Spectral gap for measure-valued diffusion processes

Ren, Panpan; Wang, Feng‐Yu

doi:10.1016/j.jmaa.2019.123624

Cited by 8 publications

(4 citation statements)

References 22 publications

(27 reference statements)

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“…Among the most influential works, we mention those by Caputo et al [CLR10] on the Aldous' spectral gap conjecture, and by Lacoin [Lac16] on the TV cutoff on one-dimensional domains, both for the Symmetric Simple Exclusion process (SSEP). More recently, versions of Aldous' spectral gap identity have been shown for other models, e.g., the Zero Range process (ZRP) [HS19], Beta and Gibbs samplers on the segment [CLL20a,CLL20b], Wright-Fisher and Fleming-Viot processes and multi-allelic Moran models [Shi77,Gri14,RW20,Cor20]. Similarly, we refer, among others, to [LL11, Jon12, LL19, LL20, Sch19, Sch21, MS19, HS20] for recent developments on the cutoff for SSEP, its asymmetric variants and ZRP.…”

Section: Introductionmentioning

confidence: 99%

Mixing of the Averaging process and its discrete dual on finite-dimensional geometries

Quattropani,

2021

Preprint

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We analyze the L 1 -mixing of a generalization of the Averaging process introduced by Aldous [Ald11]. The process takes place on a growing sequence of graphs which we assume to be finitedimensional, in the sense that the random walk on those geometries satisfies a family of Nash inequalities. As a byproduct of our analysis, we provide a complete picture of the total variation mixing of a discrete dual of the Averaging process, which we call Binomial Splitting process. A single particle of this process is essentially the random walk on the underlying graph. When several particles evolve together, they interact by synchronizing their jumps when placed on neighboring sites. We show that, given k the number of particles and n the (growing) size of the underlying graph, the system exhibits cutoff in total variation if k → ∞ and k = O(n 2 ). Finally, we exploit the duality between the two processes to show that the Binomial Splitting satisfies a version of Aldous' spectral gap identity, namely, the relaxation time of the process is independent of the number of particles.

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Section: Introductionmentioning

confidence: 99%

Mixing of the Averaging process and its discrete dual on finite-dimensional geometries

Quattropani,

2021

Preprint

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“…The construction of Dirichlet forms in these papers heavily relies on the one-dimensional property. See also [15,22,27,28] and references within for the study of different type measure-valued diffusion processes using Dirichlet forms. Next, following the idea of Konarovskyi (see e.g.…”

Section: Some Related Studiesmentioning

confidence: 99%

Image-dependent conditional McKean–Vlasov SDEs for measure-valued diffusion processes

Wang

2021

J. Evol. Equ.

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We consider a special class of mean field SDEs with common noise which depend on the image of the solution (i.e. the conditional distribution given noise). The strong well-posedness is derived under a monotone condition which is weaker than those used in the literature of mean field games, the Feynman-Kac formula is established to solve Schrördinegr type PDEs on P 2 , and the ergodicity is proved for a class of measurevalued diffusion processes.

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“…To study measure-valued diffusion processes, local Dirichlet forms have been constructed by establishing the integration by parts formula of derivatives in measure with respect to a reference distribution Ξ on the space of Radon measures, see [23,26,31,32] and references therein. Moreover, functional inequalities have been derived for measure-valued processes, see [13,14,15,16,28,36,38]. However, in these papers, the stationary distribution Ξ is either supported on the space of discrete measures (i.e.…”

Section: Introductionmentioning

confidence: 99%

Ornstein-Uhlenbeck Type Processes on Wasserstein Space

Ren¹,

Wang²

2022

Preprint

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The Wasserstein space P 2 consists of square integrable probability measures on R d and is equipped with the intrinsic Riemannian structure. By using stochastic analysis on the tangent space, we construct the Ornstein-Uhlenbeck (O-U) process on P 2 whose generator is formulated as the intrinsic Laplacian with a drift. This process satisfies the log-Sobolev inequality and has L 2 -compact Markov semigroup. Due to the important role played by O-U process in Malliavin calculus on the Wiener space, this measure-valued process should be a fundamental model to develop stochastic analysis on the Wasserstein space. Perturbations of the O-U process is also studied.

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Spectral gap for measure-valued diffusion processes

Cited by 8 publications

References 22 publications

Mixing of the Averaging process and its discrete dual on finite-dimensional geometries

Mixing of the Averaging process and its discrete dual on finite-dimensional geometries

Image-dependent conditional McKean–Vlasov SDEs for measure-valued diffusion processes

Ornstein-Uhlenbeck Type Processes on Wasserstein Space

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