BPE and a Noncausal Girsanov’s Theorem

Abstract

Brownian particle equation, BPE for short, is a class of stochastic partial differential equations of the first order that include the white noise as coefficients, at least in their principal part. This class of SPDEs was introduced by the author as a stochastic model for diffusive transport phenomenon that appears in various domains of mathematical sciences. As an application of the study to stochastic calculus, we have shown in the note of Ogawa (Rendi Conti Acad. Nazionale delle Sci. detta dei XL, 119, XXV, 125–139, 2001) that Girsanov’s theorem can be derived in an elementary way by using BPE models. Our objective in this note is to show some recent results on basic properties of the classical solution of Cauchy problem for BPE, then as an application of those results we are to establish a generalized form for Girsanov’s theorem.