Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homogenization of a multivariate diffusion with semipermeable interfaces

Olga Aryasova, Ilya Pavlyukevich, Andrey Pilipenko

Abstract

We study the homogenization problem for a system of stochastic differential equation with local time terms that models a multivariate diffusion in presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift.

Homogenization of a multivariate diffusion with semipermeable interfaces

Abstract

We study the homogenization problem for a system of stochastic differential equation with local time terms that models a multivariate diffusion in presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift.
Paper Structure (10 sections, 15 theorems, 151 equations, 1 figure)

This paper contains 10 sections, 15 theorems, 151 equations, 1 figure.

Table of Contents

  1. Setting and the main results
  2. Existence and uniqueness. Proof of Theorem \ref{['t:exist']}
  3. Dynamics around one membrane
  4. Properties of the coordinate transformation $(\Phi^\varepsilon,\Psi^\varepsilon)$
  5. Rough estimates of the exit time from the strip
  6. Accurate estimates for $X^\varepsilon$
  7. Accurate estimates containing $Y^\varepsilon$
  8. Proof of Theorem \ref{['t:hom']}
  9. Weak relative compactness
  10. Convergence of finite dimensional distributions

Key Result

Theorem 1.1

Let Assumptions A hold true. Then there is $\varepsilon_0\in(0,1]$ small enough such that for all $\varepsilon\in(0,\varepsilon_0]$ the system e:XY has a unique weak solution $(X^\varepsilon,Y^\varepsilon)$, which is a strong Markov process.

Figures (1)

  • Figure 1: Sample paths of the "underlying" diffusion $(X^0,Y^0)$ without membranes (left) and of the limiting homogenized diffusion $(X,Y)$ (right). The membranes with penetration probabilities $\frac{1}{2}+\frac{\varepsilon}{2}\beta(x,y)$ and penetration directions $(1,\theta(x,y))$ generate an additional drift that reverses the diffusion's rotation direction. Both samples start at the point $(2,2)$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Example 1.4
  • Theorem 2.1: Hadamard
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 20 more