| Abstract: |
| There is by now a series of rigorous results on non-equilibrium and stationary density fluctuations in the exclusion process on variants of the 1D lattice, possibly with a slow bond or connected to (up to) 2 variable-speed reservoirs.
Being in 1D, the stationary density profile has constant gradient, and the two-point correlation function is essentially the Green's function for the 1D random walk.
Unfortunately, when one studies the exclusion process in higher dimensions and without spatial symmetries, these 1D features are no longer retained.
To what extent can one describe non-equilibrium fluctuations then?
In this talk I will answer this question affirmatively in the setting of resistance spaces, using the Sierpinski gasket (with 3 boundary reservoirs) as the working example.
By carefully estimating the microscopic correlation function, and using inputs from analysis on fractals, Dirichlet forms, and $\Gamma$-calculus, we obtain explicit formulas for the scaling limits of non-equilibrium and stationary density fluctuations, including a generalized Ornstein-Uhlenbeck limit.
Between the fractal case and the 1D case there are superficial similarities and important distinctions.
If time permits, I will discuss extension to the weakly asymmetric exclusion process on the gasket, whose scaling limit is a stochastic Burgers` equation thereon. |
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