Solutions by Variational Method

Abstract

The variational method for solving stochastic partial differential equations (SPDE’s) of evolutionary type involves recasting them as SDE’s in a Gelfand triplet of Hilbert or Banach spaces V↪H↪V ^∗, where the embeddings are dense and continuous. We discuss only the case of separable Hilbert spaces. In order to construct a weak solution, we assume that the embeddings are compact, and use the “method of compact embedding” introduced in Chap. 3, together with the stochastic analogue of Lions’ theorem from Chap. 1. The solution is an H-valued stochastic process with continuous sample paths. Under the assumption of monotonicity, we obtain unique strong solution using pathwise uniqueness.

We also present the result on the existence of strong solutions, following the ideas in Prévôt and Röckner (A Concise Course on Stochastic Partial Differential Equations. LNM, vol. 1905. Springer, Berlin, 2007). Assuming that the coefficients are monotone suffices to produce a strong solution without the need for compactness of the embeddings in the Gelfand triplet. Using again the stochastic analogue of Lions’ theorem allows to put the solution in H and assure continuity of its sample paths. We also present results on Markov and strong Markov properties of strong variational solutions.