ABSTRACT
This is an expository article concerning certain aspects of generalized solutions to some stochastic partial differential equations (SPDEs). The meaning of a generalized solution to a SPDE varies widely. To be specific, we shall confine ourselves to two types of generalized solutions. For the first type, we regard the generalized solution as a generalized Brownian functional or a Hida distribution in White Noise analysis (see Hida, Kuo, Potthoff and Streit [6]). The white noise analysis was first used by us (Chow [1, 2]) to characterize the generalized solution of a parabolic SPDE with a white noise drift coefficient. This approach has been greatly extended and systematically developed by Potthoff [12]. In particular his method of S-transform has become a very effective tool in solving linear SPDEs with white noise coefficients [13]. The white noise approach to generalized solutions will be discussed in section 3, where the basic ideas are illustrated by some examples from first order SPDEs with a random drift. In the same setting, we consider a second type of generalized solutions: the weak solutions in the PDE sense. In this case the solutions are distribution-valued random fields, and we are interested in the regularity properties of such sample solutions. As an example, the regularity of a generalized solution to the wave equation in ℛ d with d ≥ 3, perturbed by a space-time white noise is studied in section 4. It is well known, in higher dimensions, that the solution of such a SPDE exists only in the distributional sense [15], but little is known about its regularity properties. In this section we will review in more details our recent results in this direction [3]. It will be shown that the spaces S p * ( ℛ d ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203910177/0082baec-0164-4123-9297-401a14f7d1e0/content/eq996.tif"/> in the white noise analysis (see section 2) are suitable for this purpose. To fix the 122notations and to review some basic facts about the White Noise analysis and the wave equation, some preliminary results are given in section 2 without proofs.
