arXiv subjects

Daniela Bertacchi

Publications and source records attributed to Daniela Bertacchi.

31 records · Page 2Linked to original sources

arXiv2013

Strong local survival of branching random walks is not monotone

The aim of this paper is the study of 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 about 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 quasi transitive). 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 non-strong local survival. Finally we show that the generating function of a irreducible BRW can have more than two fixed points; this disproves a previously known result.

math.PR↗

arXiv2013

Combination versus sequential monotherapy in chronic HBV infection: a mathematical approach

Sequential monotherapy is the most widely used therapeutic approach in the treatment of HBV chronic infection. Unfortunately, under therapy, in some patients the hepatitis virus mutates and gives rise to variants which are drug resistant. We conjecture that combination therapy is able to delay drug resistance for a longer time than sequential monotherapy. To study the action of these two therapeutic approaches in the event of unknown mutations and to explain the emergence of drug resistance, we propose a stochastic model for the infection within a patient which is treated with two drugs, either sequentially or contemporaneously, and develops a two-step mutation which is resistant to both drugs. We study the deterministic approximation of our stochastic model and give a biological interpretation of its asymptotic behaviour. We compare the time when this new strain first reaches detectability in the serum viral load. Our results show that the best choice is to start an early combination therapy, which allows to stay drug-resistance free for a longer time.

math.PR↗

arXiv2012

Local and global survival for nonhomogeneous random walk systems on Z

We study an interacting random walk system on Z where at time 0 there is an active particle at 0 and one inactive particle on each site $n \ge 1$. Particles become active when hit by another active particle. Once activated, the particle starting at n performs an asymmetric, translation invariant, nearest neighbor random walk with left jump probability $l_n$. We give conditions for global survival, local survival and infinite activation both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan (with parameter depending on the starting location of the particle). More precisely, once activated, the particle at n survives at each step with probability $p_n \in [0, 1]$. In particular, in the immortal case, we prove a 0-1 law for the probability of local survival when all particles drift to the right. Besides that, we give sufficient conditions for local survival or local extinction when all particles drift to the left. In the mortal case, we provide sufficient conditions for global survival, local survival and local extinction (which apply to the immortal case with mixed drifts as well). Analysis of explicit examples is provided: we describe completely the phase diagram in the cases $1/2-l_n \sim \pm 1/n^α$, $p_n = 1$ and $1_n-1/2 \sim \pm 1/n^α$, $1-p_n \sim 1/n^β$ (where $α,β> 0$).

math.PR↗

arXiv2012

A self-regulating and patch subdivided population

We consider an interacting particle process on a graph which, from a macroscopic point of view, looks like $\Z^d$ and, at a microscopic level, is a complete graph of degree $N$ (called a patch). There are two birth rates: an inter-patch one $λ$ and an intra-patch one $ϕ$. Once a site is occupied, there is no breeding from outside the patch and the probability $c(i)$ of success of an intra-patch breeding decreases with the size $i$ of the population in the site. We prove the existence of a critical value $λ_{cr}(ϕ, c, N)$ and a critical value $ϕ_{cr}(λ, c, N)$. We consider a sequence of processes generated by the families of control functions $\{c_i\}_{i \in \N}$ and degrees $\{N_i\}_{i \in \N}$; we prove, under mild assumptions, the existence of a critical value $i_{cr}$. Roughly speaking we show that, in the limit, these processes behave as the branching random walk on $\Z^d$ with external birth rate $λ$ and internal birth rate $ϕ$. Some examples of models that can be seen as particular cases are given.

math.PR↗

arXiv2011

The small world effect on the coalescing time of random walks

A small world is obtained from the $d$-dimensional torus of size 2L adding randomly chosen connections between sites, in a way such that each site has exactly one random neighbour in addition to its deterministic neighbours. We study the asymptotic behaviour of the meeting time $T_L$ of two random walks moving on this small world and compare it with the result on the torus. On the torus, in order to have convergence, we have to rescale $T_L$ by a factor $C_1L^2$ if $d=1$, by $C_2L^2\log L$ if $d=2$ and $C_dL^d$ if $d\ge3$. We prove that on the small world the rescaling factor is $C^\prime_dL^d$ and identify the constant $C^\prime_d$, proving that the walks always meet faster on the small world than on the torus if $d\le2$, while if $d\ge3$ this depends on the probability of moving along the random connection. As an application, we obtain results on the hitting time to the origin of a single walk and on the convergence of coalescing random walk systems on the small world.

math.PR↗

arXiv2008

Characterization of the 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. 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.

math.PR↗

arXiv2008

Approximating critical parameters of branching random walks

Given a branching random walk on a graph, we consider two kinds of truncations: by inhibiting the reproduction outside a subset of vertices and by allowing at most $m$ particles per site. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.

math.PR↗

arXiv2007

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.

math.PR↗

arXiv2007

Weak survival for branching random walks on graphs

We study weak and strong survival for branching random walks on multigraphs. We prove that, for a large class of multigraphs, weak survival is related to a geometrical parameter of the multigraph and that the existence of a pure weak phase is equivalent to nonamenability. Finally we study weak and strong critical behaviors of the branching random walk.

math.PR↗

arXiv2006

Asymptotic behaviour of the simple random walk on the 2-comb

We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $n$ steps, proving that for all these quantities the order is $n^{1/4}$ for the horizontal projection and $n^{1/2}$ for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical ones, respectively. The limit process is obtained. As a corollary of the estimate of the expected value of the maximal deviation, the walk dimension is determined, showing that the Einstein relation between the fractal, spectral and walk dimensions does not hold on the comb.

math.PR↗

arXiv2002

Classification on the average of random walks

We introduce a new method for studying large scale properties of random walks. The new concepts of transience and recurrence on the average are compared with the ones introduced by R.Burioni, D.Cassi and A.Vezzani and with the usual ones; their relationships are analyzed and various examples are provided.

math.PR↗

arXiv2001

Enhanced interface repulsion from quenched hard-wall randomness

We consider the harmonic crystal on the d-dimensional lattice, d larger or equal to 3, that is the centered Gaussian field $ϕ$ with covariance given by the Green function of the simple random walk on $Z^d$. Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition the field to be larger than an IID field $σ$ (which is also independent of $ϕ$), for every x in a large region $D_N=ND\cap \Z^d$, with N a positive integer and $D \subset\R^d$. We are mostly motivated by results for given typical realizations of the $σ$ (quenched set-up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, constrained not to go below a inhomogeneous substrate that acts as a hard wall. We consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of $σ$ is heavier than Gaussian, while essentially no effect is observed if the tail is sub--Gaussian. In the critical case, that is the one of approximately Gaussian tail, the interplay of the two sources of randomness, $ϕ$ and $σ$, leads to an enhanced repulsion effect of additive type.

math.PR↗

arXiv2000

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 Jones-type non-Gaussian estimates.

math.PR↗