NDLI: On Evolution of Random Fields with an Ultra Unbounded Stochastic Source

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Rozanov, Yu. A.
Copyright Year	1993
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 \[d\xi _t = A\xi _t dt + Bd\eta _t^0, \] 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} \ni \xi $ suggested is characterized by meansquare continuity of $(\varphi ,\xi )$ with respect to $\\| \varphi \\|_X $.
Starting Page	316
Ending Page	329
Page Count	14
File Format	PDF
ISSN	0040585X
DOI	10.1137/1138028
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	2
Volume Number	38
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-28
Access Restriction	Subscribed
Subject Keyword	stochastic boundary conditions Sobolev type spaces stochastic evolutional equations
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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