The Annals of Probability

On states of exit measures for superdiffusions

Yuan-Chung Sheu

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We consider the exit measures of $(L,\alpha)$-superdiffusions, $1 < \alpha \leq 2$, from a bounded smooth domain D in R d. By using analytic results about solutions of the corresponding boundary value problem, we study the states of the exit measures. (Abraham and Le Gall investigated earlier .this problem for a special case $L = \Delta$ and $\alpha = 2$). Also as an application of these analytic results, we give a different proof for the critical Hausdorff. dimension of boundary polarity (established earlier by Le Gall under more restrictive assumptions and by Dynkin and Kuznetsov for general situations).

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Ann. Probab., Volume 24, Number 1 (1996), 268-279.

First available in Project Euclid: 15 January 2003

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Primary: 60J60: Diffusion processes [See also 58J65] 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J25: Continuous-time Markov processes on general state spaces 31C45: Other generalizations (nonlinear potential theory, etc.) 35J60: Nonlinear elliptic equations

Exit measure superdiffusion Hausdorff dimension boundary polar set absolutely continuous state singular state


Sheu, Yuan-Chung. On states of exit measures for superdiffusions. Ann. Probab. 24 (1996), no. 1, 268--279. doi:10.1214/aop/1042644716.

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  • ABRAHAM, R. and LE GALL, J.-F. 1993. Sur la mesure de sortie du super mouvement Brownien. Preprint. Z.
  • DAUTRAY, R. and LIONS, J. L. 1990. Mathematical Analy sis and Numerical Methods for Science and Technology 1. Springer, Berlin.
  • DAWSON, D. A. 1993. Measure-valued Markov processes. Ecole d'Ete de Probabilites de Saint ´ ´ Flour 1991. Lecture Notes in Math. 1541 1 260. Springer, Berlin. Z.
  • DAWSON, D. A., FLEISCHMANN, K. and ROELLY, S. 1991. Absolute continuity of the measure states in a branching model with cataly sts. In Seminar on Stochastic Processes 1990, 117 160. Birkhauser, Boston. ¨ Z.
  • DAWSON, D. A. and HOCHBERG, K. J. 1979. The carrying dimension of a stochastic measure diffusion. Ann. Probab. 7 693 703. Z.
  • DOOB, J. L. 1984. Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York. Z.
  • DUNFORD, N. and SCHWARTZ, J. T. 1958. Linear Operators, Part I: General Theory. Interscience, New York. Z.
  • Dy NKIN, E. B. 1991. A probabilistic approach to one class of nonlinear differential equations. Probab. Theory Related Fields 89 89 115. Z.
  • Dy NKIN, E. B. 1992. Superdiffusions and parabolic nonlinear differential equations. Ann. Probab. 20 942 962. Z.
  • Dy NKIN, E. B. 1993. Superprocesses and partial differential equations. Ann. Probab. 21 1185 1262. Z.
  • Dy NKIN, E. B. 1994. An Introduction to Branching Measure-Valued Processes. Amer. Math. Soc., Providence, RI. Z.
  • Dy NKIN, E. B. and KUZNETSOV, S. E. 1994. Superdiffusions and removable singularities for quasilinear partial differential equations. Comm. Pure & Appl. Math. To appear. Z.
  • Dy NKIN, E. B. and KUZNETSOV, S. E. 1995. Solutions of Lu u dominated by L-harmonic functions. Preprint. Z.
  • FLEISCHMANN, K. 1988. Critical behavior of some measure-valued processes. Math. Nachr. 135 131 147. Z.
  • GMIRA, A. and VERON, L. 1991. Boundary singularities of solutions of some nonlinear elliptic ´ equations. Duke Math. J. 64 271 324. Z.
  • HUEBER, H. and SIEVEKING, M. 1982. Uniform bounds for quotients of Green functions on C1, 1-domains. Ann. Inst. Fourier 32 105 117. Z. 2
  • LE GALL, J.-F. 1993. Les solutions positives de u u dans le disque unite. C. R. Acad. Sci. ´ Paris Ser. I Math. 317 873 878. ´ Z.
  • LE GALL, J.-F. 1994a. Hitting probabilities and potential theory for the Brownian path-valued process. Ann. Inst. Fourier 44 277 306. Z. 2
  • LE GALL, J.-F. 1994b. The Brownian snake and solutions of u u in a domain. Probab. Theory Related Fields 104 393 432. Z. MAZ'YA, V. G. 1972. Beurling's theorem on a minimum principle for positive harmonic function. J. Soviet Math. 4 367 379. Z.
  • SHEU, Y. C. 1994. Removable boundary singularities for solutions of some nonlinear differential equations Duke Math. J. 74 701 711. Z.
  • STROOK, D. W. and VARADHAN, S. R. S. 1979. Multidimensional Diffusion Processes. Springer, Berlin.Z.
  • WENTZELL, A. D. 1981. A Course in the Theory of Stochastic Processes. McGraw-Hill, New York. Z.
  • WHEEDEN, R. L. and Zy GMUND, A. 1977. Measure and Integral: An Introduction to Real Analy sis. Dekker, New York.