An encounter-based approach for restricted diffusion with a gradient drift

Grebenkov, Denis S.

doi:10.48550/arxiv.2110.12181

Cited by 8 publications

(27 citation statements)

References 60 publications

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“…Following [14,15,17], one can develop a single-particle probabilistic version of the Robin boundary condition in terms of the boundary local time ℓ t . Let X t ∈ Ω represent the position of the particle at time t. The boundary local time for a totally reflecting surface ∂U is then defined according to [13]…”

Section: Diffusion-mediated Surface Reactionsmentioning

confidence: 99%

“…where Θ is the Heaviside function. Note that ℓ t has units of length due to the additional factor of D. Let P (x, ℓ, t|x 0 ) denote the joint probability density or propagator for the pair (X t , ℓ t ) and introduce the stopping time [14,15,17]…”

Section: Diffusion-mediated Surface Reactionsmentioning

confidence: 99%

“…Recently, Grebenkov [14,15,17] has used the boundary local time to develop a theoretical framework for investigating more general forms of diffusion-mediated absorption by partially reactive surfaces. The basic idea is to consider the joint probability density or propagator P (x, ℓ, t|x 0 ) for the pair (X t , ℓ t ) in the case of a perfectly reflecting boundary, where X t and ℓ t denote the particle position and local time, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Since the Robin boundary condition maps to an exponential law for the stopping local time ℓ t , the probability density p(x, t|x 0 ) can be expressed in terms of the Laplace transform of the (full) propagator P (x, ℓ, t|x 0 ) with respect to the local time ℓ. The advantage of this formulation is that one can consider a more general probability distribution Ψ(ℓ) = P[ l > ℓ] for the stopping local time l such that [14,15,17] p(x, t|x 0 ) = ´∞ 0 Ψ(ℓ)P (x, ℓ, t|x 0 )dℓ. This accommodates a wider class of surface reactions where, for example, the reactivity κ(ℓ) depends on the local time ℓ (or the number of surface encounters).…”

Section: Introductionmentioning

confidence: 99%

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Diffusion-mediated surface reactions, Brownian functionals and the Feynman-Kac formula

Bressloff¹

2022

Preprint

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Many processes in cell biology involve diffusion in a domain Ω that contains a target U whose boundary ∂U is a chemically reactive surface. Such a target could represent a single reactive molecule, an intracellular compartment or a whole cell. Recently, a probabilistic framework for studying diffusion-mediated surface reactions has been developed that considers the joint probability density or propagator for the particle position and the so-called boundary local time. The latter characterizes the amount of time that a Brownian particle spends in the neighborhood of a point on a totally reflecting boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. In this paper we generalize the theory of diffusionmediated surface reactions to cases where the whole interior target domain U acts as a partial absorber rather than the target boundary ∂U . Now the particle can freely enter and exit U , and is only able to react (be absorbed) within U . The appropriate Brownian functional is then the occupation time (accumulated time that the particle spends within U ) rather than the boundary local time. We show that both cases can be considered within a unified framework by using a Feynman-Kac formula to derive a boundary value problem (BVP) for the propagator of the corresponding Brownian functional, and introducing an associated stopping condition. We illustrate the theory by calculating the mean first passage time (MFPT) for a spherical target U located at the center of a spherical domain Ω. This is achieved by solving the propagator BVP directly, rather than using spectral methods. We find that if the first moment of the stopping time density is infinite, then the MFPT is also infinite, that is, the spherical target is not sufficiently absorbing.

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Section: Diffusion-mediated Surface Reactionsmentioning

confidence: 99%

Section: Diffusion-mediated Surface Reactionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Diffusion-mediated surface reactions, Brownian functionals and the Feynman-Kac formula

Bressloff¹

2022

Preprint

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“…In section 3, we generalize the Robin boundary condition to a more general probabilistic model of diffusion-mediated surface reactions, following the encounter-based approach developed by Grebenkov [20,21,23]. The latter exploits the fact that diffusion in a domain with a totally reflecting surface can be implemented probabilistically in terms of so-called reflected Brownian motion, which involves the introduction of a Brownian functional known as the boundary local time [25,28,26].…”

Section: Introductionmentioning

confidence: 99%

Diffusion-mediated surface reactions and stochastic resetting

Bressloff

2022

Preprint

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In this paper, we investigate the effects of stochastic resetting on diffusion in R d \U, where U is a bounded obstacle with a partially absorbing surface ∂U. We begin by considering a Robin boundary condition with a constant reactivity κ0, and show how previous results are recovered in the limits κ0 → 0, ∞. We then generalize the Robin boundary condition to a more general probabilistic model of diffusion-mediated surface reactions using an encounter-based approach. The latter considers the joint probability density or propagator P (x, ℓ, t|x0) for the pair (Xt, ℓt) in the case of a perfectly reflecting surface, where Xt and ℓt denote the particle position and local time, respectively. The local time determines the amount of time that a Brownian particle spends in a neighborhood of the boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. We construct the boundary value problem (BVP) satisfied by the propagator in the presence of resetting, and use this to derive implicit equations for the marginal density of particle position and the survival probability. Finally, we show how to explicitly solve these equations in the case of a spherically symmetric surface using separation of variables. A major result of our analysis is that, although resetting is not governed by a renewal process, the survival probability with resetting can be expressed in terms of the survival probability without resetting, and can thus be calculated explicitly. This allows us to explore the dependence of the MFPT on the resetting rate r and the type of surface reactions. In addition to determining the optimal resetting rate that minimizes the MFPT, we also show that the relative increase in the MFPT compared to the case of a totally absorbing surface can itself exhibit non-monotonic variation with r.

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Diffusion in a partially absorbing medium with position and occupation time resetting

Bressloff¹

2022

J. Stat. Mech.

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In this paper we consider diffusion in a domain Ω containing a partially absorbing target M with position and occupation time resetting. The occupation time A t is a Brownian functional that determines the amount of time that the particle spends in M over the time interval [0, t]. We assume that there exists some internal state U t of the particle at time t which is modified whenever the particle is diffusing within M . The state U t is taken to be a monotonically increasing function of A t , and absorption occurs as soon as U t crosses some fixed threshold. We first show how to analyze threshold absorption in terms of the joint probability density or generalized propagator P(x, a, t|x 0) for the pair (X t , A t ) in the case of a non-absorbing substrate M , where X t is the particle position at time t and x 0 is the initial position. We then introduce a generalized stochastic resetting protocol in which both the position X t and the internal state U t are reset to their initial values, X t → x 0 and U t → 0 , at a Poisson rate r. The latter is mathematically equivalent to resetting the occupation time, A t → 0. Since resetting is governed by a renewal process, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which means that the statistics of absorption can be determined by calculating the double Laplace transform of P(x, a, t|x 0) with respect to t and a. In order to develop the basic theory, we focus on one-dimensional diffusion with M given by a finite or semi-infinite interval, and explore how the mean first passage time with resetting depends on various model parameters. We also compare the threshold mechanism with the classical case of a constant absorption rate.

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An encounter-based approach for restricted diffusion with a gradient drift

Cited by 8 publications

References 60 publications

Diffusion-mediated surface reactions, Brownian functionals and the Feynman-Kac formula

Diffusion-mediated surface reactions, Brownian functionals and the Feynman-Kac formula

Diffusion-mediated surface reactions and stochastic resetting

Diffusion in a partially absorbing medium with position and occupation time resetting

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