Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions

By S. Andres, Naotaka Kajino. 2014

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The Liouville Brownian motion (LBM), recently introduced by Garban, Rhodes and Vargas and in a weaker form also by Berestycki, is a diffusion process evolving in a planar random geometry induced by the Liouville measure$$M_\gamma $$Mγ, formally written as $$M_\gamma (dz)=e^{\gamma X(z)-{\gamma ^2} \mathbb E[X(z)^2]/2}\, dz$$Mγ(dz)=eγX(z)-γ2E[X(z)2]/2dz, $$\gamma \in (0,2)$$γ∈(0,2), for a (massive) Gaussian free field X. It is an $$M_\gamma $$Mγ-symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure $$M_\gamma $$Mγ. In this paper we provide a detailed analysis of the heat kernel $$p_t(x,y)$$pt(x,y) of the LBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian upper bound of the form $$p_t(x,y)\le C_{1} t^{-1} \log (t^{-1}) \exp \bigl (-C_{2}((|x-y|^{\beta }\wedge 1)/t)^{\frac{1}{\beta -1}}\bigr )$$pt(x,y)≤C1t-1log(t-1)exp(-C2((|x-y|β∧1)/t)1β-1) for $$t\in \bigl (0,\frac{1}{2}\bigr ]$$t∈(0,12] for each $$\beta >\frac{1}{2}(\gamma +2)^2$$β>12(γ+2)2, and an on-diagonal lower bound of the form $$p_{t}(x,x)\ge C_{3}t^{-1}\bigl (\log (t^{-1})\bigr )^{-\eta }$$pt(x,x)≥C3t-1(log(t-1))-η for $$t\in (0,t_{\eta }(x)]$$t∈(0,tη(x)], with $$t_{\eta }(x)\in \bigl (0,\frac{1}{2}\bigr ]$$tη(x)∈(0,12]heavily dependent onx, for each $$\eta >18$$η>18 for $$M_{\gamma }$$Mγ-almost everyx. As applications, we deduce that the pointwise spectral dimension equals 2 $$M_\gamma $$Mγ-a.e. and that the global spectral dimension is also 2.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c901e45c94c6284e51429df861fbe530e4163e60

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In Applied Mathematics

  1. Rémi Rhodes
  2. Vincent Vargas
  3. A. Kupiainen
  4. F. David

In Mathematical Physics

  1. Rémi Rhodes
  2. Vincent Vargas
  3. A. Kupiainen
  4. F. David
  5. Martin Hairer
  6. Scott Sheffield
  7. Eero Saksman
  8. Thomas Madaule

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