Investigation of properties of the solutions of hyperbolic stochastic PDEs

In this paper we investigate the compact support property of the solutions of hyperbolic Stochastic PDE (SPDE) providing that initial condition function is deterministic and has compact support property. First, to approach this problem, we consider semi-SPDE. It turns out that in the semi-SPDE case solution u (t, x) preserve compact support property. When we consider SPDE, we use the stochastic differential-difference equations (SDDE) approach. It turns out that in SPDE case solution u (t, x) does not preserve compact support property. So, if we compare the semi-SPDE and SPDE then it becomes obvious that differentiation in space in SPDE plays crucial role and influence the behavior of the solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)