Mladen Savov

THE ASYMPTOTIC BEHAVIOR OF DENSITIES RELATED TO THE SUPREMUM OF A STABLE PROCESS

P(S1 > x) v A� −1 x −� as x ! 1 and P(S1 � x) v B� −1 � −1 x �� as x # 0. [Here � = P(X1 > 0) and A and B are known constants.] It is also known that S1 has a continuous density, m say. The main point of this note is to show that m(x) v Ax −(�+1) as x ! 1 and m(x) v Bx ��−1 as x # 0. Similar results are obtained for related densities.

A Wiener-Hopf type factorization for the exponential functional of Lévy processes

For a Levy process =( t) t≥0 drifting to -∞, we define the so-called exponential functional as follows: Under mild conditions on , we show that the following factorization of exponential functionals: holds, where × stands for the product of independent random variables, H - is the descending ladder height process of and Y is a spectrally positive Levy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of I for a large class of Levy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein-Uhlenbeck processes, which is itself of independent interest. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process.

Cauchy problem of the non-self-adjoint Gauss–Laguerre semigroups and uniform bounds for generalized Laguerre polynomials

We propose a new approach to construct the eigenvalue expansion in a weighted Hilbert space of the solution to the Cauchy problem associated to Gauss-Laguerre invariant Markov semigroups that we introduce. Their generators turn out to be natural non-self-adjoint and non-local generalizations of the Laguerre differential operator. Our methods rely on intertwining relations that we establish between these semigroups and the classical Laguerre semigroup and combine with techniques based on non-harmonic analysis. As a by-product we also provide regularity properties for the semigroups as well as for their heat kernels. The biorthogonal sequences that appear in their eigenvalue expansion can be expressed in terms of sequences of polynomials, and they generalize the Laguerre polynomials. By means of a delicate saddle point method, we derive uniform asymptotic bounds that allow us to get an upper bound for their norms in weighted Hilbert spaces. We believe that this work opens a way to construct spectral expansions for more general non-self-adjoint Markov semigroups.

(Non-)Ergodicity of a Degenerate Diffusion Modeling the Fiber Lay Down Process

We analyze the large time behavior of a stochastic model for the lay down of fibers on a moving conveyor belt in the production process of nonwovens. It is shown that under weak conditions this degenerate diffusion process has a unique invariant distribution and is even geometrically ergodic. This generalizes results from previous works [M. Grothaus and A. Klar, SIAM J. Math. Anal., 40 (2008), pp. 968--983; J. Dolbeault et al., arXiv:1201.2156] concerning the case of a stationary conveyor belt, in which the situation of a moving conveyor belt has been left open.

Transience and recurrence of a Brownian path with limited local time

In this note we investigate the behaviour of Brownian motion conditioned on a growth constraint of its local time which has been previously investigated by Berestycki and Benjamini. For a class of non-decreasing positive functions

Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case

f(t); t>0

Small Time One-Sided LIL Behavior for Lévy Processes at Zero

, we consider the Wiener measure under the condition that the Brownian local time is dominated by the function f up to time T. In the case where

Distributional properties of exponential functionals of Levy processes

f(t)/t^{3/2}

Small time two-sided LIL behavior for Lévy processes at zero

is integrable we describe the limiting process as T goes to infinity. Moreover, we prove two conjectures in [BB10] in the case for a class of functions f, for which

Spectral expansions of non-self-adjoint generalized Laguerre semigroups

f(t)/t^{3/2}