On reflected diffusions in cones and cylinders

Oleksii  Kulyk; Andrey Pilipenko; Sylvie Rœlly

doi:10.3842/umzh.v75i11.7418

Oleksii Kulyk Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Poland
Andrey Pilipenko Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv and National Technical University of Ukraine ,,Igor Sikorsky Kyiv Polytechnic Institute''
Sylvie Rœlly Institut für Mathematik, Universität Potsdam, Germany

DOI: https://doi.org/10.3842/umzh.v75i11.7418

Keywords: Reflected diffusion; Diffusion in a cone; Oblique reflection

Abstract

UDC 519.21

Let $X$ be a diffusion in a cone with oblique reflection at the boundary. We study the question whether $X$ reaches a vertex of the cone for a finite time with positive probability. We propose new probabilistic method of investigation connected with the long-term behavior of a diffusion reflected in a cylinder.

References

R. F. Anderson, S. Orey, Small random perturbation of dynamical systems with reflecting boundary, Nagoya Math. J., 60, 189–216 (1976). DOI: https://doi.org/10.1017/S0027763000017232

O. V. Aryasova, A. Y. Pilipenko, On Brownian motion on the plane with membranes on rays with a common endpoint, Random Ope. and Stoch. Equat., 17, № 2, 139–157 (2009). DOI: https://doi.org/10.1515/ROSE.2009.010

D. W. Stroock, S. S. Varadhan, Diffusion processes with boundary conditions, Commun. Pure and Appl. Math., 24, № 2, 147–225 (1971). DOI: https://doi.org/10.1002/cpa.3160240206

P. Dupuis, H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics, 35, № 1, 31–62 (1991). DOI: https://doi.org/10.1080/17442509108833688

P.-L. Lions, A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Commun. Pure and Appl. Math., 37, № 4, 511–537 (1984). DOI: https://doi.org/10.1002/cpa.3160370408

Y. Saisho, Stochastic differential equations for multi-dimensional domain with reflecting boundary, Probab. Theory and Related Fields, 74, № 3, 455–477 (1987). DOI: https://doi.org/10.1007/BF00699100

H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J., 9, № 1, 163–177 (1979). DOI: https://doi.org/10.32917/hmj/1206135203

J. M. Harrison, M. I. Reiman, Reflected Brownian motion on an orthant, Ann. Probab., 9, № 2, 302–308 (1981). DOI: https://doi.org/10.1214/aop/1176994471

K. Ramanan, Reflected diffusions defined via the extended Skorokhod map, Electron. J. Probab., 11, 934–992 (2006). DOI: https://doi.org/10.1214/EJP.v11-360

A. Pilipenko, An introduction to stochastic differential equations with reflection, Lect. Pure and Appl. Math., I, Potsdam Univ. Press (2014).

R. Courant, D. Hilbert, Methods of mathematical physics, vol. II, Partial differential equations, Intersci. Publ., New York, London (1962).

A. Friedman, Stochastic differential equations and applications, vol. 1, Probab. and Math. Statist., 28, Acad. Press (1975). DOI: https://doi.org/10.1016/B978-0-12-268201-8.50010-4

I. I. Gikhman, A. V. Skorokhod, The theory of stochastic processes I, Grundlehren math. Wiss., 210, Springer-Verlag, Berlin (1974).

W. Kang, K. Ramanan, Characterization of stationary distributions of reflected diffusions, Ann. Appl. Probab., 24, № 4, 1329–1374 (2014). DOI: https://doi.org/10.1214/13-AAP947

S. R. S. Varadhan, R. J. Williams, Brownian motion in a wedge with oblique reflection, Commun. Pure and Appl. Math., 38, № 4, 405–443 (1985). DOI: https://doi.org/10.1002/cpa.3160380405

Y. Kwon, R. J. Williams, Reflected Brownian motion in a cone with radially homogeneous reflection field, Trans. Amer. Math. Soc., 327, № 2, 739–780 (1991). DOI: https://doi.org/10.1090/S0002-9947-1991-1028760-9

Y. Kwon, The submartingale problem for Brownian motion in a cone with nonconstant oblique reflection, Probab. Theory and Related Fields, 92, № 3, 351–391 (1992). DOI: https://doi.org/10.1007/BF01300561

R. D. DeBlassie, E. H. Toby, Reflecting Brownian motion in a cusp, Trans. Amer. Math. Soc., 339, 297–321 (1993). DOI: https://doi.org/10.1090/S0002-9947-1993-1149119-1

R. J. Williams, Reflected Brownian motion with skew symmetric data in a polyhedral domain, Probab. Theory and Related Fields, 75, 459–485 (1987). DOI: https://doi.org/10.1007/BF00320328

W. Kang, K. Ramanan, On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries, Ann. Probab., 45, № 1, 404–468 (2017). DOI: https://doi.org/10.1214/16-AOP1153

S. A. Nazarov, B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Exp. Math., 13, De Gruyter, Berlin, New York (1994). DOI: https://doi.org/10.1515/9783110848915

P. Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York etc. (1968).

N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Math. Library, 24, North-Holland Publ. Co., Amsterdam etc. (1981).

R. J. Williams, Reflected Brownian motion in a wedge: semimartingale property, Z. Wahrscheinlichkeitstheor. und verw. Geb., 69, № 2, 161–176 (1985). DOI: https://doi.org/10.1007/BF02450279

L. M. Taylor, R. J. Williams, Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant, Probab. Theory and Related Fields, 96, № 3, 283–317 (1993). DOI: https://doi.org/10.1007/BF01292674

Y. Kwon, Reflected Brownian motion in a cone: semimartingale property, Probab. Theory and Related Fields, 101, № 2, 211–226 (1995). DOI: https://doi.org/10.1007/BF01375825

A. Kulik, Ergodic behavior of Markov processes: with applications to limit theorems, De Gruyter Stud. Math., 67, De Gryuter, Berlin (2018). DOI: https://doi.org/10.1515/9783110458930