Note on local mixing techniques for stochastic differential equations
Volume 8, Issue 1 (2021), pp. 1–15
Pub. online: 16 March 2021
Type: Research Article
Open Access
Open Access
Received
13 October 2020
13 October 2020
Revised
25 February 2021
25 February 2021
Accepted
25 February 2021
25 February 2021
Published
16 March 2021
16 March 2021
Abstract
The paper discusses several techniques which may be used for applying the coupling method to solutions of stochastic differential equations (SDEs). The coupling techniques traditionally consist of two components: one is local mixing, the other is recurrence. Often in the articles they do not split. Yet, they are quite different in their nature, and this paper separates them, concentrating only on the former.
Most of the techniques discussed here work in dimension $d\ge 1$, although, in $d=1$ there is one additional option to use intersections of trajectories, which requires nothing but the strong Markov property and nondegeneracy of the diffusion coefficient. In dimensions $d>1$ it is possible to use embedded Markov chains either by considering discrete times $n=0,1,\dots $, or by arranging special stopping time sequences and to use the local Markov–Dobrushin (MD) condition, which is one of the most efficient versions of local mixing. Further applications may be based on one or another version of the MD condition; respectively, this paper is devoted to various methods of verifying one or another form of it.
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