The Annals of Probability

Ratio Limit Theorems for a Brownian Motion Killed at the Boundary of a Benedicks Domain

Pierre Collet, Servet Martínez, and Jaime San Martín

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We consider a Brownian motion in a Benedicks domain with absorption at the boundary. We show ratio limit theorems for the associated heat kernel. When the hole is compact, therefore the Martin boundary is two dimensional; we obtain sharp estimates on the lifetime probabilities and we identify, in probabilistic terms, the various constants appearing in the theory.

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Ann. Probab. Volume 27, Number 3 (1999), 1160-1182.

First available in Project Euclid: 29 May 2002

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Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65] 35B40 35K05

Brownian motion heat kernel ratio limit theorems Benedicks domain


Collet, Pierre; Martínez, Servet; San Martín, Jaime. Ratio Limit Theorems for a Brownian Motion Killed at the Boundary of a Benedicks Domain. Ann. Probab. 27 (1999), no. 3, 1160--1182. doi:10.1214/aop/1022677443.

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  • 1 ANCONA, A. 1984. Regularite d'acces des bouts et frontiere de Martin d'un domaine ´ ´ Euclidien. J. Math. Pures Appl. 63 215 260.
  • 2 ANCONA, A. 1990. Theorie du potentiel sur les graphes et les varietes. Ecole d'ete de ´ ´ ´ ´ ´ probabilites de Saint Flour XVIII. Lecture Notes in Math. 1427. Springer, Berlin. ´
  • 3 BANUELOS, R. and DAVIS, B. 1989. Heat kernel, eigenfunctions, and conditioned Brownian, motion in planar domains. J. Funct. Anal. 84 188 200.
  • 4 BENEDICKS, M. 1980. Positive harmonic functions vanishing on the boundary of certain domains in Rn. Ark. Mat. 18 53 72.
  • 5 COLLET, P., MARTiNEZ, S. and SAN MARTiN, J. 1995. Asymptotic laws for one-dimensional ´ ´ diffusions conditioned to nonabsorption. Ann. Probab. 23 1300 1314.
  • 6 KARATZAS, I. and SHREVE, S. 1988. Brownian Motion and Stochastic Calculus. Springer, New York.
  • 7 KRYLOV, N. and SAFONOV, M. 1981. A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR-Izv. 16 151 164.
  • 8 LI, P. 1986. Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature. Ann. of Math. 124 1 21.
  • 9 PINCHOVER, Y. 1992. Large time behavior of the heat kernel and the behavior of the green function near criticality for nonsymmetric elliptic operators. J. Funct. Anal. 104 54 70.
  • 10 PINSKY, R. 1990. The lifetimes of conditioned diffusion processes. Ann. Inst. H. Poincare ´ 26 87 99.
  • 11 PINSKY, R. 1995. Positive Harmonic Functions and Diffusion. Cambridge Univ. Press.
  • 12 TRUDINGER, N. 1968. Pointwise estimates and quasilinear parabolic equations. Comm. Pure Appl. Math. 21 205 226.