Diffusions, exit time moments and Weierstrass theorems
Victor H. de la Peña and Patrick McDonald
Proc. Amer. Math. Soc. 132 (2004), 2465-2474
Primary 60J65, 40A30
March 24, 2004
Full-text PDF Free Access
Similar Articles |
Abstract: Let be a one-dimensional diffusion with infinitesimal generator given by the operator where is a smooth, positive real-valued function and the ratio of and is a constant. Given a compact interval, we prove a Weierstrass-type theorem for the exit time moments of and their corresponding (naturally weighted) first derivatives, and we provide an algorithm that produces uniform approximations of arbitrary continuous functions by exit time moments. We investigate analogues of these results in higher-dimensional Euclidean spaces. We give expansions for several families of special functions in terms of exit time moments.
L. Doob, A probability approach to the heat
equation, Trans. Amer. Math. Soc. 80 (1955), 216–280. MR 0079376
R. Z. Ha'sminskii, Probabilistic representations of the solutions of some differential equations, In: Proc. 6th All Union Conf. on Theor. Prob. and Math. Stat. (Vilnius 1960) (1960).
K. J. Kinateder, Patrick
McDonald, and David
Miller, Exit time moments, boundary value problems, and the
geometry of domains in Euclidean space, Probab. Theory Related Fields
111 (1998), no. 4, 469–487. MR 1641818
McDonald and Robert
Meyers, Dirichlet spectrum and heat content, J. Funct. Anal.
200 (2003), no. 1, 150–159. MR 1974092
G. Wendel, Hitting spheres with Brownian motion, Ann. Probab.
8 (1980), no. 1, 164–169. MR 556423
T. Whittaker and G.
N. Watson, A course of modern analysis. An introduction to the
general theory of infinite processes and of analytic functions: with an
account of the principal transcendental functions, Fourth edition.
Reprinted, Cambridge University Press, New York, 1962. MR 0178117
- J. L. Doob, A probability approach to the heat equation, Trans. Amer. Math. Soc. 80, 216-280 (1955). MR 18:76g
- R. Z. Ha'sminskii, Probabilistic representations of the solutions of some differential equations, In: Proc. 6th All Union Conf. on Theor. Prob. and Math. Stat. (Vilnius 1960) (1960).
- K. Kinateder, P. McDonald, and D. Miller, Exit time moments, boundary value problems and the geometry of domains in Euclidean space, Prob. Theory and Related Fields 111, 469-487 (1998). MR 99h:60151
- P. McDonald and R. Meyers, Dirichlet spectrum and heat content, Jour. Funct. Anal. 200, 150-159 (2003). MR 2004c:58074
- J. G. Wendel, Hitting spheres with Brownian motion, Ann. Probability 8, 164-169 (1980). MR 80m:60085
- E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, Cambridge University Press, London, 1940. MR 31:2375
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
Retrieve articles in all journals
with MSC (2000):
Victor H. de la Peña
Department of Statistics, Columbia University, New York, New York 10027
Department of Mathematics, New College of Florida, Sarasota, Florida 34243
exit time moments,
Received by editor(s):
August 13, 2002
March 24, 2004
Richard C. Bradley
© Copyright 2004
American Mathematical Society