Mutually catalytic branching processes

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Mutually catalytic branching processes

Collaborator: K. Fleischmann , A. Sturm

Cooperation with: S. Athreya (Indian Statistical Institute, New Delhi), D.A. Dawson (Carleton University Ottawa, Canada), A.M. Etheridge (University of Oxford, UK), A. Klenke (Universität Erlangen), P. Mörters (University of Bath, UK), C. Mueller (University of Rochester, USA), L. Mytnik (Technion, Haifa, Israel), E.A. Perkins (University of British Columbia, Vancouver, Canada), A. Stevens (Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig), J. Swart (Universität Erlangen), V.A. Vatutin (Steklov Mathematical Institute, Moscow, Russia), A. Wakolbinger (Johann Wolfgang Goethe-Universität Frankfurt), J. Xiong (University of Tennessee, Knoxville, USA)

Supported by: DFG Priority Program ``Interagierende Stochastische Systeme von hoher Komplexität'' (Interacting stochastic systems of high complexity)

Description: Mutually catalytic branching processes describe the evolution of two materials, which randomly move, split, and possibly disappear in space. The system is interactive in that the branching rate of each material is proportional to the local density of the other one. This interaction destroys the usual basic independence assumption in branching theory. In particular, the connection to reaction-diffusion equations is lost. The model was created by Dawson, Perkins, and Mytnik in [1], [2] in the one-dimensional Euclidean space ${\mathbb R}$ and in the lattice space ${\mathbb Z}^{d}$ . After some time of controversial discussions, the non-degenerate existence of a two-dimensional continuum model of mutually catalytic branching processes was verified in the last year, and is now comprehensively presented in the trilogy [3], [4], [5]. Here, the second part complements the first one by extending it to infinite measure states. In the third part, uniqueness in the martingale problem is resolved. One of the remarkable properties of this model is that at almost all times the states are absolutely continuous with disjoint densities. At the same time, these densities are unbounded at the interface of both types. This explains how interaction works in the model despite the segregation of densities. The long-term behavior of this process is well-understood: Only one of the types may survive in the long run.

In the case of the simpler catalytic branching processes, with only a one-sided influence of a catalyst on a reactant, further aspects have been studied. In [6] open problems from [7] are answered. It is shown that for the continuous super-Brownian reactant in ${\mathbb R}$ with stable point catalysts under a mass-time-space scaling, asymptotic clusters appear that are macroscopically isolated as in the constant medium case. They continuously change in a non-Markovian way and are distributed in the domain of attraction of a stable law of index smaller than two. These results contrast the constant medium case. For the proof of this functional limit theorem, a Brownian snake approach in this catalytic setting is established, which is meanwhile slightly generalized in [8].

For the super-Brownian reactant in ${\mathbb R}$ with super-Brownian catalyst, a universal mass-time-space scaling limit is derived ([9]): If both substances are started in uniform states, the limit reactant exists and is uniform, for all scaling indices. This is done in a setting of convergence of finite-dimensional distributions, and for certain scaling indices also in terms of a functional limit theorem.

Related models, where the branching rate of the material is affected by a continuous correlated random environment, have also been studied. Ground work on such models has been done in [10]. In [11], existence and uniqueness have been established for a model with a similar branching mechanism which, as in the mutually catalytic branching process case, allows for the absolute continuity of states in higher dimensions, in this case on $\mathbb{R}^d$ . This is of particular interest since the densities solve stochastic partial differential equations and thus extend the connections of such solutions to super-Brownian motions.

References:

D.A. DAWSON, E.A. PERKINS, Long-time behavior and coexistence in a mutually catalytic branching model, Ann. Probab., 26 (1998), pp. 1088-1138.
L. MYTNIK, Uniqueness for a mutually catalytic branching model, Probab. Theory Related Fields, 112 (1998), pp. 245-253.
D.A. DAWSON, A.M. ETHERIDGE, K. FLEISCHMANN, L. MYTNIK, E.A. PERKINS, J. XIONG, Mutually catalytic branching in the plane: Finite measure states, WIAS Preprint no. 615, 2000, submitted.
$\dito$ ,Mutually catalytic branching in the plane: Infinite measure states, WIAS Preprint no. 633, 2000, submitted.
D.A. DAWSON, K. FLEISCHMANN, L. MYTNIK, E.A. PERKINS, J. XIONG, Mutually catalytic branching in the plane: Uniqueness, WIAS Preprint no. 641, 2001, to appear in: Ann. Inst. H. Poincaré Probab. Statist.
D.A. DAWSON, K. FLEISCHMANN, P. MÖRTERS, Strong clumping of super-Brownian motion in a stable catalytic medium, WIAS Preprint no. 636, 2001, to appear in: Ann. Probab.
D.A. DAWSON, K. FLEISCHMANN, Critical branching in a highly fluctuating random medium, Probab. Theory Related Fields, 90 (1991), pp. 241-274.
A. KLENKE, Catalytic branching and the Brownian snake,manuscript, Universität Erlangen, 2001.
K. FLEISCHMANN, J. XIONG, Mass-time-space scaling of a super-Brownian catalyst reactant pair, WIAS Preprint no. 706, 2001.
L. MYTNIK, Superprocesses in random environments, Ann. Probab., 24 (1996), pp. 1953-1978.
A. STURM, On spatially structured population processes and relations to stochastic partial differential equations, PhD thesis, Oxford University, submitted 2001.