Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
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

Highlights
Master Server