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Explicit LDP for a slowed RW driven by a symmetric exclusion process

Abstract

We consider a random walk (RW) driven by a simple symmetric exclusion process (SSE). Rescaling the RW and the SSE in such a way that a joint hydrodynamic limit theorem holds we prove a joint path large deviation principle. The corresponding large deviation rate function can be split into two components, the rate function of the SSE and the one of the RW given the path of the SSE. These components have different structures (Gaussian and Poissonian, respectively) and to overcome this difficulty we make use of the theory of Orlicz spaces. In particular, the component of the rate function corresponding to the RW is explicit.

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Notes

  1. The time window [0, T] is chosen to be of finite size to avoid some technical topological considerations. Indeed, \([0,\infty )\) is not compact and one would have to be more careful in weighting the tails near infinity.

  2. Notice that we are making an abuse of notation, using the same superscript structure for \({\mathcal M}_t^{a,n}\) and \({\mathcal M}_t^{H,n}\). Later on we will introduce some more efficient way to handle multiple indices.

  3. Note that under the assumption \(c^++c^- \equiv 1\), one can take \(K_1=1\).

  4. In this section we will only use f for test functions; do not confuse with the notation local functions used in the previous section.

  5. We are considering only open sets in order to apply later the Minimax Lemma in Proposition 19.

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Acknowledgements

L.A. has been supported by NWO Gravitation Grant 024.002.003-NETWORKS. M.J. has been partially supported by the ERC Horizon 2020 grant #715734 and by NWO Gravitation Grant 024.002.003-NETWORKS.

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Correspondence to F. Völlering.

Appendix A: Orlicz spaces

Appendix A: Orlicz spaces

Orlicz spaces are a natural generalization of \(L^p\)-spaces. We recall here some basic definitions and properties (see e.g. [22] for more details). For a general measurable space \((E,{\mathcal E},\mu )\) and a Young function \(\Phi \), that is a lower-semicontinuous convex function \(\Phi :[0,\infty )\rightarrow [0,\infty ]\) with \(\Phi (0)=0\) but not identical to 0, we can define the Luxembourg norm of \(f:E\rightarrow \mathbb {R}\):

$$\begin{aligned} \left\| \, f \, \right\| _\Phi := \inf \left\{ a>0 : \int \Phi \left( \frac{|f|}{a}\right) \;d\mu \le 1 \right\} . \end{aligned}$$
(76)

The Orlicz space is then given by \(L_\Phi (\mu ):=\{f: \left\| \, f \, \right\| _\Phi <\infty \}\), and it turns out to be a Banach space. In the case that \(\Phi (x)=x^p\) we recover the \(L^p\)-space. If the measure \(\mu \) is finite it also holds that \(L_{\Phi }(\mu )\subset L^1(\mu )\).

Similar to the \(L^p\)-spaces Orlicz spaces also have a natural dual. Let \(\Phi ^*\) be the convex conjugate of \(\Phi \). Then \(\Phi ^*\) is also a Young function, and has an associated Orlicz space \(L_{\Phi ^*}\).

Based on this duality there is a Hölder inequality for dual Orlicz spaces: for all \(f\in L_\Phi (\mu )\), \(g\in L_{\Phi ^*}(\mu )\),

$$\begin{aligned} \int |fg|\;d\mu \le 2 \left\| \, f \, \right\| _{\Phi }\left\| \, g \, \right\| _{\Phi ^*}. \end{aligned}$$
(77)

In particular, \(fg\in L^1(\mu )\). Note that in contrast to the usual Hölder-inequality there is an additional factor 2.

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Avena, L., Jara, M. & Völlering, F. Explicit LDP for a slowed RW driven by a symmetric exclusion process. Probab. Theory Relat. Fields 171, 865–915 (2018). https://doi.org/10.1007/s00440-017-0797-6

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  • DOI: https://doi.org/10.1007/s00440-017-0797-6

Keywords

  • Large deviations
  • Random environments
  • Hydrodynamic limits
  • Particle systems
  • Exclusion process

Mathematics Subject Classification

  • 60F10
  • 82C22
  • 82D30