Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Weak convergence of obliquely reflected diffusions

Andrey Sarantsev

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Burdzy and Chen (Electron. J. Probab. 3 (1998) 29–33) proved results on weak convergence of multidimensional normally reflected Brownian motions. We generalize their work by considering obliquely reflected diffusion processes. We require weak convergence of domains, which is stronger than convergence in Wijsman topology, but weaker than convergence in Hausdorff topology.


Burdzy et Chen (Electron. J. Probab. 3 (1998) 29–33) ont montré des résultats portant sur la convergence faible des mouvements Browniens multidimensionnels avec réflexion normale. Nous généralisons leurs travaux dans le cas de processus de diffusion avec réflexion oblique. Notre résultat requiert la faible convergence des domaines. Notons que cette convergence est plus forte que la convergence dans la topologie de Wijsman, mais plus faible que celle de la topologie de Hausdorff.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1408-1431.

Received: 8 September 2015
Revised: 2 May 2017
Accepted: 4 May 2017
First available in Project Euclid: 11 July 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J55: Local time and additive functionals 60J65: Brownian motion [See also 58J65] 60H10: Stochastic ordinary differential equations [See also 34F05]

Reflected diffusions Oblique reflection Hausdorff convergence Wijsman convergence Weak convergence


Sarantsev, Andrey. Weak convergence of obliquely reflected diffusions. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1408--1431. doi:10.1214/17-AIHP843.

Export citation


  • [1] G. Beer. Wijsman convergence: A survey. Set-Valued Var. Anal. 2 (1–2) (1994) 77–94.
  • [2] K. Burdzy and Z.-Q. Chen. Weak convergence of reflected Brownian motions. Electron. J. Probab. 3 (4) (1998) 29–33.
  • [3] K. Burdzy and D. Marshall Hitting a boundary point with reflected Brownian motion. In Séminaire de Probabilités, XXVI 81–94. Lecture Notes in Math. 1526. Springer, 1992.
  • [4] J. G. Dai and R. J. Williams. Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedra. Theory Probab. Appl. 40 (1) (1995) 3–53.
  • [5] M. C. Delfour and J.-P. Zolesio. Shapre analysis via oriented distance functions. J. Funct. Anal. 123 (1) (1994) 129–201.
  • [6] J. M. Harrison and I. M. Reiman. Reflected Brownian motion on an orthant. Ann. Probab. 9 (2) (1981) 302–308.
  • [7] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1990.
  • [8] S. Hottovy. The Smoluchowski–Kramers approximation for stochastic differential equations with arbitrary state-dependent friction. Ph.D. Thesis, 2013.
  • [9] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, 1989.
  • [10] W. Kang and K. Ramanan. On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries, 2014. Available at arXiv:1412:0729.
  • [11] W. Kang and R. J. Williams. An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries. Ann. Appl. Probab. 17 (2) (2007) 741–779.
  • [12] T. G. Kurtz and P. Protter. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 (3) (1991) 1035–1070.
  • [13] J. R. Munkres. Topology: A First Course. Prentice-Hall, 1975.
  • [14] S. Ramasubramanian. Recurrence of projections of diffusions. Sankhya A 45 (1) (1983) 20–31.
  • [15] S. Ramasubramanian. Hitting of submanifolds by diffusions. Probab. Theory Related Fields 78 (1) (1988) 149–163.
  • [16] I. M. Reiman and R. J. Williams. A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77 (1) (1988) 87–97.
  • [17] A. Sarantsev. Triple and simultaneous collisions of competing Brownian particles. Electron. J. Probab. 20 (1) (2015) 1–29.
  • [18] A. Sarantsev. Infinite systems of competing Brownian particles, 2016. Available at arXiv:1403.4229.
  • [19] A. Sarantsev. Penalty method for obliquely reflected diffusions, 2016. Available at arXiv:1509.01777.
  • [20] L. M. Taylor and R. J. Williams. Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Related Fields 75 (4) (1993) 459–485.
  • [21] R. A. Wijsman. Convergence of sequences of convex sets, cones and functions II. Trans. Amer. Math. Soc. 123 (1966) 32–45.
  • [22] R. J. Williams. Reflected Brownian motion with skew-symmetric data in a polyhedral domain. Probab. Theory Related Fields 75 (4) (1987) 459–485.
  • [23] R. J. Williams. Semimartingale reflecting Brownian motions in the orthant. In Stochastic Networks 125–137. IMA Vol. Math. Appl. 71. Springer, 1995.
  • [24] R. J. Williams. An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Syst. 30 (1–2) (1998) 5–25.