Electronic Journal of Probability

Reflected Brownian motion: selection, approximation and linearization

Marc Arnaudon and Xue-Mei Li

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Abstract

We construct a family of SDEs with smooth coefficients whose solutions select a reflected Brownian flow as well as a corresponding stochastic damped transport process $(W_t)$, the limiting pair gives a probabilistic representation for solutions of the heat equations on differential 1-forms with the absolute boundary conditions. The transport process evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary where it is driven by the boundary local time, and with its normal part erased at the end of the excursions to the boundary of the reflected Brownian motion. On the half line, this construction selects the Skorohod solution (and its derivative with respect to initial points), not the Tanaka solution; on the half space it agrees with the construction of N. Ikeda and S. Watanabe [29] by Poisson point processes. The construction leads also to an approximation for the boundary local time, in the topology of uniform convergence but not in the semi-martingale topology, indicating the difficulty in proving convergence of solutions of a family of random ODE’s to the solution of a stochastic equation driven by the local time and with jumps. In addition, we obtain a differentiation formula for the heat semi-group with Neumann boundary condition and prove also that $(W_t)$ is the weak derivative of a family of reflected Brownian motions with respect to the initial point.

Article information

Source
Electron. J. Probab. Volume 22 (2017), paper no. 31, 55 pp.

Dates
Received: 31 March 2016
Accepted: 27 February 2017
First available in Project Euclid: 25 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1490407496

Digital Object Identifier
doi:10.1214/17-EJP41

Subjects
Primary: 60G

Keywords
Brownian motion reflection local time boundary stochastic flow heat equation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Arnaudon, Marc; Li, Xue-Mei. Reflected Brownian motion: selection, approximation and linearization. Electron. J. Probab. 22 (2017), paper no. 31, 55 pp. doi:10.1214/17-EJP41. https://projecteuclid.org/euclid.ejp/1490407496.


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References

  • [1] H. Airault, Problèmes de Dirichlet-Neumann étalés et fonctionnelles multiplicatives associées. (French) Séminaire sur les Équations aux Dérivées Partielles (1974-1975), III, Exp. No. 3, 25 pp. Collège de France, Paris, 1975.
  • [2] H. Airault, Perturbations singulières et solutions stochastiques de problèmes de D. Neumann-Spencer. (French) J. Math. Pures Appl. (9) 55 (1976), no. 3, 233–267
  • [3] S. Andres, Pathwise differentiability for SDEs in a smooth domain with reflection, Electron. J. Probab. 16 (2011), no. 28, 845-879.
  • [4] M. Arnaudon, K. A. Coulibaly and A. Thalmaier, Horizontal diffusion in $C^1$ path space Séminaire de Probabilités XLIII, 73-94, Lecture Notes in Math., 2006, Springer, Berlin, 2011
  • [5] M. Arnaudon, B. Driver and A. Thalmaier, Gradient estimates for positive harmonic functions by stochastic analysis, Stochastic Process. Appl. 117 (2007), no. 2, 202-220.
  • [6] M. Arnaudon and A. Thalmaier, Stability of stochastic differential equations in manifolds, Séminaire de Probabilités, XXXII, 188–214, Lecture Notes in Math., 1686, Springer, Berlin, 1998.
  • [7] M. Arnaudon and A. Thalmaier, Complete lifts of connections and stochastic Jacobi fields, J. Math. Pure Appl. (9) 77 (1998), no. 3, 283-315.
  • [8] M. Arnaudon, Xue-Mei Li and A. Thalmaier, Manifold-valued martingales, changes of probabilities, and smoothness of finely harmonic maps, Ann. Inst. H. Poincaré Probab. Statist. 35 (1999), no. 6, 765-791.
  • [9] A. Bensoussan and J.-L. Lions. Contrôle impulsionnel et inéquations quasi variationnelles. (French) [Impulse control and quasivariational inequalities] Méthodes Mathématiques de l’Informatique [Mathematical Methods of Information Science], 11. Gauthier-Villars, Paris, 1982
  • [10] J. M. Bismut, Large deviations and the Malliavin calculus. Progress in Mathematics, 45. Birkhäuser Boston, Inc., Boston, MA, (1984).
  • [11] K. Burdzy, Differentiability of stochastic flow of reflected Brownian motions, Electron. J. Probab. 14 (2009), no. 75, 2182–2240.
  • [12] K. Burdzy, Zhen-Qing Chen, and P. Jones. Synchronous couplings of reflected Brownian motions in smooth domains. Illinois J. Math. 50 (2006), no. 1-4, 189–268
  • [13] Zhen-Qing Chen, P. J. Fitzsimmons, and R. Song. Crossing estimates for symmetric Markov processes. Probab. Theory Related Fields 120 (2001), no. 1, 68–84.
  • [14] P. E. Conner. The Neumann’s problem for differential forms on Riemannian manifolds. Mem. Amer. Math. Soc. 1956 (1956), no. 20, 56 pp. 31.0X.
  • [15] D. Fãrst. Il caso limite del problema della rovina dei giocatori nell’ipotesi di riserva limitata. (Italian) Giorn. Ist. Ital. Attuari 20 1957 120–143.
  • [16] J.-D. Deuschel, L. Zambotti, Bismut-Elworthy’s formula and random walk representation for SDEs with reflection, Stochastic Process. Appl., 115(6):907–925, 2005.
  • [17] J. Eells and K. D. Elworthy Wiener integration on certain manifolds. Problems in non-linear analysis (C.I.M.E., IV Ciclo, Varenna, 1970), pp. 67–94. Edizioni Cremonese, Rome, 1971.
  • [18] K. D. Elworthy, Stochastic differential equations on manifolds, London Mathematical Society Lecture Notes Series, 70, Cambridge University Press, Cambridge New York, (1982).
  • [19] K. D. Elworthy, Y. Le Jan, and Xue-Mei Li. Integration by parts formulae for degenerate diffusion measures on path spaces and diffeomorphism groups. C. R. Acad. Sci. Paris Scr. I Math. 323 (1996), no. 8, 921–926.
  • [20] K.D. Elworthy and Xue- Mei Li, Differentiation of heat semigroups and applications. Probability theory and mathematical statistics (Vilnius, 1993), 239-251, Vilnius, 1994
  • [21] K. D. Elworthy and Xue-Mei Li. Formulae for the derivatives of heat semigroups. J. Funt. Anal. 125 (1994), no. 1; 252-286
  • [22] K. D. Elworthy and Xue- Mei Li, Bismut -type formulae for differential forms C.R. Acad. Sci. Paris Sér I Math. 327 (1998), no. 1, 87-92.
  • [23] M. Emery, Équations différentielles stochastiques lipschitziennes : étude de la stabilité, Séminaire de Probabilités XIII, pp 281-293, Lecture Notes in Math., 721, Springer, Berlin, 1979.
  • [24] T. Funaki and K. Ishitani Integration by parts formulae for Wiener measures on a path space between two curves. Probab. Theory Related Fields 137 (2007), no. 3-4, 289-321.
  • [25] E. P. Hsu Multiplicative functional for the heat equation on manifolds with boundary. Michigan Math. J. 50 (2002), no. 2, 351–367.
  • [26] N. Ikeda. On the construction of two-dimensional diffusion processes satisfying Wentzell’s boundary conditions and its application to boundary value problems. Mem. Coll. Sci. Univ. Kyoto Ser. A Math., 33, pp367-427, 1960/1961.
  • [27] N. Ikeda, T. Ueno, H. Tanaka and K. Satô A boundary-value problem for multi-dimensional diffusion processes. (Japanese) Sûgaku 13 1961/1962 37–53.
  • [28] N. Ikeda and S. Watanabe, Heat equation and diffusion on Riemannian manifold with boundary, In Proceedings of the International Symposium on Stochastic Differential Equations (1976), Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 75–94, 1978.
  • [29] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, second edition, North Holland Mathematical Library, 24, 1989.
  • [30] K. Itô. Stochastic parallel displacement. Probabilistic methods in differential equations (Proc. Conf., Univ. Victoria, Victoria, B.C., 1974), pp. 1–7. Lecture Notes in Math., Vol. 451, Springer, Berlin, 1975.
  • [31] Xue-Mei Li, Stochastic Flows on Noncompact Manifolds University of Warwick Ph.D. thesis (1992).
  • [32] Xue-Mei Li. Strong p-completeness of stochastic differential equations and the existence of smooth flows on non-compact manifolds. Probab. Theory Relat. Fields, 100 (4), 485-511 (1994).
  • [33] P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions Comm. Pure Appl. Math. 37(4), pp 511-537, (1984)
  • [34] P. Malliavin. Formules de la moyenne, calcul de perturbations et théorèmes d’annulation pour les formes harmoniques. J. Functional Analysis 17 (1974), 274–291.
  • [35] A. Méritet. Théorème d’annulation pour la cohomologie absolue d’une variété riemannienne à bord. (French) Bull. Sci. Math. (2) 103 (1979), no. 4, 379–400.
  • [36] P. A. Meyer, Géométrie stochastique sans larmes, Séminaire de Probabilités, XV, Lecture Notes in Mathematics, 850, Springer-Verlag, Berlin New York, 1981
  • [37] I. Shigekawa, N. Ueki, and S. Watanabe, Shinzo, A probabilistic proof of the Gauss-Bonnet-Chern theorem for manifolds with boundary. Osaka J. Math. 26 (1989), no. 4, 897–930.
  • [38] A. V. Skorohod. Stochastic equations for diffusion processes with a boundary. (Russian) Teor. Verojatnost. i Primenen. 6 (1961) 287–298.
  • [39] D. W. Stroock and S. R. S. Varadhan, Diffusion Processes with boundary conditions, Comm. Pure Appl. Math. 245, pp147-225 (1971).
  • [40] H. Tanaka Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979), no. 1, 163–177.
  • [41] A. Thalmaier and F. Y. Wang, Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal. 155 (1998), no. 1, 109-124.
  • [42] S. R. S. Varadhan and R. J. Williams. Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 (1985), no. 4, 405–443.
  • [43] F. Y. Wang, Analysis for diffusion processes on Riemannian manifolds, Advanced Series on Statistical Science and Applied Probability, Vol. 18, World Scientific (2014)
  • [44] S. Watanabe. Construction of diffusion processes with Wentzell’s boundary conditions by means of Poisson point processes of Brownian excursions. Probability theory (Papers, VIIth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1976), pp. 255–271, Banach Center Publ., 5, PWN, Warsaw, 1979.
  • [45] L. Zambotti, Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection. Probab. Theory Relat. Fields 123, 579-600 (2002).