A Probabilistic Solution Of Dirichlet Boundary Value Problems For Fractional Laplacian Operators With Gradient Perturbation

Fractional Laplace operators are closely related to diffusion processes.For any fractional Laplacian operator,there exists a diffusion process X.And the infinitesimal generator of X is this Laplacian operator.In this paper,we proved the existence of weak solutions for the following equa-tions.And ?^α/2+?·μ is the infinitesimal generator of certain process,μ is signed measure.In this dissertation,we use probabilistic approach to prove that there exists a weak solution to the zero boundary value problem for fractional Laplacian equtions whose coefficients are signed measures.We will give a probabilistic representation of the solutions.The heat kernel estimates play a crucial role in our approach.