Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

Lőrinczi, József; Hiroshima, Fumio; Betz, Volker

doi:10.1515/9783110203738

Cited by 97 publications

(103 citation statements)

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“…In the seminal paper [12] it was considered for the operators H 0 = (− ) α/2 , 0 < α < 2, and H 0 = √ − + m 2 − m, m > 0, using martingale and optional stopping methods combined with precise estimates of the corresponding resolvent kernels. For an extension of these ideas to other operators of interest in mathematical physics see [27,39].…”

Section: Introductionmentioning

confidence: 99%

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

Kaleta

Lőrinczi

2016

Potential Anal

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We study the spatial decay of eigenfunctions of non-local Schrödinger operators whose kinetic terms are generators of symmetric jump-paring Lévy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the Lévy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the Lévy intensity. We furthermore prove that under reasonable conditions the Lévy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the Lévy intensity is varied from subexponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition Lévy intensity-driven decay becomes slower than the rate of decay of the Lévy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.

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Section: Introductionmentioning

confidence: 99%

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

Kaleta

Lőrinczi

2016

Potential Anal

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“…It has next been expanded in various directions, with a special emphasis on so-called Lévy-Schrödinger semigroup reformulation of the original probability density function (pdf) dynamics, [6,15,16] and [8]- [13], c.f. also [14][15][16]. We note in passing that the familiar Fokker-Planck equation can be equally well formulated in terms of the Schrödinger semigroup and this property is universally valid in the standard theory of Brownian motion, [1,14].…”

Section: Introductionmentioning

confidence: 99%

“…also [14][15][16]. We note in passing that the familiar Fokker-Planck equation can be equally well formulated in terms of the Schrödinger semigroup and this property is universally valid in the standard theory of Brownian motion, [1,14]. Its generalization to Lévy flights is neither immediate nor obvious.…”

Section: Introductionmentioning

confidence: 99%

Lévy flights in confining environments: Random paths and their statistics

Żaba

Garbaczewski

Stephanovich

2013

Physica A: Statistical Mechanics and its Applications

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We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise. In view of the Lévy noise sensitivity to the confining "potential landscape" where jumps take place (in other words, to environmental inhomogeneities), the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) ρ * (x) ∼ exp[−Φ(x)]. Since there is no Langevin representation of the dynamics in question, our main goal here is to establish the appropriate path-wise description of the underlying jump-type process and next infer the ρ(x, t) dynamics directly from the random paths statistics. A priori given data are jump transition rates entering the master equation for ρ(x, t) and its target pdf ρ * (x). We use numerical methods and construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. The generated sample trajectories show up a qualitative typicality, e.g. they display structural features of jumping paths (predominance of small vs large jumps) specific to particular stability indices µ ∈ (0, 2).

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“…In particular, quantum field theory provides a large number of interesting problems which can be addressed by using functional integration; see [1] and [2,Chapter 6]. In a more recent development we have extended this method to operators originating from relativistic quantum theory such as the Schrödinger operator…”

Section: Fractional Boson Number Operatormentioning

confidence: 99%

A probabilistic representation of the ground state expectation of fractional powers of the boson number operator

Hiroshima

Lőrinczi

Takaesu

2012

Journal of Mathematical Analysis and Applications

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Submitted by Stephen Fulling Keywords: Gibbs measure on path space Random-time Poisson process Ground state Boson number operator Nelson model a b s t r a c tWe give a formula in terms of a joint Gibbs measure on Brownian paths and the measure of a random-time Poisson process of the ground state expectations of fractional (in fact, any real) powers of the boson number operator in the Nelson model. We use this representation to obtain tight two-sided bounds. As applications, we discuss the polaron and translation invariant Nelson models.

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Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

Cited by 97 publications

References 0 publications

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

Lévy flights in confining environments: Random paths and their statistics

A probabilistic representation of the ground state expectation of fractional powers of the boson number operator

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