Pseudo-differential operators and Markov processes

Given a L\'evy type generator L acting on test functions on Rᵈ , there are various formulations of Markov processes associated with L . One is a weak solution of the stochastic differential equation of jump type with coefficients corresponding to the local characteristics of the operator L . Another is a Markov process whose resolvent λ\ satisfies that Rλ(λ-L)f=f for any test function f . These formulations are unified as the martingale problem for the operator L .