Pseudo-differential operators and Markov processes

Abstract

Given a L\'evy type generator $L$ acting on test functions on $R^{d}$ , 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 $\{R_{\lambda}\}$ satisfies that $R_{\lambda}(\lambda-L)f=f$ for any test function $f$ . These formulations are unified as the martingale problem for the operator $L$ .