## The Annals of Applied Probability

### Large deviations for the exclusion process with a slow bond

#### Abstract

We consider the one-dimensional symmetric simple exclusion process with a slow bond. In this model, whilst all the transition rates are equal to one, a particular bond, the slow bond, has associated transition rate of value $N^{-1}$, where $N$ is the scaling parameter. This model has been considered in previous works on the subject of hydrodynamic limit and fluctuations. In this paper, assuming uniqueness for weak solutions of hydrodynamic equation associated to the perturbed process, we obtain dynamical large deviations estimates in the diffusive scaling. The main challenge here is the fact that the presence of the slow bond gives rise to Robin’s boundary conditions in the continuum, substantially complicating the large deviations scenario.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3547-3587.

Dates
Revised: February 2017
First available in Project Euclid: 15 December 2017

https://projecteuclid.org/euclid.aoap/1513328708

Digital Object Identifier
doi:10.1214/17-AAP1287

Mathematical Reviews number (MathSciNet)
MR3737932

Zentralblatt MATH identifier
06848273

#### Citation

Franco, Tertuliano; Neumann, Adriana. Large deviations for the exclusion process with a slow bond. Ann. Appl. Probab. 27 (2017), no. 6, 3547--3587. doi:10.1214/17-AAP1287. https://projecteuclid.org/euclid.aoap/1513328708

#### References

• [1] Bertini, L., Landim, C. and Mourragui, M. (2009). Dynamical large deviations for the boundary driven weakly asymmetric exclusion process. Ann. Probab. 37 2357–2403.
• [2] Faggionato, A., Jara, M. and Landim, C. (2009). Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances. Probab. Theory Related Fields 144 633–667.
• [3] Farfan, J., Landim, C. and Mourragui, M. (2011). Hydrostatics and dynamical large deviations of boundary driven gradient symmetric exclusion processes. Stochastic Process. Appl. 121 725–758.
• [4] Franco, T., Gonçalves, P. and Neumann, A. (2013). Hydrodynamical behavior of symmetric exclusion with slow bonds. Ann. Inst. Henri Poincaré Probab. Stat. 49 402–427.
• [5] Franco, T., Gonçalves, P. and Neumann, A. (2013). Phase transition in equilibrium fluctuations of symmetric slowed exclusion. Stochastic Process. Appl. 123 4156–4185.
• [6] Franco, T., Gonçalves, P. and Neumann, A. (2015). Phase transition of a heat equation with Robin’s boundary conditions and exclusion process. Trans. Amer. Math. Soc. 367 6131–6158.
• [7] Franco, T. and Landim, C. (2010). Hydrodynamic limit of gradient exclusion processes with conductances. Arch. Ration. Mech. Anal. 195 409–439.
• [8] Franco, T. and Neumann, A. (2015). Large deviations for the exclusion process with a slow bond. Available at arXiv:1501.00225v1.
• [9] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin.