Let [sigma],[delta]0,b=0. Let , be continuous, and locally of bounded variation. We develop a general analytic criterion for the pathwise uniqueness of where p[set membership, variant](0,1), and is the symmetric semimartingale local time of R-[lambda]2. The criterion is related to the existence...

Let (X(t),Y(t)) be a symmetric [alpha]-stable Lévy process on with 1[alpha]=2 and LY(t) be the local time at 0 for Y(t). A multivariate asymptotic estimate is obtained involving the first hitting time and place of the positive half of the X-axis, and LY([dot operator]) up to then. The method is...

The paper studies the rate of convergence of the weak Euler approximation for solutions to SDEs driven by Lévy processes, with Hölder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes. The equation considered has a...

We prove the metastable behavior of reversible Markov processes on finite state spaces under minimal conditions on the jump rates. To illustrate the result we deduce the metastable behavior of the Ising model with a small magnetic field at very low temperature.

We first consider the stochastic differential equations (SDE) without global Lipschitz conditions, and give sufficient conditions for the SDEs to be strictly conservative. In particular, a criteria for stochastic flows of diffeomorphisms defined by SDEs with non-global Lipschitz coefficients is...

Let X(t),t>=0,X(0)=0, be a Lévy process with a spectral Lévy measure [rho]. Assuming that and the right tail of [rho] is light, we show that in the presence of the Brownian component as u-->[infinity], while in the absence of a Brownian component these tails are not always comparable.

In this paper we describe the long time behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of the corresponding diffusion processes.

Let G={G(x),x=0} be a mean zero Gaussian process with stationary increments and set [sigma]2(x-y)=E(G(x)-G(y))2. Let f be a function with Ef2([eta])<[infinity], where [eta]=N(0,1). When [sigma]2 is regularly varying at zero and is locally integrable for some integer j0>=1, and satisfies some additional regularity conditions, in L2. Here Hj is the jth Hermite polynomial. Also :(G')j:(I[a,b]) is a jth order Wick...</[infinity],>

This paper considers extreme values attained by a centered, multidimensional Gaussian process X(t)=(X1(t),...,Xn(t)) minus drift d(t)=(d1(t),...,dn(t)), on an arbitrary set T. Under mild regularity conditions, we establish the asymptotics of for positive thresholds qi0, i=1,...,n and...

Let [sigma]>0,[delta]>=1,b>=0, 0<p<1. Let [lambda] be a continuous and positive function in . Using the technique of moving domains (see Russo and Trutnau (2005) [9]), and classical direct stochastic calculus, we construct for positive initial conditions a pair of continuous and positive semimartingales with and where the symmetric local times , of the respective semimartingales are related through the formula Well-known special cases are the (squared) Bessel processes (choose [sigma]=2, b=0, and [lambda]2[reverse not equivalent]0, or equivalently ), and the Cox-Ingersoll-Ross process (i.e. R, with [lambda]2[reverse not equivalent]0, or equivalently ). The case 0<[delta]<1 can also be handled, but is different. If p>1, then there is no solution.