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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.

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Acknowledgements

We thank the anonymous referees for useful comments that helped us improve the paper.

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Correspondence to Joe P. Chen.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

JPC thanks the US National Science Foundation (DMS-1855604), the Simons Foundation (Collaboration Grant for Mathematicians #523544), and the Research Council of Colgate University for support. PG thanks FCT/Portugal for support through the project UID/MAT/04459/2013. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovative programme (grant agreement No 715734).

Appendix A: Dirichlet-to-Neumann map on SG

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) : aV0} and {γ(a) : aV0} are given coefficients.

Let \(h^{i}: K\to \mathbb {R}\), i ∈{0, 1, 2}, denote the harmonic function with Dirichlet boundary condition hi(aj) = δij, 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 {ci}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(ai) = ci, i ∈{0, 1, 2}.

So it suffices to find {ci}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 κj and γj are shorthands for κ(aj) and γ(aj). 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)

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Chen, J.P., Gonçalves, P. Asymptotic Behavior of Density in the Boundary-Driven Exclusion Process on the Sierpinski Gasket. Math Phys Anal Geom 24, 24 (2021). https://doi.org/10.1007/s11040-021-09392-4

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Keywords

  • Exclusion process
  • Hydrodynamic limit
  • Equilibrium fluctuations
  • Heat equation
  • Ornstein-Uhlenbeck equation
  • Analysis on fractals
  • Dirichlet forms
  • Moving particle lemma

Mathematics Subject Classification (2010)

  • Primary 35K05
  • 35R60
  • 60H15
  • 60K35
  • 82C22
  • Secondary 28A80
  • 60J27