The Compact Support Property for Hyperbolic SPDEs: Two Contrasting Equations

In this Dissertation we investigate the compact support property of the solutions of two hyperbolic stochastic partial differential equations (SPDEs) whose initial condition function is deterministic and compactly supported. First, we consider the hyperbolic semi-SPDE treated by Allouba and Goodman and others in Financial mathematics modelling. This is an SPDE in both time and space in which all the derivatives (including in the noise) are only with respect to the time parameter, and hence the name semi-SPDE. It turns out that, under appropriate conditions on the diffusion coefficient, the semi-SPDE preserves the compact support property. I.e., starting from a compactly supported initial solution u₀(x), the solution u(t,x) is compactly supported in x for all times t>0.

Second, we consider a rotated wave SPDE in time-space considered by Allouba. Our approach here is to use the Allouba stochastic differential-difference equations (SDDE) approach. In this approach, we start by discretizing space, leaving time continuous, thereby obtaining a simpler version of the SPDE under question. We then resolve the question for the SDDE (or SPDE on the spatial lattice) and then use a limiting argument – as the mesh size of the spatial lattice goes to zero – to transfer regularity results to the associated SPDE. We also prove a noncompact support result for the SPDE. It turns out that in the rotated wave SPDE case, the compact support property is not preserved with positive probability. The contrast between the two SPDEs’ behaviors is due to the extra differentiation in space in the second SPDE which plays a crucial role in the behavior of solutions.