Different types of SPDEs: Existence, uniqueness, and Girsanov's theorem.

摘要

We prove Girsanov's theorem for continuous orthogonal martingale measures. We then define space-time SDEs, and use Girsanov's theorem to establish a one-to-one correspondence between solutions of two space-time SDEs differing only by a drift coefficient. For such stochastic equations, we give necessary conditions under which the laws of their solutions are absolutely continuous with respect to each other. Using Girsanov's theorem and the rotational equivalence between space-time SDEs and wave SPDEs, we prove additional existence and uniqueness results for both classes of SPDEs. The same one-to-one correspondence and absolute continuity theorems are proved for the stochastic heat and wave equations.; We also have a non-nonstandard proof of Reimers' existence theorem for heat SPDEs, under the assumptions of continuity and linear growth on the diffusion coefficient. This is accomplished by first discretizing space but leaving the time parameter continuous. We then use the tightness of the approximating interacting diffusions to extract a subsequential limit which solves a martingale problem that is equivalent to the SPDE under consideration.