Yaglom limit for stable processes in cones

Bogdan, Krzysztof; Palmowski, Zbigniew; Wang, Longmin

doi:10.48550/arxiv.1612.03548

Cited by 2 publications

(17 citation statements)

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“…A related topic is that of quasi-stationary distributions (c.f. [31] in the random walk and Brownian contexts) and the concept of so-called Yaglom limits, [33,34].…”

Section: Motivationmentioning

confidence: 99%

“…In connection with so-called Yaglom limits. [33], and in conjunction with the previous description of the conditioned random motions in a bounded domain, it is useful to say few words about the so called quasi-stationary distributions.…”

Section: Quasi-stationary Distributionsmentioning

confidence: 99%

“…relative to x ∈ (−1, 1). The eigenvalue problem for the operator (33) has been discussed in detail, for various stability parameter values, with the aid of numerical assistance, and compared with other existing solutions (analytic and computer assisted), see especially [13,39,40,43].…”

Section: Restricted Fractional Case: Hypersingular Fredholm Problemmentioning

confidence: 99%

“…A related topic is that of quasi-stationary distributions (c.f. [31] in the random walk and Brownian contexts) and the concept of so-called Yaglom limits, [33,34].Another category refers to reflecting boundary data and to reflected Lévy-stable processes (the reflected Brownian motion might be invoked at this point and set against the killed/absorbed one, c.f. [21,35,36]).…”

mentioning

confidence: 99%

“…For the second step, we invoke the hypersingular integral formula (33), here adapted to the interval…”

mentioning

confidence: 99%

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Fractional Laplacians and Levy flights in bounded domains

Garbaczewski

2018

Preprint

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We address Lévy-stable stochastic processes in bounded domains, with a focus on a discrimination between inequivalent proposals for what a boundary data-respecting fractional Laplacian (and thence the induced random process) should actually be. Versions considered are: restricted Dirichlet, spectral Dirichlet and regional (censored) fractional Laplacians. The affiliated random processes comprise: killed, reflected and conditioned Lévy flights, in particular those with an infinite life-time. The related concept of quasi-stationary distributions is briefly mentioned. I. MOTIVATIONJump-type Lévy processes in a bounded domain are a subject of an active study both in physics and mathematics communities. The physics-oriented research is conducted with some disregard to an ample coverage of the topic in the past and modern mathematical literature. The reason is rooted not only in a methodological gap between the practitioners' pragmatism and the mathematically rigorous reasoning. An important factor is a scarce (or even lack of) communication between various research groups and research streamlines. This refers not only to rather residual physics-mathematics interplay, but also to the mathematics community per se: relevant publications are scattered in a large number of highly specialized journals and easily escape the attention of potentially interested parties.Recently an attempt has been made to establish a common conceptual basis for varied frameworks in which fractional Laplacians appear. Formally looking different, but actually equivalent, definitions of fractional Laplacians, appropriate for the description of Lévy stable processes in R n , n ≥ 1, have been collected and their mutual relationships analyzed in minute detail in Ref. [1].There is a general consensus that the standard Fourier multiplier definition appears to be defective, if one passes to Lévy flights in a bounded domain. This is a consequence of an inherent nonlocality of Lévy-stable generators. Different proposals for boundary-data-respecting fractional Laplacians were given in the literature. Often, with a view towards more efficient computer-assisted calculations (that mostly in connection with nonlinear fractional differential equations, [11,12]).However, in contrast to the situation in R n , these proposals are known to be inequivalent, c.f. Refs.[2]- [16], see also [18,19]. Likewise, the induced jump-type processes are inequivalent and have different statistical characteristics. That in particular refers to a standard physical inventory, adapted directly from the Brownian motion studies [20]: the statistics of exits from the domain, e.g. first and mean first exit times, probability of survival, its long time behavior an ultimate asymptotic decay [21]- [27].Interestingly, the existence problem for jump-type processes with an infinite life-time in a bounded domain, seems to have been left aside in the physics literature (compare e.g. Ref. [28] in connection with diffusion processes and Ref. [30] for a preliminary discussion of the Cauchy process...

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“…A related topic is that of quasi-stationary distributions (c.f. [31] in the random walk and Brownian contexts) and the concept of so-called Yaglom limits, [33,34].…”

Section: Motivationmentioning

confidence: 99%

Section: Quasi-stationary Distributionsmentioning

confidence: 99%

Section: Restricted Fractional Case: Hypersingular Fredholm Problemmentioning

confidence: 99%

mentioning

confidence: 99%

“…For the second step, we invoke the hypersingular integral formula (33), here adapted to the interval…”

mentioning

confidence: 99%

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Fractional Laplacians and Levy flights in bounded domains

Garbaczewski

2018

Preprint

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Stable Lévy processes in a cone

Kyprianou¹,

Rivero²,

Satitkanitkul³

2018

Preprint

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Bañuelos and Bogdan [6] and Bogdan et al. [19] analyse the asymptotic tail distribution of the first time a stable (Lévy) process in dimension d ≥ 2 exists a cone. We use these results to develop the notion of a stable process conditioned to remain in a cone as well as the the notion of a stable process conditioned to absorb continuously at the apex of a cone (without leaving the cone). As self-similar Markov processes we examine some of their fundamental properties through the lens of its Lamperti-Kiu decomposition. In particular we are interested to understand the underlying structure of the Markov additive process that drives such processes. As a consequence of our interrogation of the underlying MAP, we are able to provide an answer by example to the open question: If the modulator of a MAP has a stationary distribution, under what conditions does its ascending ladder MAP have a stationary distribution?With the help of an analogue of the Riesz-Bogdan-Żak transform (cf. Bogdan and Żak [20], Kyprianou [40], Alili et al. [1]) as well as Hunt-Nagasawa duality theory, we show how the two forms of conditioning are dual to one another. Moreover, in the sense of Rivero [52,53] and Fitzsimmons [32], we construct the null-recurrent extension of the stable process killed on exiting a cone, showing that it again remains in the class of self-similar Markov processes. Aside from the Riesz-Bogdan-Żak transform and Hunt-Nagasawa duality, an unusual combination of the Markov additive renewal theory of e.g. Alsmeyer [2] as well as the boundary Harnack principle (see e.g. [19]) play a central role to the analysis.In the spirit of several very recent works (see [44,40,43,45,46,41]), the results presented here show that many previously unknown results of stable processes, which have long since been understood for Brownian motion, or are easily proved for Brownian motion, become accessible by appealing to the notion of the stable process as a self-similar Markov process, in addition to its special status as a Lévy processes with a semi-tractable potential analysis.

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Yaglom limit for stable processes in cones

Cited by 2 publications

References 49 publications

Fractional Laplacians and Levy flights in bounded domains

Fractional Laplacians and Levy flights in bounded domains

Stable Lévy processes in a cone

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