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Sobolev differentiable stochastic flows for SDEs with singular coefficients: applications to the transport equation. (English) Zbl 1333.60127

The authors developed a new approach for constructing a Sobolev differentiable stochastic flow for a stochastic differential equation (SDE) of the form \[ dX_t=b(t, X_t) dt+dB_t, \quad s,t\in \mathbb R, \,\, X_s=x\in \mathbb R^d, \] where \(b:\mathbb R\times\mathbb R^d\to\mathbb R^d\) is a bounded measurable coefficient and \(B\) is a \(d\)-dimensional Brownian motion.
The approach is based on Malliavin calculus ideas coupled with new probabilistic estimates on the spatial weak derivatives of solutions of SDEs. A unique feature of these estimates is that they do not depend on the spatial regularity of the drift coefficient \(b\). The existence of a Sobolev differentiable stochastic flow for an SDE is exploited to obtain a unique weak solution of the Stratonovich stochastic transport equation.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
37H10 Generation, random and stochastic difference and differential equations
37H05 General theory of random and stochastic dynamical systems
34A36 Discontinuous ordinary differential equations
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