Electronic Journal of Probability

Coupling polynomial Stratonovich integrals: the two-dimensional Brownian case

Sayan Banerjee and Wilfrid Kendall

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1519722153

Digital Object Identifier
doi:10.1214/18-EJP150

Mathematical Reviews number (MathSciNet)
MR3771761

Zentralblatt MATH identifier
1387.60120

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

Keywords
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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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


Export citation

References

  • [1] Aldous, D. J. (1983). Random walks on finite groups and rapidly mixing Markov chains. Séminaire de probabilités de Strasbourg 17, 243–297.
  • [2] Banerjee, S., M. Gordina, and P. Mariano (2016). Coupling in the Heisenberg group and its applications to gradient estimates. The Annals of Probability (to appear). arXiv:1610.06430, 33pp.
  • [3] Banerjee, S. and W. S. Kendall (2015). Coupling the Kolmogorov Diffusion: maximality and efficiency considerations. Advances in Applied Probability 48(A), 15–35.
  • [4] Banerjee, S. and W. S. Kendall (2017). Rigidity for Markovian maximal couplings of elliptic diffusions. Probability Theory and Related Fields 168(1), 55–112.
  • [5] Baudoin, F. (2004). An introduction to the geometry of stochastic flows. Imperial College Press.
  • [6] Ben Arous, G., M. Cranston, and W. S. Kendall (1995). Coupling constructions for hypoelliptic diffusions: Two examples. In M. Cranston and M. Pinsky (Eds.), Stochastic Analysis: Proceedings of Symposia in Pure Mathematics 57, Volume 57, Providence, RI Providence, pp. 193–212. American Mathematical Society.
  • [7] Burdzy, K. and W. S. Kendall (2000). Efficient Markovian couplings: examples and counterexamples. The Annals of Applied Probability 10(2), 362–409.
  • [8] Chen, M.-F. and S.-F. Li (1989). Coupling methods for multidimensional diffusion processes. The Annals of Probability, 151–177.
  • [9] Cranston, M. (1991). Gradient estimates on manifolds using coupling. Journal of Functional Analysis 99(1), 110–124.
  • [10] Cranston, M. (1992). A probabilistic approach to gradient estimates. Canad. Math. Bull 35(1), 46–55.
  • [11] Ernst, P., W. S. Kendall, G. O. Roberts, and J. S. Rosenthal (2017). MEXIT: Maximal uncoupling times for Markov processes. arXiv:1702.03917, 34pp.
  • [12] Friz, P. K. and M. Hairer (2014). A Short Course on Rough Paths: With an Introduction to Regularity Structures. Universitext. Springer International Publishing.
  • [13] Goldstein, S. (1979). Maximal coupling. Probability Theory and Related Fields 46(2), 193–204.
  • [14] Griffeath, D. (1975). A maximal coupling for Markov chains. Probability Theory and Related Fields 31(2), 95–106.
  • [15] Grubmüller, H. and P. Tavan (1994). Molecular dynamics of conformational substates for a simplified protein model. The Journal of chemical physics 101(6), 5047–5057.
  • [16] Ikeda, N. and S. Watanabe (1981). Stochastic differential equations and diffusion processes. Amsterdam: North Holland / Kodansha.
  • [17] Karatzas, I. and S. Shreve (2012). Brownian motion and stochastic calculus, Volume 113. Springer Science & Business Media.
  • [18] Kendall, W. S. (1986). Nonnegative Ricci curvature and the Brownian coupling property. Stochastics: An International Journal of Probability and Stochastic Processes 19(1–2), 111–129.
  • [19] Kendall, W. S. (2007). Coupling all the Lévy stochastic areas of multidimensional Brownian motion. The Annals of Probability, 935–953.
  • [20] Kendall, W. S. (2010). Coupling time distribution asymptotics for some couplings of the Lévy stochastic area. In N. H. Bingham and C. M. Goldie (Eds.), Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman, London Mathematical Society Lecture Note Series, Chapter 19, pp. 446–463. Cambridge: Cambridge University Press.
  • [21] Kendall, W. S. (2015). Coupling, local times, immersions. Bernoulli 21(2), 1014–1046.
  • [22] Kendall, W. S. and C. J. Price (2004). Coupling iterated Kolmogorov diffusions. Electronic Journal of Probability 9(Paper 13), 382–410.
  • [23] Kuwada, K. (2009). Characterization of maximal Markovian couplings for diffusion processes. Electron. J. Probab 14(25), 633–662.
  • [24] Lindvall, T. and L. C. G. Rogers (1986). Coupling of multidimensional diffusions by reflection. The Annals of Probability, 860–872.
  • [25] Lobry, C. (1970). Contrôlabilité des systèmes non linéaires. SIAM Journal on Control and Optimization 8(4), 573–605.
  • [26] Markus, L. and A. Weerasinghe (1988). Stochastic oscillators. Journal of Differential Equations 71(2), 288–314.
  • [27] Neuenschwander, D. (1996). Probabilities on the Heisenberg group, Volume 1630 of Lecture Notes in Mathematics. Berlin: Springer-Verlag.
  • [28] Pitman, J. W. (1976). On coupling of Markov chains. Probability Theory and Related Fields 35(4), 315–322.
  • [29] Rosenthal, J. S. (1997). Faithful Couplings of Markov Chains: Now Equals Forever. Advances in Applied Mathematics 18(3), 372–381.
  • [30] Sverchkov, M. Y. and S. N. Smirnov (1990). Maximal coupling for processes in $D[0,\infty ]$. In Soviet Math. Dokl, Volume 41, pp. 352–354.
  • [31] Villani, C. (2006). Hypocoercive diffusion operators. In International Congress of Mathematicians, Volume 3, pp. 473–498.
  • [32] Von Renesse, M.-K. (2004). Intrinsic Coupling on Riemannian Manifolds and Polyhedra. Electronic Journal of Probability 9, 411–435.