Ergodic behavior of diffusions with random jumps from the boundary

Ben-Ari, Iddo; Pinsky, Ross G.

doi:10.1016/j.spa.2008.05.002

Cited by 38 publications

(38 citation statements)

References 5 publications

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“…The principal eigenvalue of the generator of the process is 0, the rest of the spectrum is negative, and the spectral gap, which is the supremum of the real part of the nonzero spectrum, gives the exponential rate of convergence to equilibrium. See [1], [2], [8], [9], [11].…”

Section: Let D ⊂ Rmentioning

confidence: 99%

Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps

Pinsky

2009

Trans. Amer. Math. Soc.

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Abstract. Let D ⊂ R d be a bounded domain and let P(D) denote the space of probability measures on D. Consider a Brownian motion in D which is killed at the boundary and which, while alive, jumps instantaneously at an exponentially distributed random time with intensity γ > 0 to a new point, according to a distribution µ ∈ P(D). From this new point it repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator −L γ,µ , defined bywith the Dirichlet boundary condition, where V µ is a nonlocal "µ-centering" potential defined byThe operator L γ,µ is symmetric only in the case that µ is normalized Lebesgue measure; thus, only in that case can it be realized as a selfadjoint operator. The corresponding semigroup is compact, and thus the spectrum of L γ,µ consists exclusively of eigenvalues. As is well known, the principal eigenvalue gives the exponential rate of decay in t of the probability of not exiting the domain by time t. We study the behavior of the eigenvalues, our main focus being on the behavior of the principal eigenvalue for the regimes γ 1 and γ 1. We also consider conditions on µ that guarantee that the principal eigenvalue is monotone increasing or decreasing in γ.

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Section: Let D ⊂ Rmentioning

confidence: 99%

Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps

Pinsky

2009

Trans. Amer. Math. Soc.

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“…In the case where ν y = ν for all y ∈ ∂D (ν need not be a point measure), the ergodicity of BMJ (as an important special case of elliptic operators) was systematically studied in [2] using a functional analytic approach. Recently, the most general case, where ν y depends continuously on its exit point y ∈ ∂D, was studied in [3]. The key results in [2], [3] address the construction of the invariant probability measure and relate the rate of convergence of the transition distribution to the invariant measure to the spectral gap of the generator of BMJ.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the most general case, where ν y depends continuously on its exit point y ∈ ∂D, was studied in [3]. The key results in [2], [3] address the construction of the invariant probability measure and relate the rate of convergence of the transition distribution to the invariant measure to the spectral gap of the generator of BMJ.…”

Section: Introductionmentioning

confidence: 99%

“…If p(t, x, ·) represents the transition probability measure for this Markov process, then it was shown in [3] that p(t, x, ·) approaches a unique stationary invariant measure µ on D with the density…”

Section: Introductionmentioning

confidence: 99%

“…dt is the 0-potential of the transition subprobability function p D (t, x, y) of the absorbed Brownian motion on D. The article [3] also describes the rate of convergence of p(t, x, ·) to the stationary measure. Consider the eigenvalues of the non-local eigenvalue problem…”

Section: Introductionmentioning

confidence: 99%

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Spectral analysis of Brownian motion with jump boundary

Leung

Patch

2008

Proc. Amer. Math. Soc.

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Abstract. Consider a family of probability measures, indexed by ∂D, on a bounded open region D ⊂ R d with a smooth boundary. For any starting point inside D, we run a standard d-dimensional Brownian motion in R d until it first exits D, at which time it jumps to a point inside the domain D according to the jump measure at the exit point and starts a new Brownian motion. The same evolution is repeated independently each time the process reaches the boundary. We study the exponential rate at which the transition distribution of the process converges to its invariant measure, in terms of the spectral gap of the generator. In particular, we prove two conjectures of I. Ben-Ari and R. Pinsky for an interval (see J. Funct. Anal. 251 (2007), 122-140, and preprint (2007)) by studying when a combination of the sine and cosine transforms of probability measures on an interval has only real zeros.

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One dimensional Brownian motion with holding and jumping boundary

Leung

2022

Math Methods in App Sciences

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Let a particle start at some point in the unit interval and undergo Brownian motion in until it hits one of the end points. At this instant, the particle stays put for a finite holding time with an exponential distribution and then jumps back to a point inside with a probability density or parametrized by the boundary point it was at. The process starts afresh. The same evolution repeats independently each time. Many probabilistic aspects of this diffusion process are investigated in the paper by Peng and Li listed in the reference. The authors in the cited paper call this process diffusion with holding and jumping (DHJ). Our simple aim in this paper is to analyze the eigenvalues of a nonlocal boundary problem arising from this process. In particular, we answer a question on the spectral gap of the DHJ process raised at the end of the paper by Peng and Li.

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Ergodic behavior of diffusions with random jumps from the boundary

Cited by 38 publications

References 5 publications

Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps

Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps

Spectral analysis of Brownian motion with jump boundary

One dimensional Brownian motion with holding and jumping boundary

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