Heat kernel estimates for subordinate Brownian motions

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, e.g. \[ \phi(\lambda)=\frac{\lambda}{\log(1+\lambda)}-1 \quad \text{ or }\quad \phi(\lambda)=\frac{\lambda}{\log(1+\lambda^{\beta/2})},\ \beta\in (0,2)\, \] 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.