Martin boundary of random walks in convex cones

Duraj, Jetlir; Raschel, Kilian; Tarrago, Pierre; Wachtel, Vitali

doi:10.48550/arxiv.2003.03647

Cited by 2 publications

(3 citation statements)

References 41 publications

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“…In the previously cited articles, no exact asymptotics for Green functions are derived: only the Martin kernel asymptotics is considered. Building on the estimates of the local probabilities in cones derived in [10], the paper [12] obtains the asymptotics of the Green functions in this context.…”

Section: Instability Of the Martin Boundarymentioning

confidence: 99%

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Reflected random walks and unstable Martin boundary

Ignatiouk-Robert¹,

Kourkova²,

Raschel³

2020

Preprint

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We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an instability phenomenon, in the following sense: if some parameter associated to the model is rational (resp. non-rational), then the Martin boundary is discrete, homeomorphic to Z (resp. continuous, homeomorphic to R). Such instability phenomena are very rare in the literature. Along the way of proving this result, we obtain several precise estimates for the Green functions of reflected random walks with escape probabilities along the boundary axes and an arbitrarily large number of inhomogeneity domains. Our methods mix probabilistic techniques and an analytic approach for random walks with large jumps in dimension two.

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Section: Instability Of the Martin Boundarymentioning

confidence: 99%

“…. An example of domain Q as in (71), for the random walk with transition probabilities defined by (12). The points (0, 0), (u 0 , 0) = (log 2, 0) and (0, v 0 ) = (0, log 3) are represented with bullets.…”

Section: 2mentioning

confidence: 99%

Reflected random walks and unstable Martin boundary

Ignatiouk-Robert¹,

Kourkova²,

Raschel³

2020

Preprint

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“…The fact that we may take any power series F is independent of the model and therefore appears as a universal feature. • It is known from [4,25,14] that there exists a unique function which is both positive and harmonic for the Laplacian operator (1.4). We conjecture (and bring some evidence) that this unique positive harmonic function corresponds to F (t) = t in (1.17).…”

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confidence: 99%

Constructing discrete harmonic functions in wedges

Hoang¹,

Raschel²,

Tarrago³

2020

Preprint

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We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function. Contents 1. Introduction and main results 1 2. Study of the kernel 9 3. Boundary value problems for the generating functions 15 4. Proof of our main results (Theorems 1 and 2) 18 5. Various examples 25 References 35 Appendix A. Weierstrass' preparation and division theorems 38

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Martin boundary of random walks in convex cones

Cited by 2 publications

References 41 publications

Reflected random walks and unstable Martin boundary

Reflected random walks and unstable Martin boundary

Constructing discrete harmonic functions in wedges

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