert_{0,\tO}^2 \,, \qquad \forall w \in V_h\,.
\end{equation}
\end{property}

We exploit Assumption \ref{ass:d-smallness} to prove the uniform coercivity of the form $a_h$ for sufficiently refined grids; see \cite[Lemma 6]{BrDuTh72} for a similar result.
\begin{prop}\label{prop:coercvity-a}
Assume that the Nitsche penalization parameter $\gamma$ satisfies $\gamma > 2C_I$, and the mesh parameter $h_\Gamma$ satisfies $h_\Gamma^\zeta \leq \frac1{4c_dC_I}$. Then, there exists a constant $\alpha>0$ independent of the meshsize, such that
\begin{equation}\label{eq:coercivity-1}
a_h(v_h,v_h) \geq \alpha \, \Vert v_h \Vert_a^2 \,, \qquad \forall v_h \in V_h\,,
\end{equation}
where
\begin{equation}\label{eq:coercivity-2}
\Vert v \Vert_a^2 := \Vert \nabla v \Vert_{0,\tO}^2+ \Vert h^{-1/2} \tS v \Vert_{0,\tG}^2\,.
\end{equation}
\end{prop}
\begin{proof}
One has
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\!\!\!\!\!   a_h(v_h,v_h) = \Vert \nabla v_h \Vert_{0,\tO}^2 \!\!\!- 2(\tS v_h, \partial_\tns v_h)_{0,\tG} \!\!+ (\nabla v_h \cdot \bs{d}, \partial_\tns v_h)_{0,\tG} \!\!+ \gamma \, \Vert h^{-1/2} \tS v_h \Vert_{0,\tG}^2  \\
&& \hskip0.9cm = \Vert \nabla v_h \Vert_{0,\tO}^2 - 2(h^{-1/2}\tS v_h, h^{1/2}\partial_\tns v_h)_{0,\tG} \\
&& \hskip 3cm + \ (h^{-1} \Vert \bs{d}\Vert  \, h^{1/2}  \partial_{\nu} v_h, h^{1/2} \partial_\tns v_h)_{0,\tG} + \gamma \, \Vert h^{-1/2} \tS v_h \Vert_{0,\tG}^2 \,,
\end{eqnarray*}
where we have used $\bs{d}=\Vert \bs{d}\Vert\, \bs{\nu}$. Recalling \eqref{eq:d-smallness} and applying \eqref{eq:scaled-2}, we get
\begin{equation*}
\begin{split}
a_h(v_h,v_h) &\geq \Vert \nabla v_h \Vert_{0,\tO}^2 - 2 C_I^{1/2} \Vert h^{-1/2} \tS v_h \Vert_{0,\tG} \Vert \nabla v_h \Vert_{0,\tO} \\
& \hskip 2.1cm - c_d C_I h_{\G}^\zeta \Vert \nabla v_h \Vert_{0,\tO}^2 + \gamma \, \Vert h^{-1/2} \tS v_h \Vert_{0,\tG}^2 \,,
\end{split}
\end{equation*}
whence, by Young inequality,
$$
a_h(v_h,v_h) \geq \left(1- \epsilon C_I - c_d C_I h_{\G}^\zeta \right) \, \Vert \nabla v_h \Vert_{0,\tO}^2  + \left(\gamma - \tfrac1\epsilon \right)\, \Vert h^{-1/2} \tS v_h \Vert_{0,\tG}^2 \,.
$$
Choosing $\epsilon = \frac1{2C_I}$, we obtain the result with $\alpha=\min \left( \tfrac14, \gamma - 2C_I \right)$.  \end{proof}

A first consequence of Proposition \ref{prop:coercvity-a} is the well-posedness and numerical stability of the SBM discretization \eqref{eq:SBM-formulation-1} with respect to the norm $\Vert u_h \Vert_a$, which uniformly dominates the $H^1$-norm of $u_h$ (see Remark \ref{rem:H1-control} below).

We now turn to discuss the continuity of the form $a_h$. In view of the convergence analysis, we consider $a_h$ as defined on a product space $V(\tO;\tT) \times V(\tO;\tT)$, where $V(\tO;\tT)$ is an infinite-dimensional subspace of $H^1(\tO)$, depending upon the triangulation $\tT$ and containing $V_h$, made of functions for which the trace of the gradient on $\tG$ is well-defined and controlled in $L^2(\tG)$. For instance, $V(\tO;\tT)$ could be the space of functions with (broken) $H^2$-regularity in $\tO$. However, since the solution $u$ of our Poisson problem may exhibit singularities at the boundary $\Gamma$ due to the presence of corners or edges, we choose a space of weaker regularity, that we are going to define.

