Stochastic heat equation with white-noise drift
The authors study the existence and uniqueness of solutions for a one-dimensional anticipative stochastic evolution equation on the real line \[ u(t,x)=\int_{\mathbb R} p(0,t,y,x)u_0(y) dy + \int_{\mathbb R} \int_0^t p(s,t,y,x)F(s,y,u(s,y)) dW_{s,y} \] driven by a two-parameter Wiener process \(W_{t,x}\) and based on a stochastic semigroup defined by the kernel \(p(s,t,y,x)\). This kernel is supposed to be measurable w.r.t. the increments of the Wiener process on \([s,t]\times\mathbb R\). The results are based on \(L^p\)-estimates for the Skorokhod integral. As an application they establish the existence of a weak solution for the following heat equation on the real line subject to white noise drift, \[ \partial_t u(t,x)=\partial_x^2 u(t,x)+\dot{v}(t,x)\partial_x u(t,x)+F(t,x,u)\partial_t\partial_x W(t,x), \] where \(\dot{v}\) is a white noise in time.
anticipative stochastic evolution equation
scientific article; zbMATH DE number 1453271