Electronic Journal of Probability

Coupling polynomial Stratonovich integrals: the two-dimensional Brownian case

Sayan Banerjee and Wilfrid Kendall

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We show how to build an immersion coupling of a two-dimensional Brownian motion $(W_1, W_2)$ along with \(\binom{n} {2} + n= \tfrac 12n(n+1)\). 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 (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004), Kendall (2007), Kendall (2009), Kendall (2013), Banerjee and Kendall (2015), Banerjee, Gordina and Mariano (2016)) 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.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 24, 43 pp.

Received: 3 May 2017
Accepted: 11 February 2018
First available in Project Euclid: 27 February 2018

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Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]

Brownian motion coupling elliptic diffusion faithful coupling Heisenberg group hypoelliptic diffusion immersion coupling Kolmogorov diffusion Lévy stochastic area Markovian coupling monomial nilpotent diffusion parabolic Hörmander condition reflection coupling stochastic differential equation Stratonovich integral total variation distance

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Banerjee, Sayan; Kendall, Wilfrid. Coupling polynomial Stratonovich integrals: the two-dimensional Brownian case. Electron. J. Probab. 23 (2018), paper no. 24, 43 pp. doi:10.1214/18-EJP150. https://projecteuclid.org/euclid.ejp/1519722153

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