Appl Math Optim 38:95–107 (1998)
1998 Springer-Verlag New York Inc.
Generalized Solutions of Linear Parabolic Stochastic Partial
and H. Watanabe
Lehrstuhl f¨ur Mathematik V, Universit¨at Mannheim,
D-68131 Mannheim, Germany
Department of Applied Mathematics, Faculty of Science,
Okayama University of Science, Okayama, Japan
Abstract. Existence and uniqueness theorems for parabolic stochastic partial dif-
ferential equations with space–time white noise are proved. The method is a combi-
nation of the characterization theorem for Hida distributions with the Feynman–Kac
and Girsanov formulae.
Key Words. Stochastic partial differential equations, White noise analysis, S-
transform, Feynman–Kac formula, Girsanov formula.
AMS Classiﬁcation. 60H07, 60H15.
The purpose of this paper is to show that certain stochastic partial differential equations
(SPDEs) which are too singular to be solved in the more traditional frameworks have
solutions which are generalized Brownian functionals in the sense of Hida. We are
concerned with SPDEs of the following two types:
u(t, x) = Lu(t,x) + η(t, x)u(t, x) (1.1)
u(t, x) = Lu(t,x) +
The ﬁrst two authors were partially supported and supported, respectively, by the DFG.