A stochastic wave equation in two space dimensions: smoothness of the law
One considers real-valued solutions of stochastic wave equations of the type \[ \left( {\partial^2\over\partial t^2}-\Delta \right) u(t,x) =\sigma(u(t,x))F(dt,dx)+b(u(t,x)) \] with two-dimensional spatial variable \(x\) and given initial conditions \(u(0,x)\) and \({\partial u\over \partial t}u(0,x)\). The noise \(F(dt,dx)\) is assumed to be white in time, and its spatial correlation is described by a function \(f\). Under an integral condition on \(f\), the existence and uniqueness of a solution \(u(t,x)\) is proved. Then the Malliavin calculus is applied to the smoothness of its law.
stochastic partial differential equation
scientific article; zbMATH DE number 1359665