In this work, by constructing optimal Markovian couplings we investigate exponential convergence rate in the Wasserstein distance for the transmission control protocol process. Most importantly, we provide a variational formula for the lower bound of the exponential convergence rate.
This work is concerned with the continuous dependence on initial values of solutions of stochastic functional differential equations (SFDEs) with state-dependent regime-switching. Due to the state-dependence, this problem is very different to the corresponding problem for SFDEs without switching or SFDEs with Markovian switching. We provide a method to overcome the intensive interaction between the continuous component and the discrete component based on a subtle application of Skorokhod's representation for jumping processes. Furthermore, we establish the strong convergence of Euler-Maruyama's approximations, and estimate the order of error. The continuous dependence on initial values of Euler-Maruyama's approximations is also investigated in the end.
We consider the point vortex model associated to the modified Surface Quasi-Geostrophic (mSQG) equations on the two dimensional torus. It is known that this model is well posed for almost every initial conditions. We show that, when the system is perturbed by a certain space-dependent noise, it admits a unique global solution for any initial configuration. We also present an explicit example for the deterministic system on the plane where three different point vortices collapse.
We prove a general version of the stochastic Fubini theorem for stochastic integrals of Banach space valued processes with respect to compensated Poisson random measures under weak integrability assumptions, which extends this classical result from Hilbert space setting to Banach space setting.
In this paper, we establish a small time large deviation principle (small time asymptotics) for the dynamical Φ14 model, which not only involves study of the space-time white noise with intensity √ε, but also the investigation of the effect of the small (with ε) nonlinear drift.
In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework. Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.
By using the coupling method and the localization technique, we establish non-uniform gradient estimates for Markov semigroups of diffusions or stochastic differential equations driven by pure jump Lévy noises, where the coefficients only satisfy local monotonicity conditions.
In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations driven by space-time white noise on T2. First we prove that the convergence rate for stochastic 2D heat equation is of order α δ in Besov space C-α for α ∈ (0, 1) and δ > 0 arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations of order α δ in C-α for α ∈ (0, 2/9) and δ > 0 arbitrarily small.
The goal of this paper is threefold. First, we survey the existing results on Hunt's hypothesis (H) for Markov processes and Getoor's conjecture for Lévy processes. Second, we investigate (H) for multidimensional Lévy processes from the viewpoint of projections. Third, we present a few open questions for further study.
We study the averaged products of characteristic polynomials for the Gaussian and Laguerre β-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are characterised by determining the explicit functional forms of the scaled limits of the averaged products of characteristic polynomials, which are given as certain multidimensional integrals, with dimension equal to the number of products.