Pseudo-differential operators and Markov processes
A pseudo-differential operator L is said to be non-degenerate if there is a constant \(\alpha (0<\alpha \leq 2)\) and \(L=\psi^{(\alpha)}(x,\partial)+\phi^{(\alpha)}(x,\partial),\) where \(\psi^{(\alpha)}(x,\xi)\) is a homogeneