Carleman and Observability Estimates for Stochastic Wave Equations

Abstract

Based on a fundamental identity for stochastic hyperbolic-like operators, we derive in this paper a global Carleman estimate (with singular weight function) for stochastic wave equations. This leads to an observability estimate for stochastic wave equations with nonsmooth lower order terms. Moreover, the observability constant is estimated by means of an explicit function of the norm of the coefficients involved in the equation. An application to the state observation problem for semilinear stochastic wave equations is also given.