Uniqueness and absolute continuity of weak solutions for parabolic SPDE's
Let \(E\) and \(E'\) be quasicomplete locally convex linear spaces and let \(F\) be a linear subspace of the algebraic dual of \(E'\). In this very general framework the authors define a stochastic integral and introduce martingale problem on the canonical space of \(\sigma (F,E')\)-continuous functions. Then necessary and sufficient conditions are given for absolute continuity of solutions to two different martingale problems. Roughly speaking, it is shown that absolute continuity occurs if and only if the corresponding martingale problems differ in the part of finite variation only. Given absolute continuity of solutions criteria for uniqueness are also obtained. The paper extends all known results on solving martingale problem in infinite dimensions by absolutely continuous change of measure. In the last part of the paper applications to some stochastic partial differential equations are given.
stochastic partial differential equations
scientific article; zbMATH DE number 627457