Typical long-time behavior of ground state-transformed jump processes

Abstract

We consider a class of Lévy-type processes derived via a Doob-transform from Lévy processes conditioned by a control function called potential. These processes have position-dependent and generally unbounded components, with stationary distributions given by the ground states of the Lévy generators perturbed by the potential. We derive precise lower and upper envelopes for the almost sure long time behaviour of these ground statetransformed Lévy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples.

Bulk Behaviour of Ground States for Relativistic Schrödinger Operators with Compactly Supported Potentials

“…This shows that adding a potential V to L m,α gives rise to a Feller process with drift terms and biases in the jump rates, with stationary density ϕ 2 0 , which is thus no longer a Lévy process, and it gives a proper meaning to "motion in a potential landscape". For further details we refer to [36,39,47], in which also local and global sample path properties of ( X t ) t∈R are discussed.…”

Multifractal properties of sample paths of ground state-transformed jump processes

“…The results below on ground state-transformed processes complement these efforts since our approach is not through an analysis of the symbol, and apart from a direct interest in this context, our class of processes has an immediate relevance in the study of spectral properties of related self-adjoint operators and model Hamiltonians as a bonus [13,32]. Our concern in the present paper is to study sample path regularity properties of ground statetransformed processes obtained for a large class of operators H. The typical long-time behaviour of such processes has been established in [25], which is useful also in characterizing the support of the related Gibbs path measures defined by the right hand side of the Feynman-Kac formula (for perturbations of symmetric α-stable processes see also [21]). While the asymptotic behaviour on the long run is driven by the large jumps, regularity at short range depends on the ultraviolet properties of H involving the small jumps.…”

“…Our concern in the present paper is to study sample path regularity properties of ground statetransformed processes obtained for a large class of operators H. The typical long-time behaviour of such processes has been established in [25], which is useful also in characterizing the support of the related Gibbs path measures defined by the right hand side of the Feynman-Kac formula (for perturbations of symmetric α-stable processes see also [21]). While the asymptotic behaviour on the long run is driven by the large jumps, regularity at short range depends on the ultraviolet properties of H involving the small jumps.…”

Contractivity and ground state domination properties for non-local Schrödinger operators

“…In many interesting cases the specific models involve non-local Schrödinger operators based on generators of Lévy processes with jumps. Recent investigations include heat trace and spectral gap estimates [1,5,21], gradient estimates of harmonic functions [31], properties of radial solutions, ground states, eigenfunctions and eigenvalues [32,33,23,13,26,35,18], smoothing properties of evolution semigroups [25,9,22], properties of the associated transformed jump processes [24,28], as well as applications in quantum theory [34,17,16,19,15,3].…”

“…where µ(dx) = ϕ 2 0 (x)dx. The semigroup { T t : t ≥ 0} is conservative and determines a right Markov process with stationary distribution µ, which we call ground state-transformed jump process corresponding to H [28,24].…”