Daniela Bertacchi

Characterization of Critical Values of Branching Random Walks on Weighted Graphs through Infinite-Type Branching Processes

We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. Two kinds of survival can be identified: a weak survival (with positive probability there is at least one particle alive somewhere at any time) and a strong survival (with positive probability the colony survives by returning infinitely often to a fixed site). The behavior of the process depends on the value of a certain parameter which controls the birth rates; the threshold between survival and (almost sure) extinction is called critical value. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible.

Uniform asymptotic estimates of transition probabilities on combs

We investigate the asymptotical behaviour of the transition probabilities of the simple random walk on the 2-comb. In particular, we obtain space-time uniform asymptotical estimates which show the lack of symmetry of this walk better than local limit estimates. Our results also point out the impossibility of getting sub-Gaussian estimates involving the spectral and walk dimensions of the graph.

Random walks on diestel-leader graphs

We investigate various features of a quite new family of graphs, introduced as a possible example of vertex-transitive graph not roughly isometric with a Cayley graph of some finitely generated group. We exhibit a natural compactification and study a large class of random walks, proving theorems concerning almost sure convergence to the boundary, a strong law of large numbers and a central limit theorem. The asymptotic type of then-step transition probabilities of the simple random walk is determined.

Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space

Abstract We characterize the separable complete ultrametric spaces whose Wijsman hyperspace admits a continuous selection; such an investigation is closely connected to a similar result of V. Gutev about the Ball hyperspace. The characterization may be obtained in terms of a suitable property either of the base space ( X , d ) ( condition (#)) or of the Wijsman hyperspace itself (total disconnectedness). We also give a necessary and sufficient condition for the zero-dimensionality of the Wijsman hyperspace of a (separable) ultrametric space, and we provide an example where such a hyperspace turns out to be connected.

Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions

We introduce spatially explicit stochastic processes to model multispecies host- symbiont interactions. The host environment is static, modeled by the infinite percolation cluster of site percolation. Symbionts evolve on the infinite cluster through contact or voter type interactions, where each host may be infected by a colony of symbionts. In the presence of a single symbiont species, the condition for invasion as a function of the density of the habitat of hosts and the maximal size of the colonies is investigated in details. In the presence of multiple symbiont species, it is proved that the community of symbionts clusters in two dimensions whereas symbiont species may coexist in higher dimensions. The term symbiosis was coined by the mycologist Heinrich Anto de Bary to denote close and long- term physical and biochemical interactions between different species, in constrast with competition and predation that imply only brief interactions. Symbiotic relationships involve a symbiont species, smaller in size, that always benefits from the relationship, and a host species, larger in size, that may either suffer, be relatively unaffected, or also benefit from the relationship, which are referred to as parasistism, commensalism, and mutualism, respectively. Symbiotic relationships are ubiquitous in nature. For instance, more than 90% of terrestrial plants (26) live in association with mycorrhizal fungi, with the plant providing carbon to the fungus and the fungus providing nutrients to the plant, most herbivores have mutualistic gut fauna that help them digest plant matter, and almost all free-living animals are host to one or more parasite taxa (25). To understand the role of spatial structure on the persistence of host-parasite and host-mutualist associations, Lanchier and Neuhauser (17, 18, 19) have initiated the study of multispecies host- symbiont systems including local interactions based on interacting particle systems. The stochastic process introduced in (18) describes the competition among specialist and generalist symbionts evolving in a deterministic static environment of hosts. The mathematical analysis of this model showed that fine-grained habitats promote generalist strategies, while coarse-grained habitats in- crease the competitiveness of specialists. The stochastic process introduced in (17, 19) includes in addition a feedback of the hosts, which is modeled by a dynamic-host system. This process has been further extended by Durrett and Lanchier (9). The host population evolves, in the absence of symbionts, according to a biased voter model, while the symbiont population evolves in this dynamic environment of hosts according to a contact type process. The parameters of the process allow to model the effect of the symbionts on their host as well as the degree of specificity of the symbionts, thus resulting into a system of coupled interacting particle systems, each describing the evolution of a trophic level. The model is designed to understand the role of the symbionts in the spatial structure of plant communities. It is proved theoretically that generalist symbionts

Ecological equilibrium for restrained branching random walks

We study a generalized branching random walk where particles breed at a rate which depends on the number of neighboring particles. Under general assumptions on the breeding rates we prove the existence of a phase where the population survives without exploding. We construct a nontrivial invariant measure for this case.

Strong local survival of branching random walks is not monotone

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

Branching random walks and multi-type contact-processes on the percolation cluster of Z d .

In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on

Rumor Processes in Random Environment on \mathbb{N} and on Galton–Watson Trees

{{\mathbb{Z}}^d}

A self-regulating and patch subdivided population

survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster