Stochastic flows and Bismut formulas for stochastic Hamiltonian systems

We first consider the stochastic differential equations (SDE) without global Lipschitz conditions, and give sufficient conditions for the SDEs to be strictly conservative. In particular, a criteria for stochastic flows of diffeomorphisms defined by SDEs with non-global Lipschitz coefficients is obtained. We also use Zvonkin’s transformation to derive a stochastic flow of C 1 -diffeomorphisms for non-degenerate SDEs with Hölder continuous drifts. Next, we prove a Bismut type formula for certain degenerate SDEs. Lastly, we apply our results to stochastic Hamiltonian systems, which in particular covers the following stochastic nonlinear oscillator equation z ̈ t = c 0 z ̇ t − z t 3 + Θ ( z t ) w ̇ t , ( z 0 , z ̇ 0 ) = ( z , u ) ∈ R 2 , where c 0 ∈ R , Θ ∈ C ∞ ( R ) has a bounded first order derivative, and w ̇ t is a one dimensional Brownian white noise.