This article is cited in
1 paper
On evolution of random fields with an ultra unbounded stochastic source
Yu. A. Rozanov Steklov Mathematical Institute, Russian Academy of Sciences
Abstract:
The stochastic model considered is represented, in particular, by a generalized random field
$\xi _t $ on
$R^d $ the evolution of which obeys the generalized stochastic partial differential equation
$$
d\xi _t=A\xi _tdt+Bd\eta_t^0,
$$
where the elliptic operator
$A=\sum_{|k|\le 2p}a_k\partial^k\le 0$ is a drift-operator and the general differential operator
$B=\sum_{|k|\le p}b_k\partial^k$ a diffusion coefficient strengthening the stochastic source
$d\eta_t^0$ of the type of white noise. Considering this equation in a subregion
$G\subseteq R^d $ we encounter a variety of solutions, and one can be interested in identifying an appropriate
$\xi _t$,
$t\in I=(t_0,t_1)$ given an initial
$\xi_{t_0}$, say, by means of certain boundary conditions on the boundary
$\partial G$, that is, on a lateral boundary
$\partial G\times I$ of a spacetime cylinder
$G\times I$. In accordance with this we suggest an appropriate stochastic Sobolev space
$\mathbf{W}$ such that a unique solution
$\xi\in\mathbf{W}$ does exist having a boundary trace of its generalized normal derivatives
$\partial^k\xi$,
$k=0,\ldots,p-1$, on
$\partial G\times I$ which satisfy the generalized Dirichlet type boundary conditions
$$
\partial^k\xi=\partial^k\xi^+,\quad k=0,\ldots,p-1,
$$
with an arbitrary given stochastic sample
$\xi^+\in\mathbf{W}$.The generalized stochastic differential equations have been of interest for years; various approaches exist for obtaining for a given initial state
$\xi_{t_0}=0$, say, and an acceptable stochastic source, a unique solution in an appropriate function space, and this uniqueness implies that boundary conditions (if there are any) are zero for
$\xi=0$. Our approach is different and based on the application of a test function space
$X=[C_0^\infty(G\times I)]$ which appears as a closure of
$\varphi\in C_0^\infty(G\times I)$ with respect to an appropriate norm
$\|\varphi\|_X $, and the stochastic class
$\mathbf{W}\ni\xi$ suggested is characterized by meansquare continuity of
$(\varphi,\xi)$ with respect to
$\|\varphi\|_X $.
Keywords:
stochastic evolutional equations, stochastic boundary conditions, Sobolev type spaces. Received: 22.09.1992
Fulltext:
PDF file (845 kB)