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

**Donald A. Dawson**

*(Carleton University)*

**Alison M. Etheridge**

*(University of Oxford)*

**Klaus Fleischmann**

*(Weierstrass Institute for Applied Analysis and Stochastics)*

**Leonid Mytnik**

*(Technion - Israel Institute of Technology)*

**Edwin A. Perkins**

*(The University of British Columbia)*

**Jie Xiong**

*(University of Tennessee)*

#### 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.

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Pages: 1-61

Publication Date: March 15, 2002

DOI: 10.1214/EJP.v7-114

#### References

- M.T. Barlow and E.A.
Perkins.
*On the filtration of historical Brownian motion*. Ann. Probab., 22:1273-1294, 1994. Math. Review 96g:60058 - J.T. Cox and D. Griffeath.
*Diffusive clustering in the two dimensional voter model*. Ann. Probab., 14:347-370, 1986. Math. Review 91d:60250 - J.T. Cox and A. Klenke.
*Recurrence and ergodicity of interacting particle systems*. Probab. Theory Related Fields, 116(2):239-255, 2000. Math. Review 2001j:60181 - 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. Math. Review 2001h:60175 - 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. Math Review article not yet available. - D.A. Dawson and K.
Fleischmann.
*A continuous super-Brownian motion in a super-Brownian medium*. Journ. Theoret. Probab., 10(1):213-276, 1997. Math. Review 98a:60062 - D.A. Dawson and K.
Fleischmann.
*Longtime behavior of a branching process controlled by branching catalysts*. Stoch. Process. Appl., 71(2):241-257, 1997. Math. Review 99c:60187 - 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, Birkhâ°user Verlag, 2002. Math Review article not yet available. - 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 PoincarÃ Probab. Statist. (in print), 2002. Math Review article not yet available. - 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. Math. Review 99f:60167 - 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. Math. Review 911 867 428 - 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. Math. Review 98k:60149 - S.N. Ethier and T.G.
Kurtz.
*Markov Processes: Characterization and Convergence*. Wiley, New York, 1986. Math. Review 88a:60130 - S.N. Evans and E.A.
Perkins.
*Measure-valued branching diffusions with singular interactions*. Canad. J. Math., 46(1):120-168, 1994. Math. Review 94J:60099 - K. Fleischmann and A.
Greven.
*Diffusive clustering in an infinite system of hierarchically interacting diffusions*. Probab. Theory Relat. Fields, 98:517-566, 1994. Math. Review 95j:60163 - K. Fleischmann and A.
Greven.
*Time-space analysis of the cluster-formation in interacting diffusions*. Electronic J. Probab., 1(6):1-46, 1996. Math. Review 97e:60151 - K. Fleischmann and A.
Klenke.
*Smooth density field of catalytic super-Brownian motion*. Ann. Appl. Probab., 9(2):298-318, 1999. Math. Review 2000k:60168 - K. Fleischmann and A.
Klenke.
*The biodiversity of catalytic super-Brownian motion*. Ann. Appl. Probab., 10(4):1121-1136, 2000. Math. Review 2002e:60079 - 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. Math Review article not yet available. - P.-A. Meyer.
*Probability and Potentials*. Blaisdell Publishing Company, Toronto, 1966. Math. Review 34 #5119 - I. Mitoma.
*An $infty$-dimensional inhomogeneous Langevin equation*. J. Functional Analysis, 61:342-359, 1985. Math. Review 87i:60066 - L. Mytnik.
*Uniqueness for a mutually catalytic branching model*. Probab. Theory Related Fields, 112(2):245-253, 1998. Math. Review 99i:60125 - E.A. Perkins.
*Dawson-Watanabe superprocesses and measure-valued diffusions*. In â¦cole d'ÃtÃ de probabilitÃs de Saint Flour XXIX-1999, Lecture Notes in Mathematics. Springer-Verlag, Berlin, to appear 2000. Math Review article not yet available. - T. Shiga.
*Two contrasting properties of solutions for one-dimensional stochastic partial differential equations*. Can. J. Math., 46:415-437, 1994. Math. Review 95h:60099 - T. Shiga and A. Shimizu.
*Infinite-dimensional stochastic differential equations and their applications*. J. Mat. Kyoto Univ., 20:395-416, 1980. Math. Review82i:60110 - J.B. Walsh.
*An introduction to stochastic partial differential equations*. volume 1180 of Lecture Notes Math., pages 266-439. â¦cole d'ÃtÃ de probabilitÃs de Saint-Flour XIV - 1984, Springer-Verlag Berlin, 1986. Math. Review 88a:60114

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