Generalized solutions of a stochastic partial differential equation (Q1322909)

The author considers the Cauchy problem of a certain stochastic parabolic partial differential equation, where the initial data and the nonhomogeneous noise term of the equation are given by Schwartz distributions. The existence and uniqueness theorem for the generalized (distributional) solution is proved and an analogue of the Feynman-Kac formula is obtained. Applying the partial Malliavin calculus of Kusuoka- Stroock, the author shows that any generalized solution is a \(C^ \infty\)-function under a condition similar to Hörmander's hypoellipticity condition.