Let us introduce a `reference' element $\hat{T}$ for all our triangulations; as usual, we assume that $\hat{T}$ has unitary diameter. Let $\hat{E} \subset \partial\hat{T}$ any edge/face of $\hat{T}$. Let $s, p$ be real numbers satisfying $0 < s \leq 1$ and $1 \leq p \leq \infty$. Then, if $t =s-\frac1p$ satisfies $t >0$, the trace  theorem (see e.g. \cite{Grisvard}) guarantees that the trace operator $\tau_{\hat{E}} v = v_{|\hat{E}}$  maps $W^{s,p}(\hat{T})$ on $W^{t,p}(\hat{E})$ continuously. On the other hand, the Sobolev imbedding theorem guarantees that $W^{t,p}(\hat{E}) \subseteq L^2(\hat{E})$ with continuous inclusion, provided $s \geq \frac{n}p - \frac{n-1}2$; this condition holds true if $s > \frac1p$ with $p \geq 2$. Thus, $v_{|\hat{E}} \in L^2(\hat{E})$ if $v \in W^{s,p}(\hat{T})$, and the bound
$$
\Vert v \Vert_{0,\hat{E}}^2 \leq \hat{C} \left( \Vert v \Vert_{0,\hat{T}}^2 + | v |_{s,p, \hat{T}}^2 \right)\,,
$$
holds with a constant $\hat{C}>0$ independent of $v$. Here, $| v |_{s,p, \hat{T}}$ denotes the fractional seminorm $\left(\int_{\hat{T}\times\hat{T}} \frac{|v(\bs{x})-v(\bs{y})|^p}{|\bs{x}-\bs{y}|^{n+sp}} d\bs{x}d\bs{y} \right)^{1/p}$ when $s<1$, or the $L^p$-norm of $\nabla v$ when $s=1$. Next, consider an element $T \in \tT$ with edge/face $E$: a scaling argument yields after a simple computation the existence of a constant $C>0$ independent of $h$ such that
\begin{equation}\label{eq:scaled-3}
\Vert h_T^{1/2} v \Vert_{0,E}^2 \leq C \left( \Vert v \Vert_{0,T}^2  +  h_T^{2s+(1 - 2/p)n} |  v |_{s,p, T}^2  \right) \,, \qquad \forall v \in W^{s,p}({T})\,.
\end{equation}

We apply such bound to each component of the gradient of a function, yielding the existence of a constant, still denoted $C_I$ as in \eqref{eq:scaled-1}, such that for all $w \in H^1(T)$ with $\nabla w \in W^{s,p}({T})$ it holds
\begin{equation}\label{eq:scaled-4}
\Vert h_T^{1/2} \nabla w \Vert_{0,E}^2 \leq C_I \left( \Vert \nabla w \Vert_{0,T}^2  +  h_T^{2s+(1 - 2/p)n} |\nabla w |_{s,p, T}^2  \right)\! , \quad \end{equation}
(Note indeed that this inequality reduces to \eqref{eq:scaled-1} when $w \in \mathbb{P}_1(T)$.) Thereby, considering the strip of elements in $\tT$ with at least one edge/face on $\tG$, i.e., the subset
\begin{equation}\label{eq:def-cTb}
\tO^b := \bigcup \{T \in \tT^b \} \qquad \text{where \ } \tT^b:= \{ T \in \tT : \text{meas}\,(\partial T \cap \tG) > 0\}\,,
\end{equation}
we are led to the following definition of the space $V(\tO;\tT)$.

\begin{definition}\label{def:Wh}
Let  $s, p$ be real numbers satisfying $0 < s \leq 1$ and $2 \leq p \leq\infty$ with $s > \frac1p$. Then, set
\begin{equation}\label{eq:def-Wh1}
V(\tO;\tT) = V(\tO;\tT,s,p) := \{w \in H^1(\tO) : \nabla w_{|T} \in W^{s,p}(T) \ \ \forall T \in \tT^b \}\,,
\end{equation}
equipped with semi-norm
\begin{equation}\label{eq:def-Wh2}
| h^{s_p} \, \nabla w |_{s,p, \tO^b} := \left( \sum_{T \in \tT^b} | h_T^{s_p} \, \nabla w |_{s,p, T}^p \right)^{1/p} , \qquad s_p := s+\tfrac12-\tfrac1p \,,
\end{equation}
and norm
\begin{equation}\label{eq:def-Wh3}
\Vert w \Vert_{V(\tO;\tT)}^2 := \Vert w \Vert_a^2 + | h^{s_p} \, \nabla w |_{s,p, \tO^b}^2 \,.
\end{equation}
\end{definition}

