## Reflecting Brownian motion in a cusp

HTML articles powered by AMS MathViewer

- by R. Dante DeBlassie and Ellen H. Toby PDF
- Trans. Amer. Math. Soc.
**339**(1993), 297-321 Request permission

## Abstract:

Let $C$ be the cusp $\{ (x,y):x \geq 0$, $- {x^\beta } \leq y \leq {x^\beta }\}$ where $\beta > 1$. Set $\partial {C_1} = \{ (x,y):x \geq 0, y = - {x^\beta }\}$ and $\partial {C_2} = \{ (x,y):x \geq 0$, $y = {x^\beta }\}$. We study the existence and uniqueness in law of reflecting Brownian motion in $C$. The angle of reflection at $\partial {C_j}\backslash \{ 0\}$ (relative to the inward unit normal) is a constant ${\theta _j} \in \left ( { - \frac {\pi } {2},\frac {\pi } {2}} \right )$, and is positive iff the direction of reflection has a negative first component in all sufficiently small neighborhoods of $0$. When ${\theta _1} + {\theta _2} \leq 0$, existence and uniqueness in law hold. When ${\theta _1} + {\theta _2} > 0$, existence fails. We also obtain results for a large class of asymmetric cusps. We make essential use of results of Warschawski on the differentiability at the boundary of conformal maps.## References

- R. Dante DeBlassie,
*Explicit semimartingale representation of Brownian motion in a wedge*, Stochastic Process. Appl.**34**(1990), no. 1, 67–97. MR**1039563**, DOI 10.1016/0304-4149(90)90057-Y - Krzysztof Burdzy and Donald Marshall,
*Hitting a boundary point with reflected Brownian motion*, Séminaire de Probabilités, XXVI, Lecture Notes in Math., vol. 1526, Springer, Berlin, 1992, pp. 81–94. MR**1231985**, DOI 10.1007/BFb0084312 - L. C. G. Rogers,
*Brownian motion in a wedge with variable skew reflection*, Trans. Amer. Math. Soc.**326**(1991), no. 1, 227–236. MR**1008701**, DOI 10.1090/S0002-9947-1991-1008701-0 - L. C. G. Rogers,
*Brownian motion in a wedge with variable skew reflection. II*, Diffusion processes and related problems in analysis, Vol. I (Evanston, IL, 1989) Progr. Probab., vol. 22, Birkhäuser Boston, Boston, MA, 1990, pp. 95–115. MR**1110159** - L. C. G. Rogers and David Williams,
*Diffusions, Markov processes, and martingales. Vol. 2*, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987. Itô calculus. MR**921238** - Daniel W. Stroock and S. R. S. Varadhan,
*Diffusion processes with boundary conditions*, Comm. Pure Appl. Math.**24**(1971), 147–225. MR**277037**, DOI 10.1002/cpa.3160240206 - Daniel W. Stroock and S. R. Srinivasa Varadhan,
*Multidimensional diffusion processes*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin-New York, 1979. MR**532498** - 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. MR**792398**, DOI 10.1002/cpa.3160380405 - S. E. Warschawski,
*On conformal mapping of infinite strips*, Trans. Amer. Math. Soc.**51**(1942), 280–335. MR**6583**, DOI 10.1090/S0002-9947-1942-0006583-6 - S. E. Warschawski,
*On differentiability at the boundary in conformal mapping*, Proc. Amer. Math. Soc.**12**(1961), 614–620. MR**131524**, DOI 10.1090/S0002-9939-1961-0131524-8 - R. J. Williams,
*Recurrence classification and invariant measure for reflected Brownian motion in a wedge*, Ann. Probab.**13**(1985), no. 3, 758–778. MR**799421**

## Additional Information

- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**339**(1993), 297-321 - MSC: Primary 60J60; Secondary 60H99, 60J65
- DOI: https://doi.org/10.1090/S0002-9947-1993-1149119-1
- MathSciNet review: 1149119