\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 128, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/128\hfil Multipoint initial-final value problems]
{Multipoint initial-final value problems for dynamical Sobolev-type
equations \\ in the space of noises}

\author[A. Favini, S. A. Zagrebina, G. A. Sviridyuk \hfil EJDE-2018/128\hfilneg]
{Angelo Favini, Sophiya A. Zagrebina, Georgy A. Sviridyuk}

\address{Angelo Favini \newline
Department of Mathematics,
Bologna University,
Piazza di Porta San Donato 5, 40126,
Bologna (BO), Italy}
\email{angelo.favini@unibo.it}

\address{Sophiya A. Zagrebina \newline
Department of Mathematical and Computer Modelling,
South Ural State University, Lenin av., 76,
Chelyabinsk, 454080, Russian Federation}
\email{zagrebinasa@susu.ru}

\address{Georgy A. Sviridyuk \newline
Department of Mathematical Physics Equations,
South Ural State University, Lenin av., 76,
Chelyabinsk, 454080, Russian Federation}
\email{sviridyuk@susu.ru}

\thanks{Submitted March 6, 2018. Published June 19, 2018.}
\subjclass[2010]{60H30, 34K50, 34M99}
\keywords{Dynamical Sobolev-type equation; Wiener $K$-process;
\hfill\break\indent multipoint initial-final conditions;
Nelson-Gliklikh derivative;white noise; space of noises; 
\hfill\break\indent stochastic Hoff equation}

\begin{abstract}
 We prove the existence of a unique solution for a linear stochastic
 Sobolev-type equation with a relatively  $p$-bounded operator and
 a multipoint initial-final condition, in the space of ``noises".
 We apply the abstract results to specific multipoint initial-final
 and boundary value problems for the linear Hoff equation which models
 I-beam bulging under random load.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction} \label{intro}

In the simplest setup, a linear stochastic differential equation
is of the form
\begin{equation}
d\eta = (S\eta + \psi)dt + Ad\omega, \label{0.1}
\end{equation}
where $S$ and $A$ are linear operators specified below,
$\psi = \psi (t)$
is a deterministic load external action and
$\omega = \omega (t)$ is a stochastic external action,
$\eta = \eta (t)$ is the required stochastic process.
Originally $d\omega$ stood for the differential of the Wiener process
$\omega = W (t)$,
whose generalized derivative is traditionally treated as white noise.
Ito began studying the ordinary differential equations of the form \eqref{0.1}
and was joined later by Stratonovich and Skorokhod.
The Ito-Stratonovich-Skorokhod approach
in the finite-dimensional case is still popular \cite{1}.
Moreover, it has been extended successfully to the infinite-dimensional setup \cite{2},
and even to Sobolev-type equations \cite{3}.
In the framework of this direction,
the linear stochastic Hoff equation with the initial-final condition
was considered \cite{4}.


However, recently a new approach to linear stochastic equations
arose \cite{7} and is actively developing \cite{6}
in optimal measurement theory.
Namely, instead of \eqref{0.1}
we consider the linear stochastic Sobolev-type equation
\begin{equation}
L{\mathaccent"7017 \eta}  = M\eta +N\omega,\label{0.2}
\end{equation}
where $\eta = \eta (t)$ is the required stochastic process and
$\omega = \omega(t)$ is a prescribed stochastic process
corresponding to external action, ${\mathaccent"7017 \eta}$ is the
Nelson-Gliklikh derivative \cite{8,6,7} of $\eta$, the operators
$L$, $M$, and $N$ are linear and continuous. By way of example, \cite{8, 7}
consider the ``white noise" $\omega={\mathaccent"7017 W}$, while, as
shown previously \cite{6}, it is more adequate to the 
Einstein-Smoluchowski theory of Brownian motion than the traditional white
noise $d\omega=dW$ in \eqref{0.1}. (Here $W=W(t)$
is a Wiener $K$-process for a nuclear operator $K$).

Apart from the introduction, the conclusion, and the list of references,
this article consists of three sections.
The first one deals with the deterministic inhomogeneous linear Sobolev-type equation
\begin{equation}
L\dot u = Mu+f,
\label{0.3}
\end{equation}
where the operator $M$ is $(L,p)$-bounded
with $p\in \{ 0 \} \cup \mathbb{N}$.
(Note that we use the term ``Sobolev type equations" \cite{12} 
as synonymous terms ``degenerate equations" \cite{13} and 
``equations not solvable with respect to the highest-order derivative" \cite{14}).

We define  multipoint initial-final conditions
and state a theorem on the existence of a unique solution.
We borrowed all results from \cite{9, 3}
and therefore give them without proofs.
The second section extends the deterministic results of the first one
to the stochastic setup by analogy with \cite{7};
sketches of proofs complement the results.
In the third section, by way of example,
we consider the linear stochastic Hoff equation \cite{4} which models I-beam bulging.
In closing, we outline possible directions for further research.
The list of references, not intended to be complete,
reflects the authors' tastes and preferences.