With such definitions, we use  H\"older's inequality  in \eqref{eq:scaled-4} and the property that the measure $|\tG|$ of $\tG$ is uniformly bounded by $|\G|$, to get the existence of a constant $\bar{C}_I>0$ such that
\begin{equation}\label{eq:scaled-5}
\Vert h^{1/2} \nabla w \Vert_{0,\tG}^2 \leq \bar{C}_I \left( \Vert \nabla w \Vert_{0,\tO}^2  +  | h^{s_p} \, \nabla w |_{s,p, \tO^b}^2  \right) \,, \qquad \forall w \in V(\tO;\tT) \,,
\end{equation}
which extends \eqref{eq:scaled-2}; indeed, note that $V_h \subset V(\tO;\tT)$ and $\Vert v_h \Vert_{V(\tO;\tT)} = \Vert v_h \Vert_a$ for all $v_h \in V_h$. Note as well that $W^{1+s,p}(\tO) \subset V(\tO;\tT)$.

\begin{rem}\label{rem:H1-control} {\rm
The norm $\Vert w \Vert_{V(\tO;\tT)}$ uniformly controls from above the standard norm $\Vert w \Vert_{1,\tO}$. Indeed, the latter is equivalent to the norm
$\left( \Vert \nabla w \Vert_{0,\tO}^2 \!\! + \Vert w \Vert_{0,\tG}^2 \! \right)^{1/2}$ and one has (assuming without loss of generality that $h \leq 1$ in $\tO$)
\begin{equation*}
\begin{split}
\Vert w \Vert_{0,\tG} & \leq \Vert h^{-1/2} w \Vert_{0,\tG} \leq  \Vert h^{-1/2} \tS w \Vert_{0,\tG} + \Vert h^{-1/2} \nabla w \cdot \bs{d} \Vert_{0,\tG} \\
& \leq
\Vert h^{-1/2} \tS w \Vert_{0,\tG} + c_d \bar{C}_I^{1/2} h_\G^\zeta \left( \Vert \nabla w \Vert_{0,\tO} + | h^s \, \nabla w |_{s,p, \tO^b}^2 \right)^{1/2},
\end{split}
\end{equation*}
whence the result.
}
\end{rem}

\begin{prop}\label{prop:continuity-a} The bilinear form $a_h$ is defined in $V(\tO;\tT) \times V(\tO;\tT) $ and uniformly continuous therein; precisely, there exists $A>0$ independent of $h$ such that
\begin{equation}\label{eq:continuity-a}
|\, a_h(w,v)\, | \leq A \, \Vert w \Vert_{V(\tO;\tT)} \, \Vert v \Vert_{V(\tO;\tT)}  \,, \qquad \forall w, v \in V(\tO;\tT) \,.
\end{equation}
\end{prop}
\begin{proof}
One has
\begin{eqnarray*}
&& \hskip -.35cm a_h(w,v) = (\nabla w, \nabla v)_{0,\tO} - (\partial_\tns w, \tS v)_{0,\tG} + (\partial_\tns w, \nabla v \cdot \bs{d})_{0,\tG} \\
&&  \hskip 3cm -  ( \tS w, \partial_\tns v)_{0,\tG} + \gamma \, (h^{-1} \tS w, \tS v)_{0,\tG} \\
&&  \hskip 1.cm = (\nabla w, \nabla v)_{0,\tO} \!\! - (h^{1/2}\partial_\tns w, h^{-1/2} \tS v)_{0,\tG} \!\!+ (h^{1/2} \partial_\tns w, h^{-1/2} \nabla v \cdot \bs{d})_{0,\tG} \\
&& \hskip 3.cm  -  ( h^{-1/2}\tS w, h^{1/2} \partial_\tns v)_{0,\tG} + \gamma \, (h^{-1/2} \tS w, h^{-1/2}\tS v)_{0,\tG} \;.
\end{eqnarray*}
One concludes using \eqref{eq:d-smallness} and \eqref{eq:scaled-5}. \end{proof}

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