Drift Perturbation Of Subordinate Brownian Motions With Gaussian Component And Heat Kernel Estimate

The heat kernel is an important subject in modern analysis, which appears to be useful for applications in probability, fractal analysis, geometry, mathematical physics and other related fields. We describe that the heat kernel is an area of interactions between the fields of Analysis and Probability.In the first part, we study drift perturbation of subordinate Brownian motions with Gaussian component on Rd with d≥1. We establish the existence and uniqueness of the fundamental solution (also called heat kernel) of the operator Lb= △+ψ(△)+b·▽, where ψ is the Laplace exponent of a one-dimensional non-decreasing Levy process (called subordinator) and b is an RD-valued function in Kato class Kd,1. We further derive the sharp two-sided estimates of the heat kernel of Lb.In the second part, we discuss the relationship between lower bounds and upper bounds of the heat kernel on metric spaces with doubling measure. If in addition the Dirichlet form is local, then we show that a near-diagonal lower bound implies an on-diagonal upper bound. We first give the upper estimate in balls and then extend it to full spaces. Compared with previous work, our conclusion not only contains former result but reveals a more general relationship between the lower and upper bounds.