Branching processes in random media

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Branching processes in random media

Collaborator: K. Fleischmann, V. Vakhtel

Cooperation with: J. Blath, A.M. Etheridge (University of Oxford, UK), D.A. Dawson (Carleton University Ottawa, Canada), A. Klenke (Universität Mainz), P. Mörters, P. Vogt (University of Bath, UK), C. Mueller (University of Rochester, USA), L. Mytnik (Technion Haifa, Israel), A. Sturm (University of Delaware, Canada), J. Swart (Universität Erlangen), V.A. Vatutin (Steklov Mathematical Institute, Moscow, Russia), J. Xiong (University of Tennessee, Knoxville, USA)

Supported by: DFG: ``Hydrodynamische Fluktuationen von Verzweigungsmodellen in katalytischen Medien mit unendlicher Erwartung`` (Hydrodynamic fluctuations of branching models in catalytic media with infinite expectation); Alexander von Humboldt-Stiftung (Alexander von Humboldt Foundation): Fellowship

Description:

Branching models describe the evolution of materials, which randomly move, split, and possibly disappear in space. Typically, a random medium is additionally involved, often called the catalyst.

Essential progress has been made in the understanding of the spatial version of Neveu's continuous-state branching process constructed in [3]. In this super- $\alpha$ -stable motion X in $\mathbb {R}$ ^d (0 < $\alpha$ $\leq$ 2) in a constant medium, the nonlinear term in the related log-Laplace equation is the locally non-Lipschitz function u log u, hence the branching has infinite expectation. Clearly, in supercritical super- $\alpha$ -stable motions of finite mean, one expects a spread out of mass in space according to a profile described by the $\alpha$ -stable density function. But here in the infinite mean case, at macroscopic scales, the mass renormalized to a (random) probability measure is concentrated in a single space point which randomly fluctuates according to the underlying symmetric $\alpha$ -stable motion, see [4]. In the Brownian case $\alpha$ = 2, convergence is shown on the Skorohod path space, whereas for $\alpha$ < 2 only convergence of finite-dimensional distributions is established. Tightness on the Skorohod space is actually violated in the latter case, this is verified in Birkner and Blath [1] using lookdown constructions. Surprisingly, the macroscopic single-point concentration is shown by asymptotic calculations using only the first two moments of the randomly renormalized process, which can be computed based on a remarkable log-Laplace product formula which holds just by the occurrence of the logarithm in the branching term.

Superprocesses under a Brownian flow had been introduced by Skoulakis and Adler [12]. A conditional log-Laplace (CLL) approach was founded by Xiong [14]. Many of the properties of former super-Brownian motions were derived by using the log-Laplace transform which is the unique solution to a nonlinear partial differential equation (PDE). Although the nonlinear stochastic partial differential equation (SPDE) satisfied by the CLL transform is much harder to handle than the PDE, it can be expected that many properties of this superprocess under a Brownian flow can be derived using the CLL transform. As a first step in this direction, in [15] the long-term behavior of this superprocess is investigated by means of the CLL transform. Then, in [10], the CLL transform for a more general model driven by a space-time white noise is studied, and additionally the immigration of particles is allowed. Again, the long-term behavior of the superprocess is obtained. Finally in [11], a conditional excursion representation for the process from [10] is established by making use of the CLL approach.

The particle system representations [7], [8] due to Kurtz and Xiong for a class of SPDEs have been completed in [9]. This method has found applications in filtering theory. It turns out that the derivation of the filtering equation based on this method is much simpler than the classical one. Further, it is applicable to a much broader class of filtering models (cf. Kouritzin and Xiong [6] and Xiong and Zhao [17]). As a further application, in [16] a utility maximization problem in selecting optimal investment portfolio is considered when the appreciation rate for the stock is unobserved. Filtering technique is used to estimate this appreciation rate. Note that models as in [6], [16], [17] arise naturally from real-world problems, and classical filtering methods do not apply to these models.

Further understanding has been achieved for the single point catalytic super-Brownian motion. In [5], a large deviation principle is established when the branching rate tends to zero. The representation formula of [2] plays a key role. In the related model of single point catalytic branching random walk, interesting new long-term effects have been discovered, see [13].

References:

M. BIRKNER, J. BLATH, Non-tightness of rescaled $\alpha$ -spatial Neveu process for the case $\alpha$ < 2, in preparation.
K. FLEISCHMANN, J.-F. LE GALL, A new approach to the single point catalytic super-Brownian motion, Probab. Theory Related Fields, 102 (1995), pp. 63-82.
K. FLEISCHMANN, A. STURM, A super-stable motion with infinite mean branching, Ann. Inst. H. Poincaré Probab. Statist., 40 (2004), pp. 513-537.
K. FLEISCHMANN, V. VAKHTEL, Large scale localization of a spatial version of Neveu's branching process, WIAS Preprint no. 951, 2004.
K. FLEISCHMANN, J. XIONG, Large deviation principle for the single point catalytic super-Brownian motion, WIAS Preprint no. 937, 2004, submitted.
M. KOURITZIN, J. XIONG, Nonlinear filtering: From Ornstein-Uhlenbeck to white noise, to appear in: SIAM J. Control Optim., 2005.
T.G. KURTZ, J. XIONG, Particle representations for a class of nonlinear SPDEs, Stochastic Process. Appl., 83 (1999), pp. 103-126.
, Numerical solutions for a class of SPDEs with application to filtering, in: Stochastics in Finite and Infinite Dimensions, T. Hida, R. Karandikar, H. Kunita, B. Rajput, S. Watanabe, J. Xiong, eds., Birkhäuser, Boston, MA, 2001.
, A stochastic evolution equation arising from the fluctuation of a class of interacting particle systems, Commun. Math. Sci., 2 (2004), pp. 325-358.
Z. LI, H. WANG, J. XIONG, Conditional log-Laplace functionals of immigration superprocesses with dependent spatial motion, WIAS Preprint no. 900, 2004, to appear in: Acta Appl. Math., 2005.
, Conditional excursion representation for a class of interacting superprocesses, WIAS Preprint no. 935, 2004.
G. SKOULAKIS, R.J. ADLER, Superprocesses over a stochastic flow, Ann. Appl. Probab., 11 (2000), pp. 488-543.
V. VATUTIN, J. XIONG, Some limit theorems for a particle system of single point catalytic branching random walks, WIAS Preprint no. 919, 2004.
J. XIONG, A stochastic log-Laplace equation, Ann. Probab., 32 (2004), pp. 2262-2388.
, Long-term behavior for SBM over a stochastic flow, Electron. Comm. Probab., 9 (2004), pp. 36-52.
J. XIONG, Z.J. YANG, Optimal investment strategy under saving/borrowing rates spread and under partial information, WIAS Preprint no. 908, 2004.
J. XIONG, X. ZHAO, Nonlinear filtering with fractional Brownian motion noise, to appear in: Stochastic Anal. Appl., 2005