Analytic-geometric methods for finite Markov chains
with applications to quasi-stationarity

Diaconis, Persi; Houston-Edwards, Kelsey; Saloff‐Coste, Laurent

doi:10.30757/alea.v17-35

Cited by 7 publications

(10 citation statements)

References 45 publications

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“…It is well-known that the Doob-transform technique is a useful tool to study problems involving Markov processes with killing. We follow closely the notation used in our previous article [6] which will be used extensively in what follows.…”

Section: General Results Based On the Doob Transformmentioning

confidence: 99%

“…We introduce the notions of Harnack Markov chains and graphs, which allows us to treat the three-dimensional gambler's ruin starting "in the middle" in Example 5.14. Section 6 specializes to nice domains (inner-uniform domains) where the results of the authors' previous paper [6], Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity, can be harnessed. This allows uniform estimates for all starting states, in particular for the three-player gambler's ruin problem.…”

Section: Introductionmentioning

confidence: 99%

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Gambler's ruin estimates on finite inner uniform domains

Diaconis¹,

Houston-Edwards²,

Saloff‐Coste³

2019

Preprint

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Gambler's ruin estimates can be viewed as harmonic measure estimates for finite Markov chains which are absorbed (or killed) at boundary points. We relate such estimates to properties of the underlying chain and its Doob transform. Precisely, we show that gambler's ruin estimates reduce to a good understanding of the Perron-Frobenius eigenfunction and eigenvalue whenever the underlying chain and its Doob transform are Harnack Markov chains. Finite inner-uniform domains (say, in the square grid Z n ) provide a large class of examples where these ideas apply and lead to detailed estimates. In general, understanding the behavior of the Perron-Frobenius eigenfunction remains a challenge.

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Section: General Results Based On the Doob Transformmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gambler's ruin estimates on finite inner uniform domains

Diaconis¹,

Houston-Edwards²,

Saloff‐Coste³

2019

Preprint

Self Cite

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show abstract

“…In particular, it is well-known that the asymptotic behavior of this Doob transform is very linked to the one of the conditional probability measure P µ [X t ∈ •|τ 0 > t], for some µ ∈ M 1 ((0, +∞)), as it was shown for example in [5,4,10] in the context of absorbed Markov processes, or in [6] for exploding Feynman-Kac semi-groups.…”

Section: Asymptotic Behavior For the Bessel-3 Processmentioning

confidence: 92%

Polynomial rate of convergence to the Yaglom limit for Brownian motion with drift

Oçafrain¹

2020

Electron. Commun. Probab.

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This paper deals with the rate of convergence in 1-Wasserstein distance of the marginal law of a Brownian motion with drift conditioned not to have reached 0 towards the Yaglom limit of the process. In particular it is shown that, for a wide class of initial measures including probability measures with compact support, the Wasserstein distance decays asymptotically as 1/t. Likewise, this speed of convergence is recovered for the convergence of marginal laws conditioned not to be absorbed up to a horizon time towards the Bessel-3 process, when the horizon time tends to infinity.

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“…One of the main goals of this project is to understand the long-time asymptotic and ergodic properties of such Feynman-Kac semigroups and the corresponding processes evolving in the presence of the killing Schrödinger potentials. In this context, we want to mention here a recent work by Diaconis, Houston-Edwards and Saloff-Coste [18] which gave us some new insight and motivation. The two aforementioned motivations are strongly connected to each other -this is manifested via the probabilistic background lying behind the analytic approach which we undertake in this article.…”

Section: Introductionmentioning

confidence: 98%

Decay of harmonic functions for discrete time Feynman--Kac operators with confining potentials

Cygan¹,

Kaleta²,

Mateusz³

2021

Preprint

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We propose and study a certain discrete time counterpart of the classical Feynman-Kac semigroup with a confining potential in countable infinite spaces. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman-Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman-Kac operators. We include such examples as non-local discrete Schrödinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasirelativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.

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Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity

Cited by 7 publications

References 45 publications

Gambler's ruin estimates on finite inner uniform domains

Gambler's ruin estimates on finite inner uniform domains

Polynomial rate of convergence to the Yaglom limit for Brownian motion with drift

Decay of harmonic functions for discrete time Feynman--Kac operators with confining potentials

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