Nonlinear stochastic wave and heat equations. (English) Zbl 0959.60044

This is a continuation of the authors’ paper [Stochastic Processes Appl. 72, No. 2, 187-204 (1997; Zbl 0943.60048)]. Let \(\mathcal{W}\) be a homogeneous Wiener process valued in \(\mathcal{S}'(R^d)\) with a positive symmetric tempered measure as a space correlation \( \Gamma \). The stochastic wave equation \(\frac {\partial ^2}{\partial t^2}u=\Delta u(u)(u)\dot{\mathcal{W}}\) with initial conditions \(u(0,x)=u_0(x)\) and the heat equation \(\frac \partial {\partial t}u=\Delta u(u)(u)\dot{\mathcal{W}}\) 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 \(d\).


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G60 Random fields
35K05 Heat equation
35L05 Wave equation


Zbl 0943.60048
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