Diffusion in the presence of cells with semi-permeable membranes

An averaging principle for fast diffusions in domains separated by semi-permeable membranes

We prove an averaging principle which asserts convergence of diffusion processes on domains separated by semi-permeable membranes, when diffusion coefficients tend to infinity while the flux through the membranes remains constant. In the limit, points in each domain are lumped into a single state of a limit Markov chain. The limit chain's intensities are proportional to the membranes' permeability and inversely proportional to the domains' sizes. Analytically, the limit is an example of a singular perturbation in which boundary and transmission conditions play a crucial role. This averaging pr…

Modeling diffusion in thin $2D$-layers separated by a semi-permeable membrane

Motivated by models of signaling pathways in B lymphocytes, which have extremely large nuclei, we study the question of how reaction-diffusion equations in thin $2D$ domains may be approximated by diffusion equations in regions of smaller dimensions. In particular, we study how transmission conditions featuring in the approximating equations become integral parts of the limit master equation. We device a scheme which, by appropriate rescaling of coefficients and finding a common reference space for all Feller semigroups involved, allows deriving the form of the limit equation formally. The res…

A probabilistic model of diffusion through a semi-permeable barrier

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 $κ_0$ is equivalent to snapping out Brownian motion (BM). The latter sews together successive rou…

从弹离布朗运动到星形图上的Walsh蜘蛛过程

From snapping out Brownian motions to Walsh's spider processes on star-like graphs

该研究通过矩阵分析证明大渗透系数的弹离布朗运动可良好近似星形图$K_{1,k}$上的Walsh蜘蛛过程，揭示后者等价于图中心受半透膜迹线扰动的布朗运动，丰富了相关随机过程的近似理论。

One-dimensional diffusion processes with moving membrane: partial reflection in combination with jump-like exit of process from membrane

We use analytical methods to construct the two-parameter Feller semigroup associated with a Markov process on a line with a moving membrane such that at the points on both sides of the membrane it coincides with the ordinary diffusion processes given there and its behavior after reaching the membrane is described by a kind of nonlocal Feller-Wentzell conjugation condition.

A numerical method for solving snapping out Brownian motion in 2D bounded domains

Diffusion in heterogeneous media partitioned by semi-permeable interfaces has a wide range of applications in the physical and life sciences, including gas permeation in soils, diffusion magnetic resonance imaging (dMRI), drug delivery, thermal conduction in composite media, synaptic receptor trafficking, and intercellular gap junctions. At the single particle level, diffusion across a semi-permeable interface can be formulated in terms of so-called snapping out Brownian motion (SNOBM). The latter sews together successive rounds of reflected BM, each of which is restricted to one side of the i…