\section{Deterministic linear equations}
\label{sec:1}

Given two Banach spaces $\mathfrak{ U}$ and $\mathfrak{F}$,
take two operators:
$L\in \mathcal{ L} (\mathfrak{U}; \mathfrak{F})$,
that is, a linear and continuous one;
and $M\in \mathcal {C}l (\mathfrak{ U}; \mathfrak{F})$,
that is, a linear, closed, and densely defined one.
Set 
$$
\rho^L (M) = \{\mu\in\mathbb{C}: (\mu L-M)^{-1}\in\mathcal{L} (\mathfrak{F}; 
\mathfrak{U})\}
$$
is called a $L$-resolvent set of an operator $M$. The set 
$\sigma^L(M)=\mathbb{C}\setminus\rho^L(M)$ is called
 $L$-spectrum of an operator $M$. It is easy to show 
\cite[Chapter 4]{9} that the $L$-resolvent set of the operator $M$ 
is always open, and, consequently, the $L$-spectrum of the operator $M$ 
is always closed. An operator $M$ is called $(L,\sigma)$-bounded, 
if $L$-spectrum is a bounded set
(for the terminology and results, see \cite[Chapter 4]{9}).
So, if the operator $M$ is $(L,\sigma)$-bounded, then there exist degenerate 
analytic groups of solving  operators
$$
U^t=\frac{1}{2\pi i}\int_\gamma R_\mu^L (M) e^{\mu t} d\mu
\quad\text{and}\quad
F^t=\frac{1}{2\pi i}\int_\gamma L_\mu^L (M) e^{\mu t} d\mu
$$
defined on the spaces $\mathfrak{U}$ and $\mathfrak{F}$ respectively;
moreover, $U^0\equiv P$ and $F^0\equiv Q$ are projections.
Here $\gamma$ is the contour bounding a domain $D$
which contains the $L$-spectrum $\sigma^L (M)$ of the operator $M$; also, 
$R_{\mu}^L(M) = (\mu L - M)^{-1} L$
is the  right $L$-resolvent of $M$, while
$L_{\mu }^L(M) = L (\mu L - M)^{-1}$
is the  left one.
For a degenerate analytic group
the concepts of  kernel
$\ker U^{\bf .}=\ker P=\ker U^t$
and the  image $\operatorname{im } U^{\bf .}=\operatorname{im} P=\operatorname{im} U^t$
for all $t\in\mathbb{R}$ are well-defined.
Put $\mathfrak{ U}^0=\ker U^{\bf .}$,
$\mathfrak{U}^1=\operatorname{im} U^{\bf .}$,
$\mathfrak{F}^0=\ker F^{\bf .}$,
and $\mathfrak{F}^1=\operatorname{im} F^{\bf .}$.
Then $\mathfrak{U}^0\oplus\mathfrak{U}^1=\mathfrak{U}$
and $\mathfrak{F}^0\oplus\mathfrak{F}^1=\mathfrak{F}$.
Denote also by $L_k$ the restriction of $L$ to $\mathfrak{U}^k$
and by $M_k$ the restriction of $M$ to
$\operatorname{dom} M\cap\mathfrak{U}^k$, for $k=0, 1$.

\begin{theorem}[{Splitting theorem \cite[Chapter 4]{9}}]  \label{thm1.1} 
 If the operator $M$ is   $(L,\sigma )$-\\ bounded then
\begin{itemize}
\item[(i)] $L_k\in\mathcal{L} (\mathfrak{U}^k; \mathfrak{F}^k)$ for $k=0,1$;

\item[(ii)] $M_0\in \mathcal{C}l (\mathfrak{U}^0; \mathfrak{F}^0)$
and $M_1\in \mathcal{L} (\mathfrak{U}^1; \mathfrak{F}^1)$;


\item[(iii)] the operators $L^{-1}_1\in \mathcal{L} (\mathfrak{F}^1; \mathfrak{U}^1)$
and $M_0^{-1}\in \mathcal{L} (\mathfrak{F}^0; \mathfrak{U}^0)$
exist.
\end{itemize}
\end{theorem}

Put
$H=M_0^{-1}L_0\in \mathcal{L}(\mathfrak{U}^0)$
and
$S=L_1^{-1}M_1\in \mathcal{L}(\mathfrak{U}^1)$.


\begin{corollary}[{ \cite[Chapter 4]{9}}]  \label{coro1.1} 
If the operator $M$ is  $(L,\sigma )$-bounded, then
\[
(\mu L-M)^{-1}=
-\sum_{k=0}^\infty \mu^kH^kM_0^{-1} (\mathbb{I}-Q)
+\sum_{k=1}^\infty \mu^{-k}S^{k-1}L_1^{-1} Q
\]
for every $\mu \in\mathbb{C}\setminus\bar{D}$.
\end{corollary}

The operator $M$ is called $(L,p)$-bounded with
$p\in\{0\}\cup\mathbb{N}$ whenever $H^p\neq\mathbb{O}$
but $H^{p+1}=\mathbb{O}$.

