## Electronic Journal of Probability

### Mutually Catalytic Branching in The Plane: Infinite Measure States

#### Abstract

A two-type infinite-measure-valued population in $R^2$ is constructed which undergoes diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a collision rate sufficiently small compared with the diffusion rate, the model is constructed as a pair of infinite-measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit (in law), local extinction of one type is shown. Moreover the surviving population is uniform with random intensity. The process constructed is a rescaled limit of the corresponding $Z^2$-lattice model studied by Dawson and Perkins (1998) and resolves the large scale mass-time-space behavior of that model under critical scaling. This part of a trilogy extends results from the finite-measure-valued case, whereas uniqueness questions are again deferred to the third part.

#### Article information

Source
Electron. J. Probab., Volume 7 (2002), paper no. 15, 61 pp.

Dates
Accepted: 15 March 2002
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.ejp/1463434888

Digital Object Identifier
doi:10.1214/EJP.v7-114

Mathematical Reviews number (MathSciNet)
MR1921744

Zentralblatt MATH identifier
1016.60075

Rights

#### Citation

Dawson, Donald; Etheridge, Alison; Fleischmann, Klaus; Mytnik, Leonid; Perkins, Edwin; Xiong, Jie. Mutually Catalytic Branching in The Plane: Infinite Measure States. Electron. J. Probab. 7 (2002), paper no. 15, 61 pp. doi:10.1214/EJP.v7-114. https://projecteuclid.org/euclid.ejp/1463434888

#### References

• M.T. Barlow and E.A. Perkins. On the filtration of historical Brownian motion. Ann. Probab., 22:1273-1294, 1994.
• J.T. Cox and D. Griffeath. Diffusive clustering in the two dimensional voter model. Ann. Probab., 14:347-370, 1986.
• J.T. Cox and A. Klenke. Recurrence and ergodicity of interacting particle systems. Probab. Theory Related Fields, 116(2):239-255, 2000.
• J.T. Cox, A. Klenke, and E.A. Perkins. Convergence to equilibrium and linear systems duality. In Luis G. Gorostiza and B. Gail Ivanoff, editors, Stochastic Models, volume 26 of CMS Conference Proceedings, pages 41-66. Amer. Math. Soc., Providence, 2000.
• D.A. Dawson, A.M. Etheridge, K. Fleischmann, L. Mytnik, E.A. Perkins, and J. Xiong. Mutually catalytic branching in the plane: Finite measure states. WIAS Berlin, Preprint No. 615, 2000, Ann. Probab., to appear 2002.
• D.A. Dawson and K. Fleischmann. A continuous super-Brownian motion in a super-Brownian medium. Journ. Theoret. Probab., 10(1):213-276, 1997.
• D.A. Dawson and K. Fleischmann. Longtime behavior of a branching process controlled by branching catalysts. Stoch. Process. Appl., 71(2):241-257, 1997.
• D.A. Dawson and K. Fleischmann. Catalytic and mutually catalytic super-Brownian motions. In Ascona 1999 Conference, volume 52 of Progress in Probability, pages 89-110, Birkhäuser Verlag, 2002.
• D.A. Dawson, K. Fleischmann, L. Mytnik, E.A. Perkins, and J. Xiong. Mutually catalytic branching in the plane: Uniqueness. WIAS Berlin, Preprint No. 641, Ann. Inst. Henri Poincaré Probab. Statist. (in print), 2002.
• D.A. Dawson and E.A. Perkins. Long-time behavior and coexistence in a mutually catalytic branching model. Ann. Probab., 26(3):1088-1138, 1998.
• J.-F. Delmas and K. Fleischmann. On the hot spots of a catalytic super-Brownian motion. Probab. Theory Relat. Fields, 121(3):389-421, 2001.
• A.M. Etheridge and K. Fleischmann. Persistence of a two-dimensional super-Brownian motion in a catalytic medium. Probab. Theory Relat. Fields, 110(1):1-12, 1998.
• S.N. Ethier and T.G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 1986.
• S.N. Evans and E.A. Perkins. Measure-valued branching diffusions with singular interactions. Canad. J. Math., 46(1):120-168, 1994.
• K. Fleischmann and A. Greven. Diffusive clustering in an infinite system of hierarchically interacting diffusions. Probab. Theory Relat. Fields, 98:517-566, 1994.
• K. Fleischmann and A. Greven. Time-space analysis of the cluster-formation in interacting diffusions. Electronic J. Probab., 1(6):1-46, 1996.
• K. Fleischmann and A. Klenke. Smooth density field of catalytic super-Brownian motion. Ann. Appl. Probab., 9(2):298-318, 1999.
• K. Fleischmann and A. Klenke. The biodiversity of catalytic super-Brownian motion. Ann. Appl. Probab., 10(4):1121-1136, 2000.
• K. Fleischmann, A. Klenke and J. Xiong. Mass-time-space scaling of a super-Brownian catalyst reactant pair. Preprint, University Koeln, Math. Instit., in preparation.
• P.-A. Meyer. Probability and Potentials. Blaisdell Publishing Company, Toronto, 1966.
• I. Mitoma. An $\infty$-dimensional inhomogeneous Langevin equation. J. Functional Analysis, 61:342-359, 1985.
• L. Mytnik. Uniqueness for a mutually catalytic branching model. Probab. Theory Related Fields, 112(2):245-253, 1998.
• E.A. Perkins. Dawson-Watanabe superprocesses and measure-valued diffusions. In École d'été de probabilités de Saint Flour XXIX-1999, Lecture Notes in Mathematics. Springer-Verlag, Berlin, to appear 2000.
• T. Shiga. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Can. J. Math., 46:415-437, 1994.
• T. Shiga and A. Shimizu. Infinite-dimensional stochastic differential equations and their applications. J. Mat. Kyoto Univ., 20:395-416, 1980.
• J.B. Walsh. An introduction to stochastic partial differential equations. volume 1180 of Lecture Notes Math., pages 266-439. École d'été de probabilités de Saint-Flour XIV - 1984, Springer-Verlag Berlin, 1986.