Conditional transformation of drift formula and potential theory for \(\Delta +b(\cdot)\cdot \nabla\) (Q1104638)

Using conditional Brownian motion and the transformation of drift formula (of Cameron-Martin, Girsanov, Maruyama) we give integral conditions on a vector field b which imply the harmonic measures and Green functions for \(\Delta\) and \(\Delta +b(\cdot)\cdot \nabla\) on a bounded Lipschitz domain D are equivalent. By equivalent we mean there exist two-sided inequalities with constants depending only on b and D. This enables one to conclude the potential theory for \(\Delta +b(\cdot)\cdot \nabla\) on D and \(\Delta\) on D are the same.