We introduce the condition
\begin{itemize}
\item[(A1)] $\sigma^L (M)= \bigcup_{j=0}^m {\sigma}^L_{j} (M)$
 for $ m\in \mathbb{N}$;
 furthermore,
$\sigma_j^L (M)\neq\emptyset$, there exists a closed contour
$\gamma_j\subset\mathbb{ C}$, bounding a domain 
$D_j\supset\sigma_j^L (M)$,  such that 
$\overline{D_j}\cap\sigma_0^L (M)=\emptyset$
 and $\overline{D_k}\cap\overline{D_l}=\emptyset$
 for all $j,k, l=\overline{1,m}$
 with $k\neq l$.
\end{itemize}

\begin{theorem}[\cite{9}] \label{thm1.2} 
If the operator $M$ is $(L, \sigma)$-bounded and condition {\rm (A1)} is 
fulfilled then
\begin{itemize}
\item[(i)] there exist degenerate analytic groups
$$
U_j^t
= \frac{1}{2\pi i}   \int_{\gamma_j }R_{\mu }^L (M) e^{\mu t}d\mu,
\quad j=\overline{1,m}.
$$

\item[(ii)] $U^t U^s_j = U^s_j U^t = U_j^{s+t}$ for all $s$,
$t\in \mathbb{ R}$ and $j=\overline{1,m}$;

\item[(iii)] $U_k^t U_l^s = U_l^s U_k^t = \mathbb{ O}$
for all $s$, $t\in \mathbb{ R}$ and $k$, $l=\overline{1,m}$
with $k\neq l$. 
\end{itemize}
\end{theorem}

Put $U^t_0=U^t- \sum_{k=1}^m U^t_k$ for $t\in\mathbb{R}$.


\begin{remark} \label{rmk1.1} \rm
Consider the identity elements $P_j\equiv U^0_j$ 
of the constructed degenerate analytic groups $\{U_j^t:t\in
\mathbb{R}\}$, for $j=\overline{0,m}$. It is obvious that $P P_j
=P_j P = P_j$ for $j=\overline{0,m}$, and
 $P_k P_l = P_l P_k=\mathbb{O}$ for $k$, $l=\overline{0,m}$ with $k\neq l$. 
Similarly, we can construct projectors $Q_j\in\mathcal{L}(\mathfrak{F})$ for
$j=\overline{0,m}$ (see \cite{9} for details) such that $Q Q_j =Q_j Q =
Q_j$ for $j=\overline{0,m}$ and $Q_k Q_l = Q_l Q_k=\mathbb{O}$ for $k$,
$l=\overline{0,m}$ with $k\neq l$.
\end{remark} 

We refer to $P_j$ and $Q_j$ for $j=\overline{0,m}$ as
\textit{relatively spectral projectors}.

We introduce the subspaces $\mathfrak{U}^{1j} =\operatorname{im} P_j$ and 
$\mathfrak{F}^{1j}=\operatorname{im} Q_j$ for $j=\overline{0,m}$. 
By construction,
$$
\mathfrak{U}^1=\oplus_{j=0}^m \mathfrak{U}^{1j}\quad
\text{and}\quad
\mathfrak{F}^1=\oplus_{j=0}^m \mathfrak{F}^{1j}.
$$
We denote by $L_{1j}$ the restriction of $L$ to $\mathfrak{U}^{1j}$
and by $M_{1j}$ the restriction of $M$ to
$\operatorname{dom} M\cap\mathfrak{U}^{1j}$,
for $j=\overline{0,m}$.
It is not difficult to show that
$P_j\,\varphi\in\operatorname{dom} M$;
therefore, if
$\varphi\in\operatorname{dom} M$ then the domain
$\operatorname{dom} M_{1j}=\operatorname{dom}\, M\cap\mathfrak{U}^{1j}$
is dense in $\mathfrak{U}^{1j}$,
for $j=\overline{0,m}$.

\begin{theorem}[Generalized spectral theorem \cite{9}] \label{thm1.3} 
 Suppose that $L\in \mathcal{ L} (\mathfrak{U}; \mathfrak{F})$
and $M\in \mathcal{C}l (\mathfrak{U}; \mathfrak{F})$,  operator
$M$ is $(L,\sigma)$-bounded, and condition {\rm (A1)} is satisfied,
then
\begin{itemize}
\item[(i)]
$L_{1j}\in\mathcal{L} (\mathfrak{U}^{1j}; \mathfrak{F}^{1j})$
and
$M_{1j}\in \mathcal{L}(\mathfrak{U}^{1j}; \mathfrak{F}^{1j})$
for
$j=\overline{0,m}$;

\item[(ii)] the operators
$L^{-1}_{1j}\in\mathcal{L} (\mathfrak{F}^{1j};\mathfrak{U}^{1j})$
exist, for $j=\overline{0,m}$.
\end{itemize}
\end{theorem}

Thus, we assume that condition (A1) is fulfilled. Fix
$\tau_j\in\mathbb{R}$ with $\tau_j<\tau_{j+1}$, vectors 
$u_j\in\mathfrak{U}$ for $j=\overline{0,m}$, and vector-function $f\in C^\infty
(\mathbb{R}; \mathfrak{F})$. Consider the linear inhomogeneous
Sobolev-type equation
\begin{equation}
L\dot u = Mu +f.\label{1.1}
\end{equation}
We refer to a vector-function
$u\in C^\infty (\mathbb{R}; \mathfrak{ U})$
satisfying \eqref{1.1} as a {\it solution} to  \eqref{1.1}.
We refer to a solution $u=u(t)$, for $t\in \mathbb{R}$,
to \eqref{1.1} satisfying the conditions
\begin{equation}
P_j (u(\tau_j)-u_j)=0,\quad
j=\overline{0,m},\label{1.2}
\end{equation}
as a {\it solution to the multipoint initial-final value problem}
for \eqref{1.1}.

\begin{theorem}[\cite{9}] \label{thm1.4} 
 If the operator $M$ is $(L,p)$-bounded for $p\in\{0\}\cup\mathbb{N}$ and
condition (A1) holds then for all 
$f\in C^\infty (\mathbb{R}; \mathfrak{F})$ and $u_{j}\in\mathfrak{ U}$, 
for $j=\overline{0,m}$, there exists a
unique solution to problem \eqref{1.1}, \eqref{1.2}; furthermore, it
is of the form
\begin{equation}
\begin{aligned}
u(t)&=- \sum_{q=0}^p H^q M_0^{-1} (\mathbb{I}-Q)
f^{(q)} (t) \\
&\quad + \sum_{j=0}^m U_j^{t-\tau_j} u_j +
 \sum_{j=0}^m \int_{\tau_j}^t U^{t-\tau_j-s}_j L_{1j}^{-1} Q_j f(s)ds.
\end{aligned} \label{1.3}
\end{equation}
\end{theorem}
An example is presented in Section 4 of this article.


\section{Stochastic linear equations}
\label{sec:2}

For a real separable Hilbert space
$\mathfrak{U}\equiv (\mathfrak{U}, \langle\cdot,\cdot\rangle)$,
take an operator
$K\in \mathcal{L} (\mathcal{U})$
whose spectrum $\sigma (K)$ is nonnegative, discrete, with finite multiplicities
and accumulates only to zero.
Denote by $\{\lambda_j \}$ the sequence of eigenvalues of
$K$ enumerated in the non-increasing order
taking the multiplicities into account.
The linear span of the set $\{\varphi_j\}$
of associated orthonormal eigenvectors of $K$ is dense in $\mathfrak{U}$.
Assume also that $K$ is a nuclear operator, that is,
its trace satisfies  $\operatorname{Tr} K = \sum_{j=1}^\infty \lambda_j <+\infty$.

Take a sequence $\{\eta_j\}$ of independent stochastic processes
$\eta_j:\Omega\times\mathcal{I}\to\mathbb{R}$,
a complete probability space $\Omega$, and an interval
$\mathcal{I}\subset\mathbb{R}$.
Equip $\mathbb{R}$ with the Borel $\sigma$-algebra.
The set of random variables with zero mean and
finite variances constitutes a Hilbert space
with the inner product
$(\xi_1,\xi_2) = {\bf E}\xi_1\xi_2$.
Denote this Hilbert space by  $\mathbf{L}_2$. Assume that
the random variables $\eta_j (\omega,t)\in \mathbf{L}_2$
are Gaussian for all $\omega\in \mathcal{A}$ and
$t\in\mathcal{I}$, where $\mathcal{A}$ is a
$\sigma$-algebra on $\Omega$.
In addition, the sample trajectory $\eta_j (\omega,\cdot)$
is almost surely continuous, that is, $\eta_j\in\mathbf{C L}_2$.
(For a detailed description of the spaces
$\mathbf{C}^l \mathbf{L}_2$ for $l\in\{0\}\cup\mathbb{N}$,
see \cite{8,7}.)
Define the $\mathfrak{U}$-valued stochastic $K$-process
\begin{equation}
\Theta_K(t)=\sum_{j=1}^\infty\sqrt{\lambda_j} \eta_j(t)\varphi_j
\label{2.1}
\end{equation}
on assuming that the series \eqref{2.1} converges uniformly on every 
compact subset of $\mathcal{I}$. Observe that if
$\{\eta_j\}\subset\mathbf{C}\mathbf{L}_2$ then the existence of a
stochastic $K$-process $\Theta_K$ implies that
its trajectories are almost surely (a.s.) continuous.
Introduce the Nelson-Gliklikh derivatives
\begin{equation}
{\mathaccent"7017 \Theta}_K^{(l)}(t)
=\sum_{j=1}^\infty\sqrt{\lambda_j} {\mathaccent"7017 \eta}_j^{(l)}(t)
\varphi_j\label{2.2}
\end{equation}
of the stochastic $K$-process on assuming that
the derivatives in the right-hand side 
up to order $l$ exist and all series converge uniformly on every compact subset of
$\mathcal{I}$.
(For a detailed description of the Nelson-Gliklikh derivative,
see \cite{8, 1,7}).
As in  \cite{8,7} we introduce the \textit{space of differentiable ``noises"}
$\mathbf{C}_K^l \mathbf{L}_2$
of stochastic $K$-processes
whose trajectories are a.s. continuously differentiable on
$\mathcal{I}$ in the sense of Nelson-Gliklikh up to order
$l\in\{0\}\cup\mathbb{N}$.

