Sobolev Differentiable Stochastic Flows of SDE`s with Singular Drift

Details Sobolev Differentiable Stochastic Flows of SDE`s with Singular Drift Sobolev Differentiable Stochastic Flows of SDE`s with Singular Drift

2012-04-17

arxiv.org/abs/1204.3867v1

Mathematics

Probability

26 pages

Scientific paper

In this paper, we establish the existence of a Sobolev differentiable stochastic flow for a stochastic differential equation (SDE) of the form dX_t =b(t,X_t)dt+dB_t, s,t in R; X_s =x in R^d, driven by a bounded measurable drift coefficient b :RxR^d-->R^d and a d-dimensional Brownian motion B. More specifically, we show that the stochastic flow lives in the space L^2(Omega; W^1,p(R^d,w)) for all s, t and all p>1, where W^1,p(R^d,w) denotes a weighted Sobolev space with weight w possessing a p-th moment with respect to Lebesgue measure on R^d. As a consequence of our approach we obtain a new existence theorem of Sobolev differentiable flows for classical one-dimensional ODE`s driven by discontinuous vector fields.

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