The finite speed of propagation for solutions to nonlinear stochastic wave equations driven by multiplicative noise

We prove that the solutions to the stochastic wave equation in O⊂Rd, , for 1⩽d<∞, where g is a continuous function with polynomial growth of order less or equal to and σ is Lipschitz with σ(0)=0, propagate with finite speed. This result resembles the classical finite speed of propagation result for the solution to the Klein–Gordon equation and extends to equations with dissipative damping. A similar result follows for the equation with additive noise of the form , where F(t)=F(t,ξ) has compact support (in ξ) for each t>0.