As an example, let us present ``black noise'',
a stochastic $K$-process whose trajectories
a.s. coincide with the zero
(that is, absolute silence),
as well as ``white noise''
\begin{equation}
{\mathaccent"7017 W}_K (t)= \frac{W_K (t) }{2t},\label{91}
\end{equation}
the Nelson-Gliklikh derivative of the
Wiener $K$-process
$$
W_{K}(t)=\sum_{j=1}^\infty
\sqrt{\lambda_j }\beta_j (t)\varphi_j,
\quad t \in\overline{\mathbb{R}}_+.
$$
Here $\beta_j=\beta_j (t)$ is the Brownian motion of the form
$$
\beta_j (t)=\sum_{k=1}^\infty
\xi_{jk }\sin \frac{\pi (2k+1)}{2}t,\quad t \in\overline{\mathbb{R}}_+,
$$
where $\xi_{jk }$ are pairwise independent Gaussian random variables
such that $\mathbf{E}\xi_{jk }=0$ and
$\mathbf{D}\xi_{jk } = [\frac{\pi (2k+1)}{2}]^{-2}$,
here
$\xi_{jk }\in \mathbf{L}_2$, $\mathbf{E}$ is mathematical expectation and 
$\mathbf{D}$ is dispersion.

Having considered the deterministic equation \eqref{0.3}
in the previous section, we now proceed to 
the stochastic equation \eqref{0.2}.
Assume that the operator
$M$ is $(L, p)$-bounded, with $p\in\{0\}\cup\mathbb{N}$, and
condition (A1) is satisfied. Consider the linear stochastic
Sobolev-type equation
\begin{equation}
L{\mathaccent"7017 \eta} = M\eta +N\omega,
\label{2.4}
\end{equation}
where $\eta = \eta (t)$ is the required stochastic $K$-process
and $\omega=\omega (t)$ is a known stochastic $K$-process,
and the operator $N\in \mathcal{L}(\mathfrak{U}; \mathfrak{F})$.

Take $\tau_0=0$ and $\tau_j\in\mathbb{R}_+$ with
$\tau_{j-1} <\tau_{j}$ for $j=\overline{1, m}$.
Complement  \eqref{2.4} with the multipoint initial-final conditions
\begin{equation}
P_j (\eta(\tau_j)-\xi_{j})=0,\quad j=\overline{0,m}, \label{2.5}
\end{equation}
where $P_j$ are the relatively spectral projectors from Remark \ref{rmk1.1}.
Below, in view of \eqref{91},
we also have to consider the {\it weak}
(in the sense of S. Krein) {\it multipoint initial-final conditions}
\begin{equation}
\lim_{t\to \tau_0+}P_0(\eta \left(t \right)-\xi_0)=0,\quad 
P_j (\eta(\tau_j)-\xi_j)=0,\quad j=\overline{1,m}. \label{2.6}
\end{equation}
Here
\begin{equation}
\xi_j=\sum_{k=1}^\infty \sqrt{\lambda_k}\xi_{jk}\varphi_k,\,\,j=\overline{0,m},
\label{2.6.1}
\end{equation}
where $\xi_{jk}\in \mathbf{L}_2$ is a Gaussian random variable such that series
\eqref{2.6.1} is convergent.
(For instance $\mathbf{D}\xi_{jk}\leq C_j$, $k\in\mathbb{N}$, $j=\overline{0,m}$).
Call a stochastic $K$-process
$\eta\in \mathbf{C}^1_K\mathbf{L}_2$
a ({\it classical}) {\it solution} to \eqref{2.4}
whenever a.s. all its trajectories satisfy \eqref{2.4} for some
stochastic $K$-process $\omega\in \mathbf{C}_K\mathbf{L}_2$,
some operator $N\in \mathcal{L}(\mathfrak{U}; \mathfrak{F})$,
and all $t\in\mathcal{I}$. (Here and henceforth
$\mathcal{I}=(0, +\infty)$). Call a solution
$\eta=\eta(t)$ to \eqref{2.4} a ({\it classical}) {\it solution} to problem
\eqref{2.4}, \eqref{2.5} (problem \eqref{2.4}, \eqref{2.6}) whenever
in addition condition \eqref{2.5} (condition \eqref{2.6}) is
satisfied.

\begin{theorem} \label{thm2.1}
 For $p\in\{0\}\cup\mathbb{N}$ take an $(L,p)$-bounded operator
$M$ and assume that condition (A1) holds. 
Given $\tau_j\in \mathbb{R}_+$ for $j=\overline{1,m}$, an operator 
$N\in\mathcal{L} (\mathfrak{U}; \mathfrak{F})$, a nuclear operator 
$K\in\mathcal{L} (\mathfrak{U})$ with real spectrum $\sigma (K)$,
a stochastic $K$-process $\omega=\omega (t)$ such that
$(\mathbb{I}-Q)N\omega\in\mathbf{C}^{p+1}_K\mathbf{L}_2$
and $QN\omega\in\mathbf{C}_K\mathbf{L}_2$, and random variables
$\xi_j\in\mathbf{L}_2$, for $j=\overline{0, m}$,
such that \eqref{2.6.1} are fulfilled,
there exists a unique solution $\eta\in\mathbf{C}_K^1 \mathbf{L}_2$
to problem \eqref{2.4}, \eqref{2.5}; moreover,
it is of the form 
\begin{equation}
\begin{aligned}
\eta (t)&=- \sum_{q=0}^p H^q M_0^{-1} (\mathbb{I}-Q)
{\mathaccent"7017 \omega}^{(q)}(t) \\
&\quad + \sum_{j=0}^m
\Big[U_j^{t-\tau_j}\xi_j+
\int_{\tau_j}^t U^{t-\tau_j-s}_j L_{1j}^{-1} Q_jN\omega(s) ds\Big],
\quad t\in\mathcal{I}.
\end{aligned} \label{2.7}
\end{equation}
\end{theorem}

