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Flow of homeomorphisms and stochastic transport equations. (English) Zbl 1125.60083

Summary: We consider Stratonovich stochastic differential equations with drift coefficient \(A_0\) satisfying only the condition of continuity
\[ |A_0(x)-A_0(y)|\leq C|x-y|r(|x-y|^2)\text{ for all }|x-y|\leq c_0, \]
where \(r\) is a positive \(C^1\) function defined on a neighborhood \(]0, c_0]\) of 0 such that \(\int^{c_0}_0\frac{ds}{sr(s)}=+\infty\) (Osgood condition), and \(s\to r(s)\) is decreasing while \(s\to sr(s^2)\) is increasing. We prove that the equation defines a flow of homeomorphisms if the diffusion coefficients \(A_1,\dots, A_N\) are in \(C^{3+\delta}_b(\mathbb R^d,\mathbb R^d)\). If \(r(s)=(\log\,\frac1s)\cdots(\log_k\frac1s)\), we prove limit theorems for the Wong-Zakai approximation as well as for regularizing the drift \(A_0\). As an application, we solve a class of stochastic transport equations.

MSC:

60J60 Diffusion processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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