## The Annals of Probability

### Measure-valued branching processes associated with random walks on $p$-adics

#### Abstract

Measure-valued branching random walks (superprocesses) on $p$-adics are introduced and investigated. The uniqueness and existence of solutions to associated linear and nonlinear heat-type (parabolic) equations are proved, provided some condition on the parameter of the random walks is satisfied. The solutions of these equations are shown to be locally constant if their initial values are. Moreover, the heat-type equations can be identified with a system of ordinary differential equations. Conditions for the measure-valued branching stable random walks to possess the property of quasi-self-similarity are given, as well as a sufficient and necessary condition for these processes to be locally extinct. The latter result is consistent with the Euclidean case in the sense that the critical value for measure-valued branching stable processes to be locally extinct is the Hausdorff dimension of the image of the underlying processes divided by the dimension of the state space.

#### Article information

Source
Ann. Probab., Volume 28, Number 4 (2000), 1680-1710.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160503

Digital Object Identifier
doi:10.1214/aop/1019160503

Mathematical Reviews number (MathSciNet)
MR1813839

Zentralblatt MATH identifier
1044.60036

#### Citation

Albeverio, Sergio; Zhao, Xuelei. Measure-valued branching processes associated with random walks on $p$-adics. Ann. Probab. 28 (2000), no. 4, 1680--1710. doi:10.1214/aop/1019160503. https://projecteuclid.org/euclid.aop/1019160503

#### References

• Albeverio, S. (1985). Some points of interaction between stochastic analysis and quantum theory. In Stochastic Differential Systems (N. Christopeit, K. Helmes and M. Kolman, eds.) 1-26. Springer, Berlin.
• Albeverio, S. and Karwowski, W. (1991). Diffusion on p-adic numbers. In Gaussian Random Fields (K. It o and H. Hida, eds.) 86-99. World Scientific, Singapore.
• Albeverio, S. and Karwowski, W. (1994). A random walkon p-adics: the generator and its spectrum. Stochastic Process.Appl.53 1-22.
• Albeverio, S., Karwowski, W. and Zhao, X. (1999). Asymptotics and spectral results for random walks on p-adics. Stochastic Process.Appl.83 39-59. Albeverio, S. and Zhao, X. (1999a). A decomposition theorem of L´evy processes on local fields. J.Theoret.Probab.To appear. Albeverio, S. and Zhao, X. (1999b). A remarkon the relation between different constructions of random walks on p-adics. Preprint 594/1999, Univ. Bonn, Institut f ¨ur Angewandte Mathematik.
• Albeverio, S. and Zhao, X. (2001). A remarkon the construction of nonsymmetric stochastic processes on p-adics. Stoch.Anal.Appl.To appear. Albeverio, S. and Zhao, X. (1999d). L´evy processes on p-adics: hitting probabilities and oscialltion. Unpublished manuscript.
• Blumenthal, R. and Getoor, R. (1968). Markov Processes and Potential Theory. Academic Press, New York.
• Brekke, L. and Olson, M. (1989). p-adic diffusion and relaxation in glasses. Preprint EFI, Chicago.
• Dawson, D. (1977). The critical measure diffusion.Wahr.Verw.Gebiete 40 125-145.
• Dawson, D. (1993). Measure-valued Markov processes. Lecture Notes in Math. 1541. Springer, Berlin.
• Dawson, D., Fleischmann, K., Foley, R. and Peletier, L.(1986). A critical measure valued branching processes with infinite mean. Stochastic Anal.Appl.4 117-129.
• Dawson, D., Iscoe, I. and Perkins, E. (1989). Super-Brownian motion: path properties and hitting probabilities. Probab.Theory Related Fields 83 135-205.
• Dynkin, E. (1994). An Introduction to Branching Measure-Valued Processes. Amer. Math. Soc., Providence, RI.
• Evans, S. (1989). Local properties of L´evy processes on a totally disconnected group. J.Theoret. Probab. 2 209-259.
• Evans, S. (1995). p-adic white noise, chaos expansions, and stochastic integration. In Probability Measures on Groups and Related Structures XI. World Scientific, Singapore.
• Evans, S. (1998). Local fields, Gaussian measures, and Brownian motions. In Topics in Lie Groups and Probability: Boundary Theory (J. Taylor, ed.) Amer. Math. Soc., Providence, RI. To appear.
• Evans, S. and Fleischmann, K. (1996). Cluster formation in a stepping stone model with continuous, hierarchically structured sites. Ann.Probab.24 1926-1952.
• Fig a-Talamanca, A. (1994). Diffusion on compact ultrametric spaces. In Noncompact Lie Groups and Some of Their Applications (E. A. Tanner and R. Wilson, ed.) 157-167. Kluwer, Dordrecht.
• Fleischmann. K. (1988). Critical behavior of some measure-valued processes. Math.Nachr.135 131-147.
• Gel'fand, I. M., Graev, M. I. and Pyatetskii-Shapiro, I. (1969). Representation Theory and Automorphic Functions. Saunders, Philadelphia.
• Hussmann, S. (1997). Random walks on a p-adic tree. SFB 237 Preprint 359'97.
• Karwowski, W. and Vilela-Mendes, R. (1994). Hierarchical structures and asymmetric processes on p-adics and adeles. J.Math.Phys.35 4637-4650.
• Koblitz, N. (1984). p-adic Numbers, p-adic Analysis and Zeta-functions. Springer, New York.
• Kochubei, A. (1992). Parabolic equations over the field of p-adic numbers. Math.USSR Izvestiya 39 1263-1280. Kochubei, A. (1993a). A Schr¨odinger type equation over the field of p-adic numbers. J.Math. Phys. 34 3420-3428. Kochubei, A. (1993b). On p-adic Green functions. Theoret.Math.Phys.96 854-865.
• Konno, N. and Shiga, T. (1988). Stochastic partial differential equations for some measurevalued diffusions. Probab.Theory Related Fields 79 201-225.
• Perkins, E. (1988). A space-time property of a class of measure-valued branching diffusions. Trans.Amer.Math.Soc.305 743-795.
• Pinsky, R. (1996). Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions. Ann.Probab.24 237-267.
• Taibleson, M. (1975). Fourier Analysis on Local Fields. Princeton Univ. Press.
• Vladimirov, V. (1988). Generalized functions over the field of p-adic numbers. Russian Math. Surveys 43 19-64.
• Vladimirov, V., Volovich, I. and Zelenov, E. (1993). p-adic Numbers in Mathematical Physics. World Scientific, Singapore.
• Walsh, B. (1986). An introduction to stochastic partial differential equations. Lecture Notes in Math. 1180 266-348. Springer, New York.
• Yasuda, K. (1996). Additive processes on local fields. J.Math.Sci.Univ.Tokyo 3 629-654.
• Zhao, X. (1994). Some absolute continuity of superdiffusions and super-stable processes. Stochastic Process.Appl.50 21-36.
• Zhao, X. (1995). On a class of measure-valued processes with nonconstant branching rates. In Dirichlet Forms and Stochastic Processes (M. Ma, M. R¨ockner and J. Yan, eds.). de Gruyter, Berlin.
• Zhao, X. (1999). Introduction to Measure-valued Branching Processes (in Chinese). Science Press, Beijing.