Let us sketch the proof. It is straightforward to verify that
\eqref{2.7} is a solution to problem \eqref{2.4}, \eqref{2.5}.
To establish the uniqueness,
reduce the problem to the equivalent system
$$
L{\mathaccent"7017 \eta} = M\eta,\,\, P_j\,\eta^{j}(\tau_j)=0,\quad
j=\overline{0,m}.
$$
By Theorem \ref{thm1.1} the first equation here 
is equivalent to the system
\begin{equation}
H{\mathaccent"7017 \eta}^0 = \eta^0,\,\,
{\mathaccent"7017 \eta}^{1} = S\eta^{1},\label{2.9}
\end{equation}
where $\eta^0=(\mathbb{I}-P)\eta$ and $\eta^{1}=P\eta$.
Taking now the Nelson-Gliklikh derivative of the first equation
and multiplying on the left by $H$ we obtain in succession
$$
0=H^{p+1}{\mathaccent"7017 \eta} ^{0(p+1)}
=\ldots=H^{2}{\mathaccent"7017 \eta}\!{}^{0(2)}
=\dots
=H{\mathaccent"7017 \eta}^{0}=\eta^{0}.
$$
By Theorem \ref{thm1.2} and the initial-final conditions \eqref{2.5},
the second equation of \eqref{2.9} yields
$\eta^1= \sum_{j=0}^m U^{t-\tau_j} 0=0$.

In view of \eqref{91}, problem \eqref{2.4}, \eqref{2.5} is not solvable
when the right-hand side of \eqref{2.4} is the ``white noise"
$\omega (t)={\mathaccent"7017 W}_K\!(t)$.
In this case instead of conditions \eqref{2.5}
we should consider conditions \eqref{2.6}.

\begin{corollary} \label{coro2.1}
If all the hypotheses of Theorem \ref{thm2.1} hold and
$\omega (t)={\mathaccent"7017 W}_K (t)$ then,
given random variables $\xi_j\in\mathbf{L}_2$  as in \eqref{2.6.1},
there exists a unique solution to problem \eqref{2.4}, \eqref{2.6};
furthermore, it has the form
\begin{equation}
\begin{aligned}
\eta (t)&=  \sum_{j=0}^m \Big[U_j^{t-\tau_j} \xi_{j}
-S_jP_j\int_{\tau_j}^t U_j^{t-\tau_j-s} L_{1j}^{-1} Q_jN W_K(s) ds\\
&\quad +L_{1j}^{-1}Q_jNW_K(t)\Big]
 - \sum_{q=0}^p H^q M_0^{-1} (\mathbb{I}-Q)
\stackrel {\circ}{W}\!{}_K^{(q+1)}(t),\quad t\in\overline{\mathbb{R}}_+.
\end{aligned}\label{2.10}
\end{equation}
\end{corollary}

The proof of the above corollary is similar to that of Theorem \ref{thm2.1}.
The difference in the additive terms
is caused by an application of integration ``by parts",
\begin{align*}
&\int_{\tau_j}^t U^{t-\tau_j-s}_j L_{1j}^{-1} Q_jN {\mathaccent"7017 W}_K(s) ds \\
&= L_{1j}^{-1}Q_jN(W_K(t)-W_K(\tau_j))
 -S_jP_j \int^t_{\tau_j}U_j^{t-\tau_j-s}L_{1j}^{-1}Q_jNW_K(s)ds,
\end{align*}
which follows from the properties of Nelson-Gliklikh derivative.
Here  $S_j = L_{1j}^{-1}M_{1j}$
for $j=\overline{0,m}$.


\section{Linear Hoff equation with additive ``white noise"}
\label{sec:3}

Consider a bounded domain $D\subset\mathbb{R}^d$ ($d\in \mathbb{N}$)
with boundary $\partial D$ of class $C^{\infty}$.
Denote by $\mathfrak{U}$ and $\mathfrak{ F}$ the function spaces 
$\mathfrak{U}=\{u \in W_2^{l+2}(D): u(x)=0, x \in \partial D\}$
and $\mathfrak{F}=W_2^l(D)$, where $l\in \{0\}\cup\mathbb{ N}$.
Evidently, $\mathfrak{U}$ is a real separable Hilbert space
densely and continuously embedded into $\mathfrak{F}$.
Fixing $\alpha, \mu\in \mathbb{R}$, construct the operators
$L=\mu\mathbb{I}+\Delta$ and $M=\alpha\mathbb{I}$, where
$\Delta$ is the Laplace operator, and the symbol $\mathbb{I}$
stands for the embedding operator
$\mathbb{I}:\mathfrak{U}\hookrightarrow\mathfrak{F}$;
we also emphasize that here $M$ is not invertible.
Consider also the spectral problem
\begin{equation}
-\Delta u = \nu u \text{ in }  D\text{ and } u(x)=0\text{ for }\,
x\in\partial D.
\label{3.1}
\end{equation}
Its solution is a family $\{\nu_j\}\subset\mathbb{R}_+$
of eigenvalues enumerated in the nondecreasing order
taking their multiplicities into account and accumulating only to
$+\infty$, as well as the associated orthonormal (in the sense of
$\mathcal{U}$) family of eigenfunctions $\{\varphi_j\}$.
It is not difficult to show (see \cite{4} for instance)
that for all $\mu\in\mathbb{R}$ and
$\alpha\in\mathbb{R}\setminus\{0\}$ the operator $M$ is
$(L,0)$-bounded; moreover, its $L$-spectrum is
\begin{equation}
\sigma^L (M)=\big\{\mu_k=\frac{\alpha}{\mu-\nu_k},\,\,
k\in\mathbb{N}\setminus\{l:\mu=\nu_l\}\big\}\cup\{0\}. \label{3.2}
\end{equation}

