Abstract

We consider a stochastic flow on $\mathbb{R}$ generated by an SDE with its drift being a function of bounded variation. We show that the flow is differentiable with respect to the initial conditions. Asymptotic properties of the flow are studied.

Highlights

  • Consider an SDE of the form dφt(x) = α(φt(x))dt + σ(φt(x))dw(t), φ0(x) = x, (0.1)where x ∈ R, (w(t))t≥0 is a one-dimensional Wiener process

  • It is well known that if the coefficients of (0.1) are continuously differentiable and the derivatives are bounded and Hölder continuous there exists a flow of diffeomorphisms for equation (0.1)

  • Under the condition of Lipschitz continuity of the coefficients it was shown the existence of a flow of homeomorphysms

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Summary

Introduction

Where x ∈ R, (w(t))t≥0 is a one-dimensional Wiener process. It is well known (cf. [12]) that if the coefficients of (0.1) are continuously differentiable and the derivatives are bounded and Hölder continuous there exists a flow of diffeomorphisms for equation (0.1). Where x ∈ R, (w(t))t≥0 is a one-dimensional Wiener process It is well known (cf [12]) that if the coefficients of (0.1) are continuously differentiable and the derivatives are bounded and Hölder continuous there exists a flow of diffeomorphisms for equation (0.1). The essential improvement of the results was obtained by Flandoli et al [6] They proved the existence of a flow of diffeomorphysms in the case of a smooth non-degenerate noise and a possibly unbounded Hölder continuous drift term. Note that sometimes the strong solution may exist even if α is a measure In this case the flow may be discontinuous in x.

The main results
Approximation of the SDE by SDEs with smooth coefficients
Local times
Stationary distribution
Example
Full Text
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