Let Xj denote a fair gambler’s ruin process on Z∩[−N,N] started from X0=0, and denote by RN the number of runs of the absolute value, |Xj|, until the last visit j=LN by Xj to 0. Then, as N→∞, N−2RN converges in distribution to a density with Laplace transform: tanh(λ)/λ. In law, we...

In general, gradient estimates are very important and necessary for deriving convergence results in different geometric flows, and most of them are obtained by analytic methods. In this paper, we will apply a stochastic approach to systematically give gradient estimates for some important...

In this paper we study Jensen’s inequality under quadratic g-expectation, i.e., the expectation generated by backward stochastic differential equations (BSDEs) with generator of quadratic growth in its component z. In particular, we define a new kind of convexity, the C-convexity, via a second...

We study a class of ergodic BSDEs related to PDEs with Neumann boundary conditions. The randomness of the driver is given by a forward process under weakly dissipative assumptions with an invertible and bounded diffusion matrix. Furthermore, this forward process is reflected in a convex subset...

We discuss the construction and approximation of solutions to a nonlinear McKean–Vlasov equation driven by a singular self-excitatory interaction of the mean-field type. Such an equation is intended to describe an infinite population of neurons which interact with one another. Each time a...

We consider the Bouchaud trap model on the integers in the case that the trap distribution has a slowly varying tail at infinity. Our main result is a functional limit theorem for the model under the annealed law, analogous to the functional limit theorems previously established in the...

This article is concerned with the study of fractal properties of thick points for a 4-dimensional Gaussian Free Field. We adopt the definition of Gaussian Free Field on R4 introduced by Chen and Jakobson (2012) viewed as an abstract Wiener space with underlying Hilbert space H2(R4). We can...

We identify the genetic signature of a selective sweep in a population described by a birth-and-death process with density dependent competition. We study the limit behavior for large K, where K scales the population size. We focus on two loci: one under selection and one neutral. We distinguish...

We focus on the stationary distribution of a multidimensional semimartingale reflecting Brownian motion (SRBM) on a nonnegative orthant. Assuming that the stationary distribution exists and is decomposable—equal to the product of two marginal distributions, we prove that these marginal...

We consider a continuous time Markov chain on a countable state space. We prove a joint large deviation principle (LDP) of the empirical measure and current in the limit of large time interval. The proof is based on results on the joint large deviations of the empirical measure and flow obtained...