Furthermore, for $m\in\mathbb{N}$ construct the operator
$\Lambda=(-\Delta)^m$ with 
$$
\operatorname{dom} \Lambda
=\{u\in W^{l+2m}_2 (D): \Delta^ku(x)=0,\, x\in\partial D,
\, k=\overline{0, m-1}\}.
$$
The family of eigenfunctions of $\Lambda$ coincides with the family
$\{\varphi_j\}$, while its family of eigenvalues is $\{\nu^m_j\}$.
Since their asymptotics is $\nu^m_j\sim j^{\frac{2m}{d}}\to \infty$
as $j\to \infty$, we can choose $m\in\mathbb{N}$ so that,
firstly, the dimension $d$ of the domain $D$ has some acceptable
physical meaning, and secondly, the series
$ \sum_{j=1}^\infty \nu^{-1}_j$ converges.
Then the Green operator of $\Lambda$ is nuclear,
and we take it as $K$. Therefore,
consider the linear stochastic Hoff equation in the form
\begin{equation}
L{\mathaccent"7017 \eta}=M\eta+{\mathaccent"7017 W}_K,
\label{3.3}
\end{equation}
where $L$ and $M$ are defined above, while $N$
is the embedding operator
$\mathbb{I}:\mathfrak{U}\hookrightarrow\mathfrak{F}$
and ${\mathaccent"7017 W}_K={\mathaccent"7017 W}_K(t)$
is the Nelson-Gliklikh derivative of the
$\mathfrak{U}$-valued Wiener $K$-process $W_K=W_K (t)$, for
$t\in\mathbb{R}_+$.

To state initial-final conditions, we need relatively
spectral projectors. In this example we confine the discussion, for
the sake of simplicity, to just two initial-final conditions.
Furthermore, here we present the initial-final conditions satisfying
condition (A1), while in Remark \ref{rmk3.1} below we verify that in this
case, thanks to the structure of $\sigma^L (M)$ in \eqref{3.2}, we
can avoid condition (A1). Thus, take the projectors
$$
P (Q)=\begin{cases}
\mathbb{I}_\mathfrak{U}  (\mathbb{I}_\mathfrak{F})
&\text{if } \mu \neq \nu_j \;\forall  j\in \mathbb{N};\\
\mathbb{I}_\mathfrak{U} -  \sum_{j:\mu 
= \nu_j} \langle \cdot, \varphi_j \rangle_\mathfrak{U} \varphi_j
\Big(\mathbb{ I}_\mathfrak{F} -  \sum_{j:\mu = \nu_j}
\langle \cdot, \psi_j\rangle_\mathfrak{F} \psi_j \Big),
\end{cases} 
$$
where $\{\psi_j\}$ is a family of eigenfunctions $\{\varphi_j\}$
orthonormal in the sense of the inner product
$\langle \cdot, \cdot\rangle_\mathfrak{F}$ in
$\mathfrak{F}$. Furthermore, choose 
$h\in\mathbb{R}_+$ with $h< \max_{j\in\mathbb{N}}
\{|\nu_j|\}$ and construct the projectors
\begin{equation}
\begin{gathered}
P_1 = \mathbb{ I}_\mathfrak{U} -  \sum_{h< |\nu_j|}
\langle \cdot, \varphi_j\rangle_\mathfrak{U} \varphi_j,\quad
Q_1=\mathbb{I}_\mathfrak{F} -  \sum_{h<|\nu_j|}
\langle \cdot, \psi_j\rangle_\mathfrak{F} \psi_j ;\\
P_0 = P-P_1,\quad Q_0 = Q-Q_1.
\end{gathered} \label{3.5}
\end{equation}
Observe that in the construction of these projectors 
condition (A1) holds because
$\sigma_0^L (M)=\{\mu_j\in\sigma^L (M) : |\nu_j|\leq h\}$
and
$\sigma_1^L (M)=\{\mu_j\in\sigma^L (M) : |\nu_j|> h\}$;
hence, $\sigma_0^L (M)\cap\sigma_1^L (M)=\emptyset$.
Finally, choose $\tau_1\in \mathbb{R}_+$ as well as random variables
$\xi_0$ and $\xi_1$ independent of each other and of
stochastic $K$-processes $\eta$ and pose the initial-final conditions
\begin{equation}
\lim_{t\to 0+}P_0(\eta \left(t \right)-\xi_0)=0,
\quad P_1 (\eta(\tau_1)-\xi_1)=0, \label{3.6}
\end{equation}
where 
\begin{equation}\xi_0= \sum_{k=1}^\infty \sqrt{\nu_k}\xi_{0k}\varphi_k,\quad
\xi_1= \sum_{k=1}^\infty \sqrt{\nu_k}\xi_{1k}\varphi_k.\label{3.6.1}
\end{equation}
Applying the results of Section 2 to problem \eqref{3.3}, \eqref{3.6}, 
we obtain the following theorem.

\begin{theorem} \label{thm3.1}
If condition {\rm (A1)} is satisfied then for all numbers
$\mu \in \mathbb{R}$, $\alpha \in \mathbb{R}\setminus\{0\}$
and $\tau_1\in\mathbb{R}_+$, as well as random variables
$\xi_{0k}$ and$\xi_{1k}$ such as 
$\mathbf{D}\xi_{0k}\leq C_0$ and $\mathbf{D}\xi_{1k}\leq C_1$
for some $C_0$, $C_1\in\mathbb{R}_+$ there exists a unique solution
$\eta=\eta(t)$, for $t\in\mathbb{R}_+$,
to problem \eqref{3.3}, \eqref{3.6}; furthermore,
it is of the form
\begin{equation}
\begin{aligned}
\eta (t)
&=(L_{10}^{-1}Q_0+ L_{11}^{-1}Q_1) W_K (t) -L_{11}^{-1}Q_1 W_K (\tau_1)\\
&\quad -S_0P_0 \int^t_0
U_0^{t-s}L_{10}^{-1}Q_0 W_K(s)ds + U_0^t\xi_0+U_1^{t-\tau_1}\xi_1 \\
&\quad -S_1P_1 \int^t_{\tau_1}
U_1^{t-\tau_1-s}L_{11}^{-1}Q_1 W_K(s)ds-M_0^{-1}(\mathbb {I}-Q)N\stackrel
{\circ}{W_K }(t),
\end{aligned} \label{3.7}
\end{equation}
for $t\in\mathbb{R}_+$.
\end{theorem}

Here
\begin{equation}
\begin{gathered}
U_0^t =  \sum_{\nu_j \in\sigma_0^L (M)} e^{t\mu_j}
 \langle \cdot, \varphi_j\rangle_\mathcal{U} \varphi_j, \quad
U_1^t =  \sum_{\nu_j \in\sigma_1^L (M)} e^{t\mu_j}\langle \cdot,
 \varphi_j\rangle_\mathcal{U}\varphi_j,
\\
L_{10}^{-1} =  \sum_{\nu_j \in\sigma_0^L (M)}(\mu-\nu_j)^{-1}
 \langle\cdot, \varphi_j\rangle_\mathcal{U} \varphi_j, 
\\
L_{11}^{-1} =  \sum_{\nu_j \in\sigma_1^L (M)}(\mu-\nu_j)^{-1}
 \langle\cdot, \varphi_j\rangle_\mathcal{U} \varphi_j, 
\\
 S_{10} = \alpha \sum_{\nu_j \in\sigma_0^L (M)}(\mu-\nu_j)^{-1}
 \langle\cdot, \varphi_j\rangle_\mathcal{U} \varphi_j, \\
S_{11} = \alpha \sum_{\nu_j \in\sigma_1^L (M)}(\mu-\nu_j)^{-1}
 \langle\cdot, \varphi_j\rangle_\mathcal{U} \varphi_j, \\
 M_{0}^{-1} = \alpha^{-1} \sum_{\nu_j =\mu}
 \langle\cdot, \psi_j\rangle_\mathcal{F} \psi_j.
 \end{gathered} \label{3.8}
\end{equation}

\begin{remark} \label{rmk3.1} \rm
Verify that in this concrete case condition (A1) could not be satisfied;
however, Theorem \ref{thm3.1} remains valid.
Let all eigenvalues be simple, put 
$\sigma_0^L (M)=\{\mu_j\in\sigma^L (M): \,j=2n\}$
and $\sigma_1^L (M)=\{\mu_j\in\sigma^L (M):
\,j=2n-1\}$, $n\in\mathbb{N}$.
Then $\sigma_0^L (M) \cap \sigma_1^L(M)=\emptyset$.
Nevertheless, \eqref{3.5} and \eqref{3.8} remain valid,
and so \eqref{3.7} holds.
The uniqueness of this solution is proved in the standard fashion
(see Section 2).
\end{remark}

\subsection*{Conclusion}
The next stage of our studies is to carry over the ideas and methods of
the theory of multipoint initial-final problems
for linear Sobolev-type equations from relatively $p$-bounded setup
to relatively $p$-sectorial setup by analogy with \cite{10,7}.
In addition, it would be interesting to extend these ideas and methods
to Sobolev-type equations of high order \cite{8},
and also apply them to inverse problems as in \cite{11}.

\subsection*{Acknowledgments}
The work was supported by Act 211 Government of the Russian Federation, 
contract 02.A03.21.0011.

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\end{document}