Quasi-invariance of Lebesgue measure under the homeomorphic flow generated by SDE with non-Lipschitz coefficient

AbstractWe consider the stochastic flow generated by Stratonovich stochastic differential equations with non-Lipschitz drift coefficients. Based on the author's previous works, we show that if the generalized divergence of the drift is bounded, then the Lebesgue measure on Rd is quasi-invariant under the action of the stochastic flow, and the explicit expression of the Radon–Nikodym derivative is also presented. Finally we show in a special case that the unique solution of the corresponding Fokker–Planck equation is given by the density of the stochastic flow