Girsanov Transformations for Non-Symmetric Diffusions
Abstract.Let X be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form (ℰ, 𝓓 (ℰ)) on L2(E ;m). For u ∈ 𝓓(ℰ)e, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process Y of X . First, let be the dual process of X and Ŷ the Girsanov transformed process of . We give a necessary and sufficient condition for (Y , Ŷ to be in duality with respect to the measure e2um. We also construct a counterexample, which shows that this condition may not be satisfied and hence (Y , Ŷ ) may not be dual processes. Then we present a sufficient condition under which Y is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.