The Annals of Probability

On states of exit measures for superdiffusions

Yuan-Chung Sheu

Abstract

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).

Article information

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

Dates
First available in Project Euclid: 15 January 2003

https://projecteuclid.org/euclid.aop/1042644716

Digital Object Identifier
doi:10.1214/aop/1042644716

Mathematical Reviews number (MathSciNet)
MR1387635

Zentralblatt MATH identifier
0854.60079

Citation

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

References

• 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.
• HSINCHU, TAIWAN E-mail: sheu@math.nctu.edu.tw