Stochastic evolution systems with constant coefficients

Abstract

While solvability of a single stochastic hyperbolic or parabolic equation is well known, the problem remains mostly open for stochastic evolution systems. The paper investigates well-posedness and stability in Sobolev spaces on $$\mathbb {R}^{\hbox {d}}$$ of the initial value problem for systems of stochastic evolution equations with constant coefficients and multiplicative time-only Gaussian white noise. A general criterion for well-posedness is derived in terms of sums of certain Kronecker products of the system matrices, and a stochastic analogue of the Petrowski parabolicity condition is proposed.