A class of path-valued Markov processes and its applications to superprocesses. (English) Zbl 0794.60076

Le Gall, Jean-François

Probab. Theory Relat. Fields 95, No. 1, 25-46 (1993).

Let \((\xi_ s)\) be a continuous Markov process satisfying certain regularity assumptions. We introduce a path-valued strong Markov process associated with \((\xi_ s)\), which is closely related to the so-called superprocess with spatial motion \((\xi_ s)\). In particular, a subset \(H\) of the state space of \((\xi_ s)\) intersects the range of the superprocess if and only if the set of paths that hit \(H\) is not polar for the path-valued process. The latter property can be investigated by using the tools of the potential theory of symmetric Markov processes: A set is not polar if and only if it supports a measure of finite energy. The same approach can be applied to study sets that are polar for the graph of the superprocess. In the special case when \((\xi_ s)\) is a diffusion process, we recover certain results recently obtained by Dynkin.

Reviewer: J.F.Le Gall (Paris)

MSC:

60J45	Probabilistic potential theory
60J25	Continuous-time Markov processes on general state spaces
60G57	Random measures
60J60	Diffusion processes
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

continuous Markov process; superprocess; potential theory; graph of the superprocess; diffusion process

References:

[1]	Baras, P., Pierre, M.: Probl?mes paraboliques semi-lin?aires avec donn?es mesures. Appl. Anal.18, 111-149 (1984) · Zbl 0582.35060 · doi:10.1080/00036818408839514
[2]	Dawson, D.A., Iscoe, I., Perkins, E.A.: Super-Brownian motion: Path properties and hitting probabilities. Probab. Theory Relat. Fields83, 135-205 (1989) · Zbl 0692.60063 · doi:10.1007/BF00333147
[3]	Dawson, D.A., Perkins, E.A.: Historical processes. Mem. Am. Math. Soc.93, 454 (1991) · Zbl 0754.60062
[4]	Doob, J.L.: Classical potential theory and its probabilistic counterpart. Berlin Heidelberg New York: Springer 1984 · Zbl 0549.31001
[5]	Dynkin, E.B.: Green’s and Dirichlet space associated with fine Markov processes. J. Funct. Anal.47, 381-418, (1982) · Zbl 0488.60083 · doi:10.1016/0022-1236(82)90112-4
[6]	Dynkin, E.B.: Branching particle systems and superprocesses. Ann. Probab.19, 1157-1194 (1991) · Zbl 0732.60092 · doi:10.1214/aop/1176990339
[7]	Dynkin, E.B.: A probabilistic approach to one class of non linear differential equations. Probab. Theory Relat. Fields89, 89-115 (1991) · Zbl 0722.60062 · doi:10.1007/BF01225827
[8]	Dynkin, E.B.: Superdiffusions and parabolic nonlinear differential equations. Ann. Probab.20, 942-962 (1992) · Zbl 0756.60074 · doi:10.1214/aop/1176989812
[9]	Ethier, S.N., Kurtz, T.G.: Markov processes: Characterization and convergence, New York: Wiley 1986 · Zbl 0592.60049
[10]	Fitzsimmons, P.J., Getoor, R.K.: On the potential theory of symmetric Markov process. Math. Ann.281, 495-512 (1988) · Zbl 0627.60067 · doi:10.1007/BF01457159
[11]	Friedman, A.: Partial differential equations of parabolic type. Englewood Cliffs, NJ: Prentice-Hall 1964 · Zbl 0144.34903
[12]	Le Gall, J.F.: Marches al?atoires, mouvement brownien et processus de branchement. In: Az?ma, J. (et al. (eds.), Seminaire de Probabilit?s XXIII. (Lect. Notes Math., vol. 1372, pp. 447-464) Berlin Heidelberg New York. Springer 1989
[13]	Le Gall, J.F.: Brownian excursions, trees and measure-valued branching processes. Ann. Probab.19, 1399-1439 (1991) · Zbl 0753.60078 · doi:10.1214/aop/1176990218
[14]	Le Gall J.F.: A path-valued Markov process and its connections with partial differential equations. Proceedings of the first European Congress of Mathematics (to appear) · Zbl 0812.60058
[15]	Meyers, N.G.: A theory of capacities for potentials of functions in Lebesgue classes. Math. Scand.26 255-292 (1970) · Zbl 0242.31006
[16]	Miranda, C.: Partial differential equations of elliptic type, 2nd edn. Berlin Heidelberg New York: Springer 1970 · Zbl 0198.14101
[17]	Neveu, J., Pitman, J.W.: The branching process in a Brownian excursion. In: Az?ma, J. et al. (eds.) S?minaire de Probabilit?s XXIII. (Lect. Notes Math.,. vol. 1372, pp. 248-257) Berlin Heidelberg New York: springer 1989 · Zbl 0741.60081
[18]	Perkins, E.A.: Polar sets and multiple points for super-Brownian motion. Ann. Probab.18, 453-491 (1990) · Zbl 0721.60046 · doi:10.1214/aop/1176990841
[19]	Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979 · Zbl 0426.60069
[20]	Watanabe, S.: A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ.8, 141-167 (1968) · Zbl 0159.46201