On Evolution of Random Fields with an Ultra Unbounded Stochastic Source
Yu. A. Rozanov · Theory of Probability and Its Applications · 1994
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 \[d\xi _t = A\xi _t dt + Bd\eta _t^0, \] where the elliptic operator $A = \sum_{| k | \leq 2p} {a_k \partial ^k } \leq 0$ 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${\bf W}$ such that a unique solution $\xi \in {\bf W}$ 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 $\xi ^ + \in {\bf W}$. 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 ${\bf W} i \xi $ suggested is characterized by meansquare continuity of $(\varphi ,\xi )$ with respect to $\| \varphi \|_X $.