Drift Transformations of Symmetric Diffusions, and Duality

Starting with a symmetric Markov diffusion process X (with symmetry measure m and L(m) infinitesimal generator A) and a suitable core C for the Dirichlet form of X, we describe a class of derivations defined on C. Associated with each such derivation B is a drift transformation of X, obtained through Girsanov’s theorem. The transformed process X is typically non-symmetric, but we are able to show that if the “divergence” of B is positive, then m is an excessive measure for X , and the L(m) infinitesimal generator of X is an extension of f 7→ Af + B(f). The methods used are mainly probabilistic, and involve the notions of even and odd continuous additive functionals, and Nakao’s stochastic divergence.