Fluctuations in Markov Processes

Landim, Cláudio; Olla, Stefano; Komorowski, Tomasz

doi:10.1007/978-3-642-29880-6

Cited by 161 publications

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“…This ends the proof of (3.2), modulo the fact that estimate (3.17) still requires to be shown. Its proof is an adaptation to the present case of an argument used in [12], see also Section 2.7.4 of [5]. Define a bounded operator T : 2 (Z) → 2 (Z) by the formula T f x := t x f x , where…”

Section: Proposition 31 Under the Assumptions Of Theorem 21 For Anymentioning

confidence: 99%

A Criterion for Essential Self-Adjointness of a Symmetric Operator Defined by Some Infinite Hermitian Matrix with Unbounded Entries

Komorowski

2015

Integr. Equ. Oper. Theory

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Abstract. We shall consider a double infinite, Hermitian, complex entry matrix A = [ax,y] x,y∈Z . In the present note we give a criterion, expressed in terms of the entries of the matrix, for the corresponding symmetric operator defined on compactly supported sequences, to be essentially self-adjoint in the space 2(Z). Roughly speaking, assuming that x denotes the row number, we require that: (1) there exist γ ∈ [0, 1) and n > 0 for which the entries that are at distance larger than n(|x| 2 +1) γ/2 from the diagonal vanish and (2) the 1 norm of the xth row grows slower that |x| γ−1 , as |x| → +∞.

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Section: Proposition 31 Under the Assumptions Of Theorem 21 For Anymentioning

confidence: 99%

A Criterion for Essential Self-Adjointness of a Symmetric Operator Defined by Some Infinite Hermitian Matrix with Unbounded Entries

Komorowski

2015

Integr. Equ. Oper. Theory

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“…In this paper we approach this problem probabilistically, and in Section 3 we formulate this system as a stochastic homogenization problem using the framework of [26]. Using methods from the theory of stochastic homogenization for SDEs one can then identify the limiting behaviour of the diffusion process.…”

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confidence: 99%

“…In this section we will rigorously state the assumptions on the random field h(x) which are necessary for the problem to be well-defined and for a homogenization limit of both the SDE (2.4) and the PDE (2.7) to exist. The approach described here is a direct application of the results in [26,Chapter 9], whose approach we will follow very closely.…”

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“…The Homogenization Result. In this section we state the homogenization result for the SDE (2.4) and PDE (2.7) making use of standard stochastic homogenization techniques such as [7,24,26]. The approach adopted here closely follows that of [26].…”

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“…In this section we state the homogenization result for the SDE (2.4) and PDE (2.7) making use of standard stochastic homogenization techniques such as [7,24,26]. The approach adopted here closely follows that of [26]. The assumption that the random field h is stationary and ergodic with respect to spatial translations is essential to obtaining a limiting diffusion process in the limit as ε → 0.…”

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Homogenization of Lateral Diffusion on a Random Surface

Duncan¹

2015

Multiscale Model. Simul.

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Abstract. We study the problem of lateral diffusion on a static, quasi-planar surface generated by a stationary, ergodic random field possessing rapid small-scale spatial fluctuations. The aim is to study the effective behaviour of a particle undergoing Brownian motion on the surface viewed a projection on the underlying plane. By formulating the problem as a diffusion in a random medium, we are able to use known results from the theory of stochastic homogenization of SDEs to show that, in the limit of small scale fluctuations, the diffusion process behaves quantitatively like a Brownian motion with constant diffusion tensor D. While D will not have a closed-form expression in general, we are able to derive variational bounds for the effective diffusion tensor, and using a duality transformation argument, obtain a closed form expression for D in the special case where D is isotropic. We also describe a numerical scheme for approximating the effective diffusion tensor numerically and illustrate this scheme with two examples.Key words. stochastic homogenisation, random media, Laplace-Beltrami, diffusion, multiscale-analysis. AMS subject classifications. 35Q92, 60H30, 35B271. Introduction. Lateral diffusion of particles along interfaces is a frequently occurring phenomenon in cellular biology. In the case of lipid bilayer membranes, the lipid molecules and integral membrane proteins which constitute the cell membrane themselves undergo lateral diffusion as a result of thermal agitation [4]. The mobility of membrane proteins has far-reaching implications for many cellular processes, in particular protein transport, signalling and morphology [2, 6, 9], thus there has been considerable interest in measuring how the mobility of these proteins is affected by the membrane and its surrounding environment. The dynamics of protein diffusion within a fluid membrane was first considered by Saffman and Delbrück [43] who proposed a continuum hydrodynamic model for a laterally diffusing particle in a flat, homogeneous fluid membrane. The model predicted a relationship for the diffusion tensor of a particle in the membrane in terms of the particle radius, the thickness and viscosity of the membrane and the viscosity of the bulk medium. Continuum models for studying the influence of shape fluctuations of the membrane on the macroscopic protein diffusion rate were subsequently considered in [18,19,32,41] in which the proteins undergo Brownian motion laterally along an infinitesimally thin two dimensional surface embedded in R 3 . The equilibrium fluctuations of the surface are characterised by the Canhan-Helfrich Hamiltonian [10,20]. In [32], an expression for the effective diffusion tensor was derived by considering the joint Markov process for the coupled particle and surface, applying an adiabatic elimination to average out the rapid temporal surface fluctuations. Another approach, based on a path-integral formulation was considered in [41]. More recently,in [13], a multiscale approach to the problem of lateral diffusion on rapi...

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