Stochastic Processes and their Applications最新文献

Scale-free configuration models are intimately connected to power law Galton–Watson trees. It is known that contact process epidemics can propagate on these trees and therefore these networks with arbitrarily small infection rate, and this continues to be true after uniformly immunizing a small positive proportion of vertices. So, we instead immunize those with largest degree: above a threshold for the maximum permitted degree, we discover the epidemic with immunization has survival probability similar to without, by duality corresponding to comparable metastable density. With maximal degree below a threshold on the same order, this survival probability is severely reduced or zero.

In this paper, we explore the metastable behavior of the Glauber dynamics associated with the three-state Potts model with an asymmetrical external field at a low-temperature regime. The model exhibits three monochromatic configurations: a unique stable state and two metastable states with different stability levels. We investigate metastable transitions, which are transitions from each metastable state to the ground state, separately and verify that the two transitions exhibit different behaviors.

For the metastable state with greater stability, we derive large deviation-type results for metastable transition time, both in terms of probability and expectation. On the other hand, a particularly intriguing phenomenon emerges when starting from the other metastable state: the process may fall into the deep valley of another metastable state with a low probability but remains trapped there for an exponentially long time. We identify specific conditions on the external field under which this rare event contributes to the mean hitting time. To this end, we conduct a detailed analysis of the energy landscape, revealing a sharp saddle configuration analogous to the Ising model.

We quantify superdiffusive transience for a two-dimensional random walk in which the vertical coordinate is a martingale and the horizontal coordinate has a positive drift that is a polynomial function of the individual coordinates and of the present time. We describe how the model was motivated through an heuristic connection to a self-interacting, planar random walk which interacts with its own centre of mass via an excluded-volume mechanism, and is conjectured to be superdiffusive with a scale exponent $3/4$. The self-interacting process originated in discussions with Francis Comets.

The present paper is devoted to the study of diagonally quadratic backward stochastic differential equation with oblique reflection. Using a penalization approach, we show the existence of a solution by providing some delicate a priori estimates. We further obtain the uniqueness by verifying the first component of the solution is indeed the value of a switching problem for quadratic BSDEs. Moreover, we provide an extension for the solvability and apply our results to study a risk-sensitive switching problem for functional stochastic differential equations.

The aim of this article is to study the impact of resistance acquisition on the distribution of neutral mutations in a cell population under therapeutic pressure. The cell population is modeled by a bi-type branching process. Initially, the cells all carry type 0, associated with a negative growth rate. Mutations towards type 1 are assumed to be rare and random, and lead to the survival of cells under treatment, i.e. type 1 is associated with a positive growth rate, and thus models the acquisition of a resistance. Cells also carry neutral mutations, acquired at birth and accumulated by inheritance, that do not affect their type. We describe the expectation of the ”Site Frequency Spectrum” (SFS), which is an index of neutral mutation distribution in a population, under the asymptotic of rare events of resistance acquisition and of large initial population. Precisely, we give asymptotically-equivalent expressions of the expected number of neutral mutations shared by both a small and a large number of cells. To identify the influence of relatives on the SFS, our work also lead us to study in detail subcritical binary Galton–Watson trees, where each leaf is marked with a small probability. As a by-product of this study, we thus provide the law of the generation of a randomly chosen leaf in such a Galton–Watson tree conditioned on the number of marks.

In order to understand the cost of a potentially high infectiousness of symptomatic individuals or, on the contrary, the benefit of social distancing, quarantine, etc. in the course of an infectious disease, this paper considers a natural variant of the popular contact process that distinguishes between asymptomatic and symptomatic individuals. Infected individuals all recover at rate one but infect nearby individuals at a rate that depends on whether they show the symptoms of the disease or not. Newly infected individuals are always asymptomatic and may or may not show the symptoms before they recover. The analysis of the corresponding mean-field model reveals that, in the absence of local interactions, regardless of the rate at which asymptomatic individuals become symptomatic, there is an epidemic whenever at least one of the infection rates is sufficiently large. In contrast, our analysis of the interacting particle system shows that, when the rate at which asymptomatic individuals become symptomatic is small and the asymptomatic individuals are not infectious, there cannot be an epidemic even when the symptomatic individuals are highly infectious.

We extend the central limit theorem under the Dedecker–Rio condition to adapted stationary and ergodic sequences of random variables taking values in a class of smooth Banach spaces. This result applies to the case of random variables taking values in $L^p\left(\mu \right)$, with $2⩽p<\infty$ and $\mu$ a $\sigma$-finite real measure. As an application we give a sufficient condition for empirical processes indexed by Sobolev balls to satisfy the central limit theorem, and discuss about the optimality of these conditions.

We derive subexponential tail asymptotics for the distribution of the maximum of a compound renewal process with linear component and of a Lévy process, both with negative drift, over random time horizon $\tau$ that does not depend on the future increments of the process. Our asymptotic results are uniform over the whole class of such random times. Particular examples are given by stopping times and by $\tau$ independent of the processes. We link our results with random walk theory.

Focusing on a class of regime-switching functional diffusion processes with infinite delay, a Freidlin–Wentzell type large deviations principle (LDP) is established by using an extended contraction principle and an exponential approximation argument under a local one-side Lipschitz condition. The result is new even for functional diffusion processes with infinite delay without regime-switching. Several interesting examples are given to illustrate our results.

We introduce a new metric for collections of aged paths and a robust set of conditions implying compactness for set of collections of aged paths in the topology corresponding to this metric. We show that the distribution of stable webs ($1<\alpha \le 2$) made up of collections of stable paths is tight in this topology. We then show convergence to stable webs for coalescing systems of random walks(suitably normalized). We obtain some path results in the Brownian case.