János Engländer

Law of large numbers for a class of superdiffusions

Abstract Pinsky [R.G. Pinsky, Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions, Ann. Probab. 24 (1) 237–267] proved that the finite mass superdiffusion X corresponding to the semilinear operator L u + β u − α u 2 exhibits local extinction if and only if λ c ⩽ 0 , where λ c : = λ c ( L + β ) is the generalized principal eigenvalue of L + β on R d . For the case when λ c > 0 , it has been shown in Englander and Turaev [J. Englander, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] that in law the superdiffusion locally behaves like exp [ t λ c ] times a non-negative non-degenerate random variable, provided that the operator L + β − λ c satisfies a certain spectral condition (‘product-criticality’), and that α and μ = X 0 are ‘not too large’. In this article we will prove that the convergence in law used in the formulation in [J. Englander, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] can actually be replaced by convergence in probability. Furthermore, instead of R d we will consider a general Euclidean domain D ⊆ R d . As far as the proof of our main theorem is concerned, the heavy analytic method of [J. Englander, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] is replaced by a different, simpler and more probabilistic one. We introduce a space–time weighted superprocess (H-transformed superprocess) and use it in the proof along with some elementary probabilistic arguments.

Law of large numbers for superdiffusions: The non-ergodic case

In previous work of D. Turaev, A. Winter and the author, the Law of Large Numbers for the local mass of certain superdiffusions was proved under an ergodicity assumption. In this paper we go beyond ergodicity, that is we consider cases when the scaling for the expectation of the local mass is not purely exponential. Inter alia, we prove the analog of theWatanabe–Biggins LLN for super-Brownian motion.

Uniqueness/nonuniqueness for nonnegative solutions of second-order parabolic equations of the form ut=Lu+Vu−γup in Rn

In this paper we investigate uniqueness and nonuniqueness for solutions of the equation (NS)ut=Lu+Vu−γupinRn×(0,∞),u(x,0)=f(x),x∈Rn,u⩾0, where γ>0,p>1,γ,V∈Cα(Rn),0⩽f∈C(Rn) and L=∑i,j=1nai,j(x)∂2∂xi∂xj+∑i=1nbi(x)∂∂xi with ai,j,bi∈Cα(Rn).

Quenched law of large numbers for branching Brownian motion in a random medium

We study a spatial branching model, where the underlying motion is d-dimensional (d≥1) Brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all d≥1. We also show that the branching Brownian motion with mild obstacles spreads less quickly than ordinary branching Brownian motion by giving an upper estimate on its speed. When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the quenched local growth that is independent of the Poissonian intensity. More general offspring distributions (beyond the dyadic one considered in the main theorems) as well as mild obstacle models for superprocesses are also discussed.We study a spatial branching model, where the underlying motion is d-dimensional (d ≥ 1) Brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all d≥ 1. We also show that the branching Brownian motion with mild obstacles spreads less quickly than ordinary branching Brownian motion by giving an upper estimate on its speed. When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the quenched local growth that is independent of the Poissonian intensity. More general offspring distributions (beyond the dyadic one considered in the main theorems) as well as mild obstacle models for superprocesses are also discussed. Résumé. Nous étudions un modèle de branchement spatial où le mouvement de base est Brownien d-dimensionnel (d≥ 1) et le taux de branchement est modifié par une collection aléatoire d’ensembles sur lesquels la reproduction n’a pas lieu (obstacles moux). Le résultat principal de cet article est l’asymptotique (en probabilité) des taux de croissance globaux “quenchés” pour tout d≥ 1, et nous identifions les termes de correction sous-exponentielle. Nous montrons aussi que le branchement Brownien avec obstacles moux diffuse moins vite que le branchement Brownien classique en donnant une borne supérieure de sa vitesse. Dans le cas où le mouvement de base est un processus de diffusion arbitraire nous obtenons une dichotomie pour la croissance locale “quenchée” qui est indépendante de l’intensité Poissonnienne. Le cas de distributions plus générales du nombre de descendants (autre que le cas dyadique considéré dans les théorème principaux), ainsi que des modèles d’obstacles moux pour des superprocessus, sont aussi discutés. MSC: Primary 60J65; secondary 60J80; 60F10; 82B44

Extinction properties of super-Brownian motions with additional spatially dependent mass production

Consider the finite measure-valued continuous super-Brownian motion X on corresponding to the log-Laplace equation where the coefficient [beta](x) for the additional mass production varies in space, is Holder continuous, and bounded from above. We prove criteria for (finite time) extinction and local extinction of X in terms of [beta]. There exists a threshold decay rate kdx-2 as x-->[infinity] such that X does not become extinct if [beta] is above this threshold, whereas it does below the threshold (where for this case [beta] might have to be modified on a compact set). For local extinction one has the same criterion, but in dimensions d>6 with the constant kd replaced by Kd>kd (phase transition). h-transforms for measure-valued processes play an important role in the proofs. We also show that X does not exhibit local extinction in dimension 1 if [beta] is no longer bounded from above and, in fact, degenerates to a single point source [delta]0. In this case, its expectation grows exponentially as t-->[infinity].

On the volume of the supercritical super-Brownian sausage conditioned on survival

Let [alpha] and [beta] be positive constants. Let X be the supercritical super-Brownian motion corresponding to the evolution equation in and let Z be the binary branching Brownian-motion with branching rate [beta]. For t[greater-or-equal, slanted]0, let , that is R(t) is the (accumulated) support of X up to time t. For t[greater-or-equal, slanted]0 and a>0, let We call Ra(t) the super-Brownian sausage corresponding to the supercritical super-Brownian motion X. For t[greater-or-equal, slanted]0, let , that is is the (accumulated) support of Z up to time t. For t[greater-or-equal, slanted]0 and a>0, let We call the branching Brownian sausage corresponding to Z. In this paper we prove that for all d[greater-or-equal, slanted]2 and all a,[alpha],[nu]>0.

The compact support property for measure-valued processes

The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued processes corresponding to semi-linear equations of the form ut = Lu + βu− αu p in R d × (0, ∞), p ∈ (1, 2]; u(x, 0) = f( x) in R d ; u(x, t) 0i nR d ×[ 0, ∞).

Weak extinction versus global exponential growth of total mass for superdiffusions

Consider a superdiffusion X on R d corresponding to the semi-linear operator A(u) = Lu + βu − ku 2 ,w hereL is a second order elliptic operator, β(·) is in the Kato class, and k(·) ≥ 0 is bounded on compact subsets of R d and is positive on a set

Spatial branching in random environments and with interaction

Preliminaries The Spine Construction and the Strong Law of Large Numbers Examples of the Strong Law The Strong Law for a Type of Self-Interaction The Center of Mass Branching Brownian Motion in a Poissonian Field of Traps Branching Brownian Motion in a Field of Mild Obstacles Critical Branching Random Walk in Random Environment.

Critical branching random walk in an IID environment

Abstract Using a high performance computer cluster, we run simulations regarding an open problem about d-dimensional critical branching random walks in a random IID environment The environment is given by the rule that at every site independently, with probability p ∈ [0, 1], there is a cookie, completely suppressing the branching of any particle located there. The simulations suggest self averaging: the asymptotic survival probability in n steps is the same in the annealed and the quenched case; it is , where q ≔ 1 –p. This particular asymptotics indicates a non-trivial phenomenon: the tail of the survival probability (both in the annealed and the quenched case) is the same as in the case of non-spatial unit time critical branching, where the branching rule is modified: branching only takes place with probability q for every particle at every iteration.