Accumulation time of stochastic processes with resetting

Bressloff, Paul C.

doi:10.1088/1751-8121/ac16e5

Cited by 15 publications

(14 citation statements)

References 33 publications

(64 reference statements)

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“…In light of this, we consider an alternative way of characterizing the approach to steady state that is based on the so-called accumulation time. The latter was originally developed within the context of diffusion-based morphogenesis [25][26][27][28], but has more recently been applied to intracellular protein gradient formation [29] and to diffusion processes with stochastic resetting [30].…”

Section: Introductionmentioning

confidence: 99%

Accumulation time of diffusion in a two-dimensional singularly perturbed domain

Bressloff

2022

Proc. R. Soc. A.

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A general problem of current interest is the analysis of diffusion problems in singularly perturbed domains, within which small subdomains are removed from the domain interior and boundary conditions imposed on the resulting holes. One major application is to intracellular diffusion, where the holes could represent organelles or biochemical substrates. In this paper, we use a combination of matched asymptotic analysis and Green’s function methods to calculate the so-called accumulation time for relaxation to steady state. The standard measure of the relaxation rate is in terms of the principal non-zero eigenvalue of the negative Laplacian. However, this global measure does not account for possible differences in the relaxation rate at different spatial locations, is independent of the initial conditions, and relies on the assumption that the eigenvalues have sufficiently large spectral gaps. As previously established for diffusion-based morphogen gradient formation, the accumulation time provides a better measure of the relaxation process.

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Section: Introductionmentioning

confidence: 99%

Accumulation time of diffusion in a two-dimensional singularly perturbed domain

Bressloff

2022

Proc. R. Soc. A.

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“…Since the trajectories contributing to the transient region are rare events, one can establish the existence of the phase transition by carrying out an asymptotic expansion of the exact solution [24]. It turns out that this transition can also be understood in terms of the spatial variation of the accumulation time for relaxation to the NESS [25]. That is, T (x) ∼ |x − x 0 |/ √ 4rD for |x − x 0 | ≫ D/r.…”

Section: Discussionmentioning

confidence: 99%

Accumulation time of diffusion in a 3D singularly perturbed domain

Bressloff

2022

Preprint

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Boundary value problems for diffusion in singularly perturbed domains (domains with small holes removed from the interior) is a topic of considerable current interest. Applications include intracellular diffusive transport and the spread of pollutants or heat from localized sources. In a previous paper, we introduced a new method for characterizing the approach to steady-state in the case of two-dimensional (2D) diffusion. This was based on a local measure of the relaxation rate known as the accumulation time T (x). The latter was calculated by solving the diffusion equation in Laplace space using a combination of matched asymptotics and Green's function methods. We thus obtained an asymptotic expansion of T (x) in powers of ν = −1/ ln ǫ, where ǫ specifies the relative size of the holes. In this paper, we develop the corresponding theory for three-dimensional (3D) diffusion. The analysis is a non-trivial extension of the 2D case due to differences in the singular nature of the Laplace transformed Green's function. In particular, the asymptotic expansion of the solution of the 3D diffusion equation in Laplace space involves terms of order O((ǫ/s) n ), where s is the Laplace variable. These s-singularities have to be removed by partial series resummations in order to obtain an asymptotic expansion of T (x) in powers of ǫ.

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“…In the case of homogeneous diffusion in R d , one can use large deviation theory to show that the approach to the stationary state exhibits a dynamical phase transition, which can be interpreted as a traveling front separating spatial regions for which the probability density has relaxed to the NESS from those where transients persist [42]. Recently, we introduced an alternative method for characterizing the relaxation process, which is based on the notion of an accumulation time [43]. We proceeded by decomposing the probability density into decreasing and accumulating components, and showed how the latter evolved in an analogous fashion to the formation of a concentration gradient in diffusion-based morphogenesis.…”

Section: Relaxation Timementioning

confidence: 99%

A probabilistic model of diffusion through a semi-permeable barrier

Bressloff¹

2022

Preprint

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Diffusion through semipermeable structures arises in a wide range of processes in the physical and life sciences. Examples at the microscopic level range from artificial membranes for reverse osmosis to lipid bilayers regulating molecular transport in biological cells to chemical and electrical gap junctions. There are also macroscopic analogs such as animal migration in heterogeneous landscapes. It has recently been shown that one-dimensional diffusion through a barrier with constant permeability κ0 is equivalent to snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either the left or right of the barrier. Each round is killed when its Brownian local time exceeds an exponential random variable parameterized by κ0. A new round is then immediately started in either direction with equal probability. In this paper we use a combination of renewal theory, Laplace transforms and Green's function methods to show how an extended version of snapping out BM provides a general probabilistic framework for modeling diffusion through a semipermeable barrier. This includes modifications of the diffusion process away from the barrier (eg. stochastic resetting) and non-Markovian models of membrane absorption that kill each round of partially reflected BM. The latter leads to timedependent permeabilities.

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Accumulation time of stochastic processes with resetting

Cited by 15 publications

References 33 publications

Accumulation time of diffusion in a two-dimensional singularly perturbed domain

Accumulation time of diffusion in a two-dimensional singularly perturbed domain

Accumulation time of diffusion in a 3D singularly perturbed domain

A probabilistic model of diffusion through a semi-permeable barrier

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