Interacting catalytic branching processes

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Interacting catalytic branching processes

Collaborator: K. Fleischmann, A. Sturm, V. Vakthel

Cooperation with: D.A. Dawson (Carleton University Ottawa, Canada), A.M. Etheridge (University of Oxford, UK), A. Klenke (Universität Köln and Universität Mainz), 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), J. Swart (Friedrich-Alexander-Universität Erlangen-Nürnberg), 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); Alexander von Humboldt-Stiftung (Alexander von Humboldt Foundation): Fellowship

Description: Pairs of interacting catalytic branching processes describe the evolution of two types of materials, which randomly move, split, and possibly disappear in space. The point is the interaction in the system: The branching behavior of each material is dependent on the other one.

A main result this year is the construction and investigation of a so-called symbiotic branching process, [3]. This model generalizes three well-known interacting models: the mutually catalytic branching model in $\mathbb {R}$ of Dawson and Perkins (1998), [2], and Mytnik (1998), [11], the continuous stepping-stone model of Shiga (1988), [13], and the continuous space Anderson model (see, for instance, Mueller (1991), [10]). Compared with mutually catalytic branching, the two types might now be somehow correlated. Basic tools such as self-duality, particle system moment duality, measure case moment duality, and moment equations are shown to be still available in this generalized context. As an application, the compact interface property is derived: Starting from complementary Heaviside states, the interface is compact almost surely at each time. For the stepping-stone model, such property was known earlier by Tribe (1995), [15], but for mutually catalytic branching, for instance, this is the first result on the interface at all. Techniques from [15] are essentially used, except the ones which have been based on the boundedness of states. Instead of this, some exponential grow of certain moments could be established, which are now used as a replacement for the original boundedness of states.

Superprocesses under a Brownian flow are studied in [16]. Such processes were first considered by Skoulakis and Adler (2001), [14], by a moment duality method. As they indicated, it is natural to study the process under a fixed environment and to make use of the related conditional log-Laplace transform. This idea is now confirmed in [16]. The conditional log-Laplace functional is in fact shown to be the unique solution to a nonlinear stochastic partial differential equation by making use of a particle system representation developed by Kurtz and Xiong (1999), [8]. This approach has many potential applications.

In [9] the model is governed by a white noise in space-time. Therefore, it also involves spatial interaction. Further, an immigration mechanism is introduced to the model. In this case, the conditional log-Laplace functional is the unique solution to a nonlinear SPDE driven by space-time white noise. As an application, the long-term limit is obtained.

A uniqueness problem raised in Fleischmann and Xiong (2001), [6], for critical cyclically catalytic super-Brownian motions is solved in [1] in the simplified spaceless case, that is, for cyclically catalytic branching diffusions X, where, moreover, any correlation between the components is allowed (in the sense of symbiotic branching). More precisely, X is characterized as the unique strong solution of a singular stochastic equation.

A spatial version of Neveu's (1992), [12], continuous-state branching process is constructed in [4]. Opposed to earlier superprocesses, here the branching has infinite mean. Construction is provided by starting from certain supercritical ( $\alpha$ , d, $\beta$ )-superprocesses X⁽⁾ with symmetric $\alpha$ -stable motion and (1 + $\beta$ )-branching and proving convergence on path space of finite measure-valued cadlag paths as $\beta$ $\downarrow$ 0. The log-Laplace equation related to the new process X, say, has the locally non-Lipschitz function u log u as nonlinear term (instead of u¹⁺ in the case of X⁽⁾). It can nevertheless be shown to be well posed. X behaves quite differently from usual supercritical spatial branching processes. In fact, it is immortal at all finite times, propagates mass instantaneously everywhere in space also in the Brownian case $\alpha$ = 2, and it has locally countably infinite biodiversity.

The phenomenon of multi-scale clustering is verified in [5] for a non-Markovian branching particle system in $\mathbb {R}$ ^d in the critical dimension d = $\alpha$ / $\beta$ , where particles move according to a symmetric $\alpha$ -stable motion, have a lifetime distribution of finite mean, and branch with an offspring law of index 1 + $\beta$ , 0 < $\beta$ $\leq$ 2. This is expressed in an fdd scaling limit theorem, where initially one starts with an increasing localized population or with an increasing homogeneous Poissonian population. The limit state is uniform, but its intensity varies in line with the scaling index according to a continuous-state branching process of index 1 + $\beta$ . The case $\alpha$ = 2 of Brownian particles was due to Klenke (1998), [7], where PDE methods had been used which are not available in the present setting.

References:

D.A. DAWSON, K. FLEISCHMANN, J. XIONG, Strong uniqueness of cyclically symbiotic branching diffusions, WIAS Preprint no. 853, 2003.
D.A. DAWSON, E.A. PERKINS, Long-time behavior and coexistence in a mutually catalytic branching model, Ann. Probab., 26 (1998), pp. 1088-1138.
A.M. ETHERIDGE, K. FLEISCHMANN, Compact interface property for symbiotic branching, WIAS Preprint no. 822, 2003, submitted.
K. FLEISCHMANN, A. STURM, A super-stable motion with infinite mean branching, WIAS Preprint no. 821, 2003, submitted.
K. FLEISCHMANN, V.A. VATUTIN, Multi-scale clustering for a non-Markovian spatial branching process, WIAS Preprint no. 889, 2003.
K. FLEISCHMANN, J. XIONG, A cyclically catalytic super-Brownian motion, Ann. Probab., 29 (2001), pp. 820-861.
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C. MUELLER, Long time existence for the heat equation with a noise term, Probab. Theory Related Fields, 90 (1991), pp. 505-518.
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J. NEVEU, A continuous state branching process in relation with the GREM model of spin glasses theory, Rapport interne no. 267, Ecole Polytechnique, Palaiseau, France, 1992.
T. SHIGA, Stepping stone models in population genetics and population dynamics, in: Stochastic Processes in Physics and Engineering, S. Albeverio et al., eds., vol. 42 of Mathematics and its Applications, D. Reidel Publishing Company, Dordrecht, 1988, pp. 345-355.
G. SKOULAKIS, R.J. ADLER, Superprocesses over a stochastic flow, Ann. Appl. Probab., 11 (2001), pp. 488-543.
R. TRIBE, Large time behavior of interface solutions to the heat equation with Fisher-Wright white noise, Probab. Theory Related Fields, 102 (1995), pp. 289-311.
J. XIONG, A stochastic log-Laplace equation, WIAS Preprint no. 859, 2003.