On evolution of random fields with an ultra unbounded stochastic source

Yu. A. Rozanov

Teoriâ veroâtnostej i ee primeneniâ, Tome 38 (1993) no. 2, pp. 356-373

Résumé

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 $.

Détail

Keywords: stochastic evolutional equations, stochastic boundary conditions, Sobolev type spaces.