Journal of Theoretical Probability最新文献

For a class of (Lambda )-Fleming–Viot processes with Brownian spatial motion in (mathbb {R}^d) whose associated (Lambda )-coalescents come down from infinity, we obtain sharp global and local moduli of continuity for the ancestral processes recovered from the associated lookdown representations. As applications, we establish both global and local moduli of continuity for the (Lambda )-Fleming–Viot support processes. In particular, if the (Lambda )-coalescent is the Beta((2-beta ,beta )) coalescent for (beta in (1,2]) with (beta =2) corresponding to Kingman’s coalescent, then for (h(t)=sqrt{tlog (1/t)}), the global modulus of continuity holds for the support process with modulus function (sqrt{2beta /(beta -1)}h(t)), and both the left and right local moduli of continuity hold for the support process with modulus function (sqrt{2/(beta -1)}h(t)).

We deal with McKean–Vlasov and Boltzmann-type jump equations. This means that the coefficients of the stochastic equation depend on the law of the solution, and the equation is driven by a Poisson point measure with intensity measure which depends on the law of the solution as well. Alfonsi and Bally (Construction of Boltzmann and McKean Vlasov type flows (the sewing lemma approach), 2021, arXiv:2105.12677) have proved that under some suitable conditions, the solution (X_t) of such equation exists and is unique. One also proves that (X_t) is the probabilistic interpretation of an analytical weak equation. Moreover, the Euler scheme (X_t^{{mathcal {P}}}) of this equation converges to (X_t) in Wasserstein distance. In this paper, under more restrictive assumptions, we show that the Euler scheme (X_t^{{mathcal {P}}}) converges to (X_t) in total variation distance and (X_t) has a smooth density (which is a function solution of the analytical weak equation). On the other hand, in view of simulation, we use a truncated Euler scheme (X^{{mathcal {P}},M}_t) which has a finite numbers of jumps in any compact interval. We prove that (X^{{mathcal {P}},M}_{t}) also converges to (X_t) in total variation distance. Finally, we give an algorithm based on a particle system associated with (X^{{mathcal {P}},M}_t) in order to approximate the density of the law of (X_t). Complete estimates of the error are obtained.

Abstract

We consider a Markovian model of an SIR epidemic spreading on a contact graph that is drawn uniformly at random from the set of all graphs with n vertices and given vertex degrees. Janson, Luczak and Windridge (Random Struct Alg 45(4):724–761, 2014) prove that the evolution of such an epidemic is well approximated by the solution to a simple set of differential equations, thus providing probabilistic underpinnings to the works of Miller (J Math Biol 62(3):349–358, 2011) and Volz (J Math Biol 56(3):293–310, 2008). The present paper provides an additional probabilistic interpretation of the limiting deterministic functions in Janson, Luczak and Windridge (Random Struct Alg 45(4):724–761, 2014), thus clarifying further the connection between their results and the results of Miller and Volz.

In this paper, we consider (upward skip-free) discrete-time and discrete-space Markov additive chains (MACs) and develop the theory for the so-called (widetilde{{textbf {W}}}) and (widetilde{{textbf {Z}}}) scale matrices, which are shown to play a vital role in the determination of a number of exit problems and related fluctuation identities. The theory developed in this fully discrete set-up follows similar lines of reasoning as the analogous theory for Markov additive processes in continuous time and is exploited to obtain the probabilistic construction of the scale matrices, identify the form of the generating function and produce a simple recursion relation for (widetilde{{textbf {W}}}), as well as its connection with the so-called occupation mass formula. In addition to the standard one- and two-sided exit problems (upwards and downwards), we also derive distributional characteristics for a number of quantities related to the one- and two-sided ‘reflected’ processes.

Let (X={ X(t), tin mathbb {R}^{N}} ) be a centered space-time anisotropic Gaussian random field in (mathbb {R}^d) with stationary increments, where the components (X_{i}(i=1,ldots ,d)) are independent but distributed differently. Under certain conditions, we not only give the Hausdorff dimension of the graph sets of X in the asymmetric metric in the recurrent case, but also determine the exact Hausdorff measure functions of the graph sets of X in the transient and recurrent cases, respectively. Moreover, we establish a uniform Hausdorff dimension result for the image sets of X. Our results extend the corresponding results on fractional Brownian motion and space or time anisotropic Gaussian random fields.

We introduce the voter model on the infinite integer lattice with a slow membrane and investigate its hydrodynamic behavior and nonequilibrium fluctuations. The voter model is one of the classical interacting particle systems with state space ({0,1}^{mathbb Z^d}). In our model, a voter adopts one of its neighbors’ opinion at rate one except for neighbors crossing the hyperplane ({x:x_1 = 1/2}), where the rate is (alpha N^{-beta }) and thus is called a slow membrane. Above, (alpha >0 textrm{and} beta ge 0) are given parameters and the positive integer N is a scaling parameter. We consider the limit (N rightarrow infty ) and prove that the hydrodynamic limits are given by the heat equation without or with Robin/Neumann conditions depending on the values of (beta ). We also consider the nonequilibrium fluctuations, where the limit is described by generalized Ornstein–Uhlenbeck processes with certain boundary conditions corresponding to the hydrodynamic equation.

We consider rough differential equations whose coefficients contain path-dependent bounded variation terms and prove the existence and a priori estimate of solutions. These equations include classical path-dependent stochastic differential equations containing running maximum processes and normal reflection terms. We apply these results to determine the topological support of the solution processes.

Abstract

We study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift.