The Research Of Dirichlet Heat Kernel Estimates For Fractional Laplacian With Gradient Perturbation

In this master degree thesis,we discuss the Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation.Suppose that α∈(1,2),D be a bounded C1,1 open set in Rd and b an Rd-valued function on Rd whose components are in a certain Kato class of the rotationally symmetric α-stable process.In this master degree thesis,we prove the transition density is comparable with that of the fractional Laplacian in D with zero exterior condition,namely,pb,D(t,x,y)(?)PD(t,x,y).