Related papers: Intrinsic Ultracontractivity for Levy processes

Let $D$ be a domain of finite Lebesgue measure in $\bR^d$ and let $X^D_t$ be the symmetric $\alpha$-stable process killed upon exiting $D$. Each element of the set $\{\lambda_i^\alpha\}_{i=1}^\infty$ of eigenvalues associated to $X^D_t$,…

We prove explicit and sharp two-sided estimates for the transition density of the Langevin process with quadratic potential, killed outside of the position interval (0,1). The long-time asymptotics of this transition density are also…

In contrast to their seemingly simple and shared structure of independence and stationarity, L\'evy processes exhibit a wide variety of behaviors, from the self-similar Wiener process to piecewise-constant compound Poisson processes.…

In this paper, we study an approximation scheme for L\'evy processes with drift in terms of a representation that is akin to the celebrated Mehler formula for L\'evy-Ornstein-Uhlenbeck processes. The approximation scheme is based on a…

Spectral theory for the transition semigroup of one-dimensional symmetric Levy process killed upon hitting the origin is studied. Under very mild assumptions, an integral-type formula for eigenfunctions is obtained, and eigenfunction…

We construct intrinsic on-and off-diagonal upper and lower estimates for the transition probability density of a L\'evy process in small time. By intrinsic we mean that such estimates reflect the structure of the characteristic exponent of…

In this paper we introduce a new class of L\'evy processes which we call hypergeometric-stable L\'evy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric…

We derive explicitly the coupling property for the transition semigroup of a L\'{e}vy process and gradient estimates for the associated semigroup of transition operators. This is based on the asymptotic behaviour of the symbol or the…

We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for L\'{e}vy processes. The proof uses a…

We study heat equations $\frac{\partial u}{\partial t} - \operatorname{div} \left( A \nabla u \right) = 0$ on bounded Lipschitz domains $\Omega$ in $\mathbb{R}^{d}$ for $d \in \mathbb{N}$, where $-\operatorname{div} \left( A \nabla \cdot…

The escape from a given domain is one of the fundamental problems in statistical physics and the theory of stochastic processes. Here, we explore properties of the escape of an inertial particle driven by L\'evy noise from a bounded domain,…

We consider the Schroedinger equation with a supersymmetric random potential, where the superpotential is a Levy noise. We focus on the problem of computing the so-called complex Lyapunov exponent, whose real and imaginary parts are,…

In this paper, we discuss hypercontractivity for the Markov semigroup $P_t$ which is generated by segment processes associated with a range of functional SDEs of neutral type. As applications, we also reveal that the semigroup $P_t$…

We show that the second term in the asymptotic expansion as t approaches 0 of the trace of the Dirichlet heat kernel on Lipschitz domains for unimodal L\'evy processes, satisfying some weak scaling conditions, is given by the surface area…

Strongly continuous semigroups of unital completely positive maps (i.e. quantum Markov semigroups or quantum dynamical semigroups) on compact quantum groups are studied. We show that quantum Markov semigroups on the universal or reduced…

We investigate the hypercontractivity property of generalized Mehler semigroups on the $L^p$-scale with respect to invariant measures. This property is first obtained in the purely theoretical setting of skew operators and, subsequently,…

The class of Levy processes for which overshoots are almost surely constant quantities is precisely characterized.

Small-space and large-time estimates and asymptotic expansion of the distribution function and (the derivatives of) the density function of hitting times of points for symmetric L\'evy processes are studied. The L\'evy measure is assumed to…

Hypercontractivity is proved for products of qubit channels that belong to self-adjoint semigroups. The hypercontractive bound gives necessary and sufficient conditions for a product of the form e^{- t_1 H_1} \ot ... \ot e^{- t_n H_n} to be…

We consider isotropic L\'evy processes on a compact Riemannian manifold, obtained from an $\mathbb{R}^d$-valued L\'evy process through rolling without slipping. We prove that the Feller semigroups associated with these processes extend to…