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Heat kernel estimates for subordinate Brownian motions
Author(s) -
Mimica Ante
Publication year - 2016
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdw043
Subject(s) - subordinator , mathematics , brownian motion , scaling , exponent , laplace transform , fractional brownian motion , mathematical analysis , diagonal , equivalence (formal languages) , heat kernel , infinity , statistical physics , pure mathematics , statistics , geometry , physics , linguistics , philosophy
In this article, we study transition probabilities of a class of subordinate Brownian motions. Under mild assumptions on the Laplace exponent of the corresponding subordinator, sharp two‐sided estimates of the transition probability are established. This approach, in particular, covers subordinators with Laplace exponents that vary regularly at infinity with index one, for example, ϕ ( λ ) = λ log ( 1 + λ ) - 1 or ϕ ( λ ) = λ log ( 1 + λ β / 2 ) , β ∈ ( 0 , 2 )that correspond to subordinate Brownian motions with scaling order that is not necessarily strictly between 0 and 2. These estimates are applied to estimate Green function (potential) of subordinate Brownian motion. We also prove the equivalence of the lower scaling condition of the Laplace exponent and the near diagonal upper estimate of the transition estimate.