Drift Transformations of Symmetric Diffusions, and
Duality
### Drift Transformations of Symmetric Diffusions, and
Duality

#### P.J. Fitzsimmons

(To appear in *Infinite Dimensional Analysis, Quantum Probability and Related Topics*)

Starting with a symmetric Markov diffusion process *X* (with symmetry measure *m* and *L*^{2}(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*^{B} 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*^{B}, and the *L*^{2}(m) infinitesimal generator of *X*^{B} is an extension of *f --> 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.

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(Version of June 19, 2006)

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June 19, 2006; July 10, 2007