Differentiability of Stochastic Flow of Reflected Brownian Motions

Krzysztof Burdzy (University of Washington)

Abstract


We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation.

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Pages: 2182-2240

Publication Date: October 6, 2009

DOI: 10.1214/EJP.v14-700

References

  1. Airault, Hélène. Perturbations singulières et solutions stochastiques de problèmes de D. Neumann-Spencer. (French) J. Math. Pures Appl. (9) 55 (1976), no. 3, 233--267. MR0501184 (58 #18606)
  2. Andres, Sebastian. Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 1, 104--116. MR2500230
  3. Andres, Sebastian. Pathwise Differentiability for SDEs in a Smooth Domain with Reflection (preprint)
  4. Burdzy, K. Multidimensional Brownian excursions and potential theory. Pitman Research Notes in Mathematics Series, 164. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. vi+172 pp. ISBN: 0-582-00573-6 MR0932248 (89d:60146)
  5. Burdzy, Krzysztof; Chen, Zhen-Qing. Coalescence of synchronous couplings. Probab. Theory Related Fields 123 (2002), no. 4, 553--578. MR1921013 (2003e:60184)
  6. Burdzy, Krzysztof; Chen, Zhen-Qing; Jones, Peter. Synchronous couplings of reflected Brownian motions in smooth domains. Illinois J. Math. 50 (2006), no. 1-4, 189--268 (electronic). MR2247829 (2007i:60102)
  7. Burdzy, Krzysztof; Lee, James M.. Multiplicative functional for reflected Brownian motion via deterministic ODE (preprint, Math Arxiv: 0805.3740)
  8. Cranston, M.; Le Jan, Y. On the noncoalescence of a two point Brownian motion reflecting on a circle. Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 2, 99--107. MR1001020 (90k:60154)
  9. Cranston, M.; Le Jan, Y. Noncoalescence for the Skorohod equation in a convex domain of $R\sp 2$. Probab. Theory Related Fields 87 (1990), no. 2, 241--252. MR1080491 (92e:60116)
  10. Deuschel, Jean-Dominique; Zambotti, Lorenzo. Bismut-Elworthy's formula and random walk representation for SDEs with reflection. Stochastic Process. Appl. 115 (2005), no. 6, 907--925. MR2134484 (2006e:60080)
  11. Dupuis, Paul; Ishii, Hitoshi. On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics Stochastics Rep. 35 (1991), no. 1, 31--62. MR1110990 (93e:60110)
  12. Dupuis, Paul; Ramanan, Kavita. Convex duality and the Skorokhod problem. I, II. Probab. Theory Related Fields 115 (1999), no. 2, 153--195, 197--236. MR1720348 (2001f:49041)
  13. Fukushima, Masatoshi; Ōshima, Yōichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. x+392 pp. ISBN: 3-11-011626-X MR1303354 (96f:60126)
  14. Hsu, Elton P. Multiplicative functional for the heat equation on manifolds with boundary. Michigan Math. J. 50 (2002), no. 2, 351--367. MR1914069 (2003f:58067)
  15. Ikeda, Nobuyuki; Watanabe, Shinzo. Heat equation and diffusion on Riemannian manifold with boundary. Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), pp. 75--94, Wiley, New York-Chichester-Brisbane, 1978. MR0536005 (81b:58046)
  16. Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3 MR1011252 (90m:60069)
  17. Kunita, H. Stochastic differential equations and stochastic flows of diffeomorphisms. École d'été de probabilités de Saint-Flour, XII---1982, 143--303, Lecture Notes in Math., 1097, Springer, Berlin, 1984. MR0876080 (87m:60127)
  18. Lions, P.-L.; Sznitman, A.-S. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984), no. 4, 511--537. MR0745330 (85m:60105)
  19. Maisonneuve, Bernard. Exit systems. Ann. Probability 3 (1975), no. 3, 399--411. MR0400417 (53 #4251)
  20. Mandelbaum, Avi; Massey, William A. Strong approximations for time-dependent queues. Math. Oper. Res. 20 (1995), no. 1, 33--64. MR1320446 (96b:60240)
  21. Mandelbaum, Avi; Ramanan, Kavita; Directional derivatives of oblique reflection maps Math. Oper. Res. (to appear)
  22. Pilipenko, A. Yu. Stochastic flows with reflection. (Russian) Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 2005, no. 10, 23--28. MR2218498 (2007j:60089)
  23. Pilipenko, A. Yu. Properties of flows generated by stochastic equations with reflection. (Russian) Ukraïn. Mat. Zh. 57 (2005), no. 8, 1069--1078; translation in Ukrainian Math. J. 57 (2005), no. 8, 1262--1274 MR2218469 (2007g:60066)
  24. Pilipenko, A. Yu. On the generalized differentiability with initial data of a flow generated by a stochastic equation with reflection. (Ukrainian) Teor. Ĭmovīr. Mat. Stat. No. 75 (2006), 127--139; translation in Theory Probab. Math. Statist. No. 75 (2007), 147--160 MR2321188 (2008e:60179)
  25. Pilipenko, A. Yu. Stochastic flows with reflection. (2008, Math Arxiv: 0810.4644)
  26. Saisho, Yasumasa. Stochastic differential equations for multidimensional domain with reflecting boundary. Probab. Theory Related Fields 74 (1987), no. 3, 455--477. MR0873889 (88b:60139)
  27. Tanaka, Hiroshi. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979), no. 1, 163--177. MR0529332 (80k:60075)
  28. Weerasinghe, A., Some properties of stochastic flows and diffusions with reflections, Ph.D.~Thesis, University of Minnesota, 1986.
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