Branching Random Walk with Catalysts
Vladas Sidoravicius (IMPA)
Abstract
Shnerb et al. (2000), (2001) studied the following system of interacting particles on $\Bbb Z^d$: There are two kinds of particles, called $A$-particles and $B$-particles. The $A$-particles perform continuous time simple random walks, independently of each other. The jumprate of each $A$-particle is $D_A$. The $B$-particles perform continuous time simple random walks with jumprate $D_B$, but in addition they die at rate $\delta$ and a $B$-particle at $x$ at time $s$ splits into two particles at $x$ during the next $ds$ time units with a probability $\beta N_A(x,s)ds +o(ds)$, where $N_A(x,s)\; (N_B(x,s))$ denotes the number of $A$-particles (respectively $B$-particles) at $x$ at time $s$. Conditionally on the $A$-system, the jumps, deaths and splittings of different $B$-particles are independent. Thus the $B$-particles perform a branching random walk, but with a birth rate of new particles which is proportional to the number of $A$-particles which coincide with the appropriate $B$-particles. One starts the process with all the $N_A(x,0),\, x \in \Bbb Z^d$, as independent Poisson variables with mean $\mu_A$, and the $N_B(x,0),\, x \in \Bbb Z^d$, independent of the $A$-system, translation invariant and with mean $\mu_B$.
Shnerb et al. (2000) made the interesting discovery that in dimension 1 and 2 the expectation $\Bbb E\{N_B(x,t)\}$ tends to infinity, no matter what the values of $\delta, \beta, D_A$, $D_B, \mu_A,\mu_B \in (0, \infty)$ are. We shall show here that nevertheless there is a phase transition in all dimensions, that is, the system becomes (locally) extinct for large $\delta$ but it survives for $\beta$ large and $\delta$ small.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1-51
Publication Date: March 24, 2003
DOI: 10.1214/EJP.v8-127
References
- Athreya, K. B. and Ney, P. E. (1972), Branching Processes. Springer-Verlag. Math. Review 51:9242
- Billingsley, P. (1968), Convergence of Probability Measures. Wiley & Sons. Math. Review 38:1718
- Billingsley, P. (1986), Probability and Measure, 2nd ed. Wiley & Sons Math. Review 87f:60001
- Breiman, L. (1968), Probability. Addison-Wesley Publ. Co. Math. Review 37:4841
- Carmona, R. A. and Molchanov, S. A. (1994), Parabolic Anderson problem and intermittency. AMS Memoir 518, Amer. Math.Soc. Math. Review 94h:35080
- Chow, Y. S. and Teicher, H. (1988), Probability Theory, 2nd ed. Springer-Verlag. Math. Review 89e:60001
- Dawson, D. A. and Fleischmann, K. (2000), Catalytic and mutually catalytic branching. pp. 145-170 in Infinite Dimensional Stochastic Ananlysis (Ph. Cl'ement, F. den Hollander, J. van Neerven and B. de Pagter eds.) Koninklijke Nederlandse Akademdie van Wetenschappen. Math. Review 2002f:60164
- Derman, C. (1955), Some contributions to the theory of denumerable Markov chains. Trans. Amer. Math. Soc. 79 , 541-555. Math. Review 17:50c
- Gärtner, J. and den Hollander,F. (2003) , Intermittancy in a dynamic random medium. In preparation, Math. Review number not available.
- Gärtner, J., König, W. and Molchanov, S. A. (2000), Almost sure asymptotics for the continuous parabolic Anderson model, Probab. Theory Rel. Fields 118 , 547-573. Math. Review 2002i:60121
- Harris, T. E. (1963), The Theory of Branching Processes. Springer-Verlag. Math. Review 29:664
- Ikeda, N. Nagasawa, M. and Watanabe, S. (1968a), Branching Markov processes I J. Math. Kyoto Univ. 8 part I, 233-278. Math. Review 38:764
- Ikeda, N. Nagasawa, M. and Watanabe, S. (1968a), Branching Markov processes II J. Math. Kyoto Univ. 8, 365-410. Math. Review 38:6677
- Jagers, P. (1975), Branching Processes with Biological Applications. Wiley & Sons. Math. Review 58:7890
- Klenke, A. (2000a), Longtime behavior of stochastic processes with complex interactions {rm (especially Ch. 3)}. Habilitations thesis, University Erlangen. Math. Review number not available.
- Klenke, A. (2000b), A review on spatial catalytic branching, pp. 245-263 in Stochastic Models, (L. G. Gorostiza and B. G. Ivanoff eds.) CMS Conference proceedings, vol. 26, Amer. Math. Soc. Math. Review 2002a:60142
- Molchanov, S. A. (1994), Lectures on random media, pp. 242-411 in Ecole d'Et'e de Probabilit'es de St Flour XXII, (P. Bernard ed.) Lecture Notes in Math, vol. 1581, Springer-Verlag. Math. Review 95m:60165
- Savits, T. H. (1969), The explosion problem for branching Markov processes. Osaka J. Math. 6, 375-395. Math. Review 43:8137
- Shnerb, N. M., Louzoun, Y., Bettelheim, E. and Solomon, S. (2000) The importance of being discrete: Life always wins on the surface Proc. Nat. Acad. Sciences 97 , 10322-10324. Math. Review number not available.
- Shnerb, N. M., Bettelheim, E., Louzoun, Y., Agam, O. and Solomon, S. (2001), Adaptation of autocatalytic fluctuations to diffusive noise Phys. Rev. E 63 , 021103. Math. Review number not available.
- Spitzer, F. (1976), Principles of Random Walk, 2nd ed. Springer-Verlag. Math. Review 52:9383
- Strassen, V. (1965), The existence of probability measures with given marginals. Ann. Math. Statist. 36 , 423-439. Math. Review 31:1693
- Tanny, D. (1977), Limit theorems for branching processes in a random environment. Ann. Probab. 5 , 100-116. Math. Review 54:14135

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