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Teoriya Veroyatnostei i ee Primeneniya, 1993, Volume 38, Issue 2, Pages 356–373 (Mi tvp3946)  

This article is cited in 1 scientific paper (total in 1 paper)

On evolution of random fields with an ultra unbounded stochastic source

Yu. A. Rozanov

Steklov Mathematical Institute, Russian Academy of Sciences
Full-text PDF (845 kB) Citations (1)
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
English version:
Theory of Probability and its Applications, 1993, Volume 38, Issue 2, Pages 316–329
DOI: https://doi.org/10.1137/1138028
Bibliographic databases:
Language: Russian
Citation: Yu. A. Rozanov, “On evolution of random fields with an ultra unbounded stochastic source”, Teor. Veroyatnost. i Primenen., 38:2 (1993), 356–373; Theory Probab. Appl., 38:2 (1993), 316–329
Citation in format AMSBIB
\Bibitem{Roz93}
\by Yu.~A.~Rozanov
\paper On evolution of random fields with an ultra unbounded stochastic source
\jour Teor. Veroyatnost. i Primenen.
\yr 1993
\vol 38
\issue 2
\pages 356--373
\mathnet{http://mi.mathnet.ru/tvp3946}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1317984}
\zmath{https://zbmath.org/?q=an:0808.60055}
\transl
\jour Theory Probab. Appl.
\yr 1993
\vol 38
\issue 2
\pages 316--329
\crossref{https://doi.org/10.1137/1138028}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1993NY72300009}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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