Reflected Brownian motion in a cone with radially homogeneous reflection field
HTML articles powered by AMS MathViewer
- by Y. Kwon and R. J. Williams PDF
- Trans. Amer. Math. Soc. 327 (1991), 739-780 Request permission
Abstract:
This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties. (i) The state space is a cone in $d$-dimensions $(d \geq 3)$, and the process behaves in the interior of the cone like ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the cone, the direction of reflection being fixed on each radial line emanating from the vertex of the cone. (iii) The amount of time that the process spends at the vertex of the cone is zero (i.e., the set of times for which the process is at the vertex has zero Lebesgue measure). The question of existence and uniqueness is cast in precise mathematical terms as a submartingale problem in the style used by Stroock and Varadhan for diffusions on smooth domains with smooth boundary conditions. The question is resolved in terms of a real parameter $\alpha$ which in general depends in a rather complicated way on the geometric data of the problem, i.e., on the cone and the directions of reflection. However, a criterion is given for determining whether $\alpha > 0$. It is shown that there is a unique continuous strong Markov process satisfying (i)-(iii) above if and only if $\alpha < 2$, and that starting away from the vertex, this process does not reach the vertex if $\alpha \leq 0$ and does reach the vertex almost surely if $0 < \alpha < 2$. If $\alpha \geq 2$, there is a unique continuous strong Markov process satisfying (i) and (ii) above; it reaches the vertex of the cone almost surely and remains there. These results are illustrated in concrete terms for some special cases. The process considered here serves as a model for comparison with a reflected Brownian motion in a cone having a nonradially homogeneous reflection field. This is discussed in a subsequent work by Kwon.References
-
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Appl. Math. Series 55, National Bureau of Standards, Washington, D.C., 1972.
- R. F. Bass and É. Pardoux, Uniqueness for diffusions with piecewise constant coefficients, Probab. Theory Related Fields 76 (1987), no. 4, 557–572. MR 917679, DOI 10.1007/BF00960074
- D. L. Burkholder, Exit times of Brownian motion, harmonic majorization, and Hardy spaces, Advances in Math. 26 (1977), no. 2, 182–205. MR 474525, DOI 10.1016/0001-8708(77)90029-9
- Kai Lai Chung, Lectures from Markov processes to Brownian motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 249, Springer-Verlag, New York-Berlin, 1982. MR 648601
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325
- Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, DOI 10.1007/BF00280445
- Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085, DOI 10.1002/9780470316658
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- P. Hall and C. C. Heyde, Martingale limit theory and its application, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 624435
- Samuel Karlin and Howard M. Taylor, A second course in stochastic processes, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 611513
- M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Translation 1950 (1950), no. 26, 128. MR 0038008
- Thomas G. Kurtz, Approximation of population processes, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 36, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. MR 610982 Y. Kwon, The submartingale problem for Brownian motion in a cone with nonconstant oblique reflection, Ph.D. Dissertation, Univ. of Washington, 1989.
- Serge Lang, Differential manifolds, 2nd ed., Springer-Verlag, New York, 1985. MR 772023, DOI 10.1007/978-1-4684-0265-0
- Jean-François Le Gall, Mouvement brownien, cônes et processus stables, Probab. Theory Related Fields 76 (1987), no. 4, 587–627 (French, with English summary). MR 917681, DOI 10.1007/BF00960076
- P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37 (1984), no. 4, 511–537. MR 745330, DOI 10.1002/cpa.3160370408
- Jürgen Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. MR 170091, DOI 10.1002/cpa.3160130308
- Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen. 1 (1956), 177–238 (Russian, with English summary). MR 0084896
- M. H. Protter and H. F. Weinberger, On the spectrum of general second order operators, Bull. Amer. Math. Soc. 72 (1966), 251–255. MR 190527, DOI 10.1090/S0002-9904-1966-11485-4
- 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
- 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
- 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
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 327 (1991), 739-780
- MSC: Primary 60J65; Secondary 35J99, 58G32, 60G44
- DOI: https://doi.org/10.1090/S0002-9947-1991-1028760-9
- MathSciNet review: 1028760