We describe the exact tail behavior of the solutions to certain nonlinear stochastic differential equations driven by Lévy motions with regularly varying tails and establish existence and uniqueness of solutions to these equations.

The solutions of Burgers equation with white noise initial velocity are closely connected to the convex hull of a Brownian excursion with parabolic drift . We derive from the law of the minimum [sigma] and the location [eta] of the minimum of s|-2es/s(1-s) a complete description of the convex...

We study long-time asymptotic behavior of the current-valued processes on compact Riemannian manifolds determined by the stochastic line integrals. Sample path large deviation estimates are proved, which induce the law of the iterated logarithm as a corollary. As their application, we give a...

Let X1,...,Xn be jointly Gaussian random variables with mean zero. It is shown that [for all]x>0 and [for all]1[less-than-or-equals, slant]k<nandwhere [rho]=([Sigma]/([Sigma]11 [Sigma]22))1/2, [Sigma],[Sigma]11 and [Sigma]22 are the covariance matrices of (X1,...,Xn), (X1,...,Xk) and (Xk+1,...,Xn), respectively. In particular, for fractional Brownian motion {X(t),t[greater-or-equal, slanted]0} of order [alpha] (0<[alpha]<1), there exists d[alpha]>0 such that for any 0<a<b, x>0 and y>0. As an application, it is proved that the small ball constant for the fractional Brownian motion of order [alpha] exists.

Let denote the space of locally finite measures on Rd and let denote the space of probability measures on . Define the mean measure [pi][nu] of byFor such a measure [nu] with locally finite mean measure [pi][nu], let f be a nonnegative, locally bounded test function satisfying <f,[pi][nu]>=[infinity]. [nu]...</f,[pi][nu]>

We study the problem of a.s. convergence of the quadratic variation of Brownian motion. We present some new sufficient and necessary conditions for the convergence. As a byproduct we get a new proof of the convergence in the case of refined partitions, a result that is due to Lévy. Our method...

A sufficient condition under which a symmetric [alpha]-stable process {X(n),n[set membership, variant]N} is a.s. bounded is given. We also show that in some sense this condition is optimal.

We formulate a stochastic differential equation describing the Lagrangian environment process of a passive tracer in Ornstein-Uhlenbeck velocity fields. We subsequently prove a local existence and uniqueness result when the velocity field is regular. When the Ornstein-Uhlenbeck velocity field is...

This note considers the taboo counterpart of stationarity. A general stochastic process in two-sided time is defined to be taboo-stationary if its global distribution does not change by shifting the origin to an arbitrary non-random time in the future under taboo, that is, conditionally on some...

We study one-dimensional resource sharing systems which can be seen as interacting particle systems taking values in . We first get, by coupling techniques, an estimate of their invariant measures. Then, for processes having a reversible measure, we show the uniqueness of the invariant measure...