Small perturbations in a hyperbolic stochastic partial differential equation
The authors investigate a hyperbolic stochastic partial differential equation \[ \frac {\partial ^2 X_{s,t}}{\partial s\partial t} = a_3(X_{s,t}) \dot W_{s,t}+ a_4(X_{s,t}) + a_1(s,t)\frac {\partial X_{s,t}} {\partial t} + a_2(s,t)\frac {\partial X_{s,t}}{\partial s}, \quad (s,t)\in [0,1]^2, \] with initial datum \(X_{s,t}=X_0\) for \(st=0\), where \(\dot W_{s,t}\) denotes the white noise on \([0,1]^2\). Solution to this problem is defined by the variation of constants formula \[ X_{s,t} = X_0 +\int _{[0,s]\times [0,t]} \gamma _{s,t}(u,v) \left [ a_3(X_{u,v}) dW_{u,v} + a_4(X_{u,v}) du dv\right ], \tag{1} \] \(\gamma \) being the Green function of the second order operator \(\partial ^2 f/\partial s\partial t -a_1(s,t)\partial f/ \partial t -a_2(s,t)\partial f/\partial s\). The functions \(a_1\), \(a_2\) are assumed to be bounded and differentiable with bounded derivatives, while the functions \(a_3\), \(a_4\) are assumed to have bounded derivatives of all orders. Under suitable non-degeneracy assumptions on \(a_3(X_0)\) the authors prove that the law of the random variable \(X_{z}\), \(z\in (0,1)^2\), is absolutely continuous with respect to Lebesgue measure on \(\mathbb R\) and its density \(p_{z}\) satisfies \(p_{z}\in \mathcal C^\infty (\mathbb R)\). Moreover, the function \(z\mapsto p_{z}(y)\) is shown to be Lipschitz continuous for any \(y\in \mathbb R\). Related results were previously established [see \textit{C. Rovira} and \textit{M. Sanz-Solé}, J. Theor. Probab. 9, No. 4, 863-901 (1996; Zbl 0878.60040)] under a different set of hypotheses. In the second part of the paper, diffusion coefficients of the form \(a_3(u)=\varepsilon \bar a_3(u)\), \(\varepsilon >0\), are considered and the behaviour of the density \(p^\varepsilon _{z}(\cdot )\) as \(\varepsilon \searrow 0\) is described.
hyperbolic stochastic partial differential equations
scientific article; zbMATH DE number 1134839