On the Asymptotic Behavior of the Hyperbolic Brownian Motion

In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4, then the distances on the surface are highly concentrated around the minimal possible value, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres ([20]) from the setting of Ramanujan graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from [27], we are able to show that the results apply to congruence subgroups of SL 2 (Z) and other arithmetic lattices, without relying on the well known conjecture of Selberg ([28]). Conceptually, we show the close relation between the cutoff phenomenon and temperedness of representations of algebraic groups over local fields, partly answering a question of Diaconis ([7]), who asked under what general phenomena cutoff exists.

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“…We can immediately see that as in Proposition 2.5, this proves the implication from (1) to (2). Denote it by p 0 (X).…”

Section: P -Boundssupporting

confidence: 70%

“…The Brownian motion was studied by many authors, and can be analyzed either by the Helgason-Fourier transform, or by the "distance to infinity" approach used to study the discrete random walk. In any case, based on [4,2], we may write…”

Section: And Hencementioning

confidence: 99%

Cutoff on hyperbolic surfaces

Golubev

Kamber

2019

Geom Dedicata

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“…Since the method in [3] is valid only when the starting point is close to the origin, we need to use a different method in other cases and the Laplace transform is useful for calculations. By using the derived representation we establish the exponential decay of the hitting probability, confirming the conjecture in [2].…”

supporting

confidence: 77%

“…The probability that d-dimensional hyperbolic Brownian motion reaches the boundary of a ball in some time is given in [3] when the starting point is outside the ball. Moreover, for 2 ≤ d ≤ 7, the asymptotic behavior of the probability is discussed in [2] as the starting point tends to infinity. The limiting value of the logarithm of the hitting probability is linear in the radius of the ball and the same is expected in other cases.…”

mentioning

confidence: 99%

Hitting times of hyperbolic Bessel processes

Hamana,

Zhang

2024

Colloq. Math.

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We investigate the first hitting time to a point of a hyperbolic Bessel process which is the generalization of the radial part of Brownian motion on a hyperbolic space. We give the Laplace transform of the hitting time and the probability that the hyperbolic Bessel process reaches a given point in some time. Moreover, the limiting behavior of the expectation of the hitting time is computed. These results are improvements of some preceding results.

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“…Similar probabilities (yet not this one in particular), of hitting some boundary, or a ball around it, have been studied for the hyperbolic Brownian motion in [6,7,14,21,22] In Section 2, we analyse the dynamics of (1.1) under different parameter configurations, and propose several space transformations to translate properties of one model configuration to the other. In Section 3, we use these maps to derive an exact formula for P for general parameter values.…”

Section: Introductionmentioning

confidence: 94%

On the probability of hitting the boundary for Brownian motions on the SABR plane

Gulisashvili¹,

Horvath

Jacquier

2016

Electron. Commun. Probab.

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Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models???related to the SABR model in mathematical finance???which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods

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On the Asymptotic Behavior of the Hyperbolic Brownian Motion

Cited by 4 publications

References 32 publications

Cutoff on hyperbolic surfaces

Cutoff on hyperbolic surfaces

Hitting times of hyperbolic Bessel processes

On the probability of hitting the boundary for Brownian motions on the SABR plane

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