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Zbl 0959.60044
Peszat, Szymon; Zabczyk, Jerzy
Nonlinear stochastic wave and heat equations. (English)
[J] Probab. Theory Relat. Fields 116, No.3, 421-443 (2000). ISSN 0178-8051; ISSN 1432-2064/e

This is a continuation of the authors' paper [Stochastic Processes Appl. 72, No. 2, 187-204 (1997; Zbl 0943.60048)]. Let $\Cal{W}$ be a homogeneous Wiener process valued in $\Cal{S}'(R^d)$ with a positive symmetric tempered measure as a space correlation $ \Gamma $. The stochastic wave equation $\frac {\partial ^2}{\partial t^2}u=\Delta u(u)(u)\dot{\Cal{W}}$ with initial conditions $u(0,x)=u_0(x)$ and the heat equation $\frac \partial {\partial t}u=\Delta u(u)(u)\dot{\Cal{W}}$ with initial condition $ u(0,x)=v_0(x)$ are considered on $ R^d $ with Lipschitz $f$ and $b$. Comparing the paper cited above, in the present paper $ \Gamma $ is extended to a generalised function and the Fourier transform of which is not necessarily absolutely continuous with respect to the Lebegue measure $\lambda $. Let Condition (H) be: there exists $ \kappa $ such that $ \Gamma \kappa \lambda \geq 0 $. Condition (G) is defined by Condition (H) together with $$\Biggl(\log \frac {1}{|y|}\Biggr) I_{|y|\leq 1} \in L( \Gamma),\ d=2; \qquad \frac {1}{|y|^{d-2}}I_{|y|\leq 1} \in L( \Gamma),\ d>2 .$$ The main results are: (1) For $ d\leq 3 $, (G) ensures the existence and uniqueness of the stochastic wave equation on $ 0 \leq t < \infty $ and when (H) is true and $|b(x)|> \varepsilon $ and there exist solutions of the stochastic wave equation of some $u_0(x)$ and $v_0(x)$ for $0 \leq t \leq T$, then (G) holds conversely. (2) The same conclusions as in (1) hold for the heat equation for any dimension $d$.
[Gong Guanglu (Beijing)]

MSC 2000:: *60H15 Stochastic partial differential equations
60G60 Random fields
35K05 Heat equation
35L05 Wave equation

Keywords: stochastic wave; heat equation; homogeneous Wiener process

Citations: Zbl 0943.60048

Cited in: Zbl 0977.60070

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