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

• Published in: Probability theory and related fields 2016-12, Vol.166 (3-4), p.713-752
• Main Authors: Andres, Sebastian, Kajino, Naotaka
• Format: Article
• Language: English
• Subjects: Brownian motion; Continuity; Continuity (mathematics); Diffusion; Economics; Evolution; Finance; Gaussian; Geometry; Gravity; Insurance; Kernels; Lower bounds; Management; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Normal distribution; Operations Research/Decision Theory; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Spectra; Statistics for Business; Studies; Texts; Theorems; Theoretical; Upper bounds
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-015-0670-4

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 γ , formally written as M γ ( d z ) = e γ X ( z ) - γ 2 E [ X ( z ) 2 ] / 2 d z , γ ∈ ( 0 , 2 ) , for a (massive) Gaussian free field X . It is an M γ -symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure M γ . In this paper we provide a detailed analysis of the heat kernel p t ( 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 ) ≤ C 1 t - 1 log ( t - 1 ) exp ( - C 2 ( ( | x - y | β ∧ 1 ) / t ) 1 β - 1 ) for t ∈ ( 0 , 1 2 ] for each β > 1 2 ( γ + 2 ) 2 , and an on-diagonal lower bound of the form p t ( x , x ) ≥ C 3 t - 1 ( log ( t - 1 ) ) - η for t ∈ ( 0 , t η ( x ) ] , with t η ( x ) ∈ ( 0 , 1 2 ] heavily dependent on x , for each η > 18 for M γ -almost every x . As applications, we deduce that the pointwise spectral dimension equals 2 M γ -a.e. and that the global spectral dimension is also 2.