Asymptotic Behavior of Density in the Boundary-Driven Exclusion Process on the Sierpinski Gasket

Abstract

We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.

Appendix A: Dirichlet-to-Neumann map on SG

In this appendix we characterize the harmonic function which satisfies the Robin boundary condition

$$ \left\{\begin{array}{ll} {\Delta} h(x) =0 ,& x\in K\setminus V_{0},\\ \partial^{\perp} h(a) + \kappa(a) h(a) = \gamma(a), & a\in V_{0}, \end{array} \right. $$

(A.1)

where {κ(a) : a ∈ V₀} and {γ(a) : a ∈ V₀} are given coefficients.

Let $h^{i}: K\to \mathbb {R}$, i ∈{0, 1, 2}, denote the harmonic function with Dirichlet boundary condition hⁱ(a_j) = δ_ij, j ∈{0, 1, 2}. By the harmonic extension algorithm described in [36, Section 1.3], $\{h^{i}\}_{i=0}^{2}$ is a basis for the space of harmonic functions on K, so we may express the solution h of (A.1) as a linear combination $h= {\sum }_{i=0}^{2} \textbf {c}_{i} h^{i}$, where the coefficients {c_i}_i are determined by the boundary condition in (A.1):

$$ \sum\limits_{i=0}^{2} \textbf{c}_{i} (\partial^{\perp} h^{i})(a_{j}) + \kappa(a_{j}) \textbf{c}_{j} = \gamma(a_{j}), \quad j\in \{0,1,2\}. $$

(A.2)

We can then conclude that h is a harmonic function satisfying the Dirichlet boundary condition h(a_i) = c_i, i ∈{0, 1, 2}.

So it suffices to find {c_i}_i. The harmonic extension algorithm [36, Section 1.3] yields

$$ (\partial^{\perp} h^{i})(a_{j}) = \left\{\begin{array}{ll} 2, & \text{if } j=i,\\ -1, & \text{if } j \neq i. \end{array} \right. $$

(A.3)

Thus we arrive at the matrix problem

$$ \left[\begin{array}{lll} 2+\kappa_{0} & -1 & -1 \\ -1 & 2+\kappa_{1} & -1 \\ -1 & -1 & 2+\kappa_{2} \end{array}\right] \left[\begin{array}{ll} \textbf{c}_{0} \\ \textbf{c}_{1} \\ \textbf{c}_{2} \end{array}\right] = \left[\begin{array}{ll} \gamma_{0} \\ \gamma_{1} \\ \gamma_{2} \end{array}\right] . $$

(A.4)

where κ_j and γ_j are shorthands for κ(a_j) and γ(a_j). It can be checked that the left-hand side matrix is invertible iff its determinant

$$ \boldsymbol{\Delta}:=3(\kappa_{0}+\kappa_{1}+\kappa_{2}) + 2(\kappa_{0} \kappa_{1}+ \kappa_{1}\kappa_{2}+\kappa_{2}\kappa_{0}) + \kappa_{0}\kappa_{1}\kappa_{2} $$

(A.5)

is nonzero. Assuming invertibility, we find

$$ \left[\begin{array}{ll} \textbf{c}_{0} \\ \textbf{c}_{1} \\ \textbf{c}_{2} \end{array}\right] = \frac{1}{\boldsymbol{\Delta}} \left[\begin{array}{lll} 3+ 2(\kappa_{1}+\kappa_{2}) + \kappa_{1}\kappa_{2} & 3+\kappa_{2} & 3+\kappa_{1} \\ 3+\kappa_{2} & 3+ 2(\kappa_{2} +\kappa_{0}) + \kappa_{2} \kappa_{0} & 3+\kappa_{0}\\ 3+\kappa_{1} & 3+\kappa_{0} & 3+2(\kappa_{0}+\kappa_{1})+\kappa_{0}\kappa_{1} \end{array}\right] \left[\begin{array}{ll} \gamma_{0} \\ \gamma_{1} \\ \gamma_{2} \end{array}\right]. $$

(A.6)