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Coupling polynomial Stratonovich integrals: the two-dimensional Brownian case. (English) Zbl 1387.60120

Summary: We show how to build an immersion coupling of a two-dimensional Brownian motion \((W_1, W_2)\) along with \(\binom{n}{2} + n= \frac{1}{2}n(n+1)\). integrals of the form \(\int W_1^iW_2^j \circ\operatorname{d}W_2\), where \(j=1,\ldots,n\) and \(i=0,\ldots, n-j\) for some fixed \(n\). The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals and is part of an ongoing research programme aimed at gaining a better understanding of when it is possible to couple not only diffusions but also multiple selected integral functionals of the diffusions.

MSC:

60J60 Diffusion processes
60J65 Brownian motion
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