Advances in Superprocesses and Nonlinear PDEs

herausgegeben von: Janos Englander, Brian Rider

Verlag: Springer US

Buchreihe : Springer Proceedings in Mathematics & Statistics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Sergei Kuznetsov is one of the top experts on measure valued branching processes (also known as “superprocesses”) and their connection to nonlinear partial diﬀerential operators. His research interests range from stochastic processes and partial diﬀerential equations to mathematical statistics, time series analysis and statistical software; he has over 90 papers published in international research journals. His most well known contribution to probability theory is the "Kuznetsov-measure." A conference honoring his 60th birthday has been organized at Boulder, Colorado in the summer of 2010, with the participation of Sergei Kuznetsov’s mentor and major co-author, Eugene Dynkin. The conference focused on topics related to superprocesses, branching diffusions and nonlinear partial differential equations. In particular, connections to the so-called “Kuznetsov-measure” were emphasized. Leading experts in the field as well as young researchers contributed to the conference. The meeting was organized by J. Englander and B. Rider (U. of Colorado).

Inhaltsverzeichnis

Frontmatter

Markov Processes and Their Applications to Partial Differential Equations: Kuznetsov’s Contributions

Abstract

We describe some directions of research in probability theory and related problems of analysis to which S. E. Kuznetsov has made fundamental contributions.A Markov process (understood as a random path X _t, 0 ≤ t < ∞ such that past before t and future after t are independent given X _t) is determined by a probability measure P on a path space. This measure can be constructed starting from a transition function and probability distribution of X ₀. For a number of applications, it is also important to consider a path in both, forward and backward directions which leads to a concept of dual processes. In 1973, Kuznetsov constructed, as a substitute for such a pair of processes, a single random process (X _t, ℙ) determined on a random time interval (α, β). The corresponding forward and backward transition functions define a dual pair of processes. A σ-finite measure ℙ became, under the name “Kuznetsov measure,” an important tool for research on Markov processes and their applications.

In 1980, Kuznetsov proved that every Markov process in a Borel state space has a transition function (a problem that was open for many years). In 1992, he used this result to obtain simple necessary and sufficient conditions for existence of a unique decomposition of excessive functions into extreme elements—a significant extension of a classical result on positive superharmonic functions.

Intimate relations between the Brownian motion and differential equations involving the Laplacian Δ were known for a long time. Applications of probabilistic tools to classical potential theory and to study of linear PDEs are more recent. Even more recent is application of such tools to nonlinear PDEs. In a series of publications, starting from 1994, Dynkin and Kuznetsov investigated a class of semilinear elliptic equations by using super-Brownian motion and more general measure-valued Markov processes called superdiffusions. The main directions of this work were (a) description of removable singularities of solutions and (b) characterization of all positive solutions. One of the principal tools for solving the second problem was the fine trace of a solution on the boundary invented by Kuznetsov.

The same class of semilinear equations was the subject of research by Le Gall who applied a path-valued process Brownian snake instead of the super-Brownian motion. A slightly more general class of equations was studied by analysts including H. Brezis, M. Marcus and L. Veron. In the opinion of Brezis: “it is amazing how useful for PDEs are the new ideas coming from probability. This is an area where the interaction of probability and PDEs is most fruitful and exiting”.

E. B. Dynkin

Stochastic Equations on Projective Systems of Groups

Abstract

We consider stochastic equations of the form X _k = ϕ_k(X _{k + 1})Z _k, k ∈ ℕ, where X _k and Z _k are random variables taking values in a compact group G _k, ϕ_k :G _{k + 1} → G _k is a continuous homomorphism, and the noise \((Z_{k})_{k\in \mathbb{N}}\) is a sequence of independent random variables. We take the sequence of homomorphisms and the sequence of noise distributions as given and investigate what conditions on these objects result in a unique distribution for the “solution” sequence \((X_{k})_{k\in \mathbb{N}}\) and what conditions permit the existence of a solution sequence that is a function of the noise alone (i.e., the solution does not incorporate extra input randomness “at infinity”). Our results extend previous work on stochastic equations on a single group that was originally motivated by Tsirelson’s example of a stochastic differential equation that has a unique solution in law but no strong solutions.

Steven N. Evans, Tatyana Gordeeva

Modeling Competition Between Two Influenza Strains

Abstract

We use spatial and nonspatial models to argue that competition alone may explain why two influenza strains do not usually coexist during a given flu season. The more virulent strain is likely to crowd out the less virulent one. This can be seen as a consequence of the Exclusion Principle of Ecology. We exhibit, however, a spatial model for which coexistence is possible.

Rinaldo B. Schinazi

Asymptotic Results for Near Critical Bienaymé–Galton–Watson and Catalyst-Reactant Branching Processes

Abstract

Near critical single-type Bienaymé–Galton–Watson (BGW) processes are considered. Results on convergence of Yaglom distributions of suitably scaled BGW processes to that of the corresponding diffusion approximation are given. Convergences of stationary distributions for Q-processes and models with immigration to the corresponding distributions of the associated diffusion approximations are established. Similar results can be obtained in a multitype setting. To illustrate this, a result on convergence of Yaglom distributions of suitably scaled multitype subcritical BGW processes to that of the associated diffusion model is presented.

In the second part, near critical catalyst-reactant branching processes with controlled immigration are considered. The reactant population evolves according to a branching process whose branching rate is proportional to the total mass of the catalyst. The bulk catalyst evolution is that of a classical continuous-time branching process; in addition, there is a specific form of immigration. Immigration takes place exactly when the catalyst population falls below a certain threshold, in which case the population is instantaneously replenished to the threshold. A diffusion limit theorem for the scaled processes is presented, in which the catalyst limit is described through a reflected diffusion, while the reactant limit is a diffusion with coefficients that are functions of both the reactant and the catalyst. Stochastic averaging under fast catalyst dynamics is considered next. In the case where the catalyst evolves “much faster” than the reactant, a scaling limit, in which the reactant is described through a one-dimensional SDE with coefficients depending on the invariant distribution of the reflected diffusion, is obtained.

Amarjit Budhiraja, Dominik Reinhold

Some Path Large-Deviation Results for a Branching Diffusion

Abstract

We give an intuitive proof of a path large-deviations result for a typed branching diffusion as found in Git, J.Harris and S.C.Harris (Ann. App. Probab. 17(2):609-653, 2007). Our approach involves an application of a change of measure technique involving a distinguished infinite line of descent, or spine, and we follow the spine set-up of Hardy and Harris (Séminaire de Probabilités XLII:281–330, 2009). Our proof combines simple martingale ideas with applications of Varadhan’s lemma and is successful mainly because a “spine decomposition” effectively reduces otherwise difficult calculations on the whole collection of branching diffusion particles down to just a single particle (the spine) whose large-deviations behaviour is well known. A similar approach was used for branching Brownian motion in Hardy and Harris (Stoch. Process. Appl. 116(12):1992–2013, 2006). Importantly, our techniques should be applicable in a much wider class of branching diffusion large-deviations problems.

Robert Hardy, Simon C. Harris

Longtime Behavior for Mutually Catalytic Branching with Negative Correlations

Abstract

In several examples, dualities for interacting diffusion and particle systems permit the study of the longtime behavior of solutions. A particularly difficult model in which many techniques collapse is a two-type model with mutually catalytic interaction introduced by Dawson/Perkins for which they proved under some assumptions a dichotomy between extinction and coexistence directly from the defining equations.In the present chapter we show how to prove a precise dichotomy for a related model with negatively correlated noises. The proof uses moment bounds on exit times of correlated Brownian motions from the first quadrant and explicit second moment calculations. Since the uniform integrability bound is independent of the branching rate our proof can be extended to infinite branching rate processes.

Leif Döring, Leonid Mytnik

Super-Brownian Motion: L p -Convergence of Martingales Through the Pathwise Spine Decomposition

Abstract

Evans [7] described the semigroup of a superprocess with quadratic branching mechanism under a martingale change of measure in terms of the semigroup of an immortal particle and the semigroup of the superprocess prior to the change of measure. This result, commonly referred to as the spine decomposition, alludes to a pathwise decomposition in which independent copies of the original process “immigrate” along the path of the immortal particle. For branching particle diffusions, the analogue of this decomposition has already been demonstrated in the pathwise sense; see, for example, [11, 10]. The purpose of this short note is to exemplify a new pathwise spine decomposition for supercritical super-Brownian motion with general branching mechanism (cf. [13]) by studying L ^p-convergence of naturally underlying additive martingales in the spirit of analogous arguments for branching particle diffusions due to Harris and Hardy [10]. Amongst other ingredients, the Dynkin–Kuznetsov ℕ-measure plays a pivotal role in the analysis.

A. E. Kyprianou, A. Murillo-Salas

Backmatter

Titel: Advances in Superprocesses and Nonlinear PDEs
herausgegeben von: Janos Englander
Brian Rider
Verlag: Springer US
Electronic ISBN: 978-1-4614-6240-8
Print ISBN: 978-1-4614-6239-2
DOI: https://doi.org/10.1007/978-1-4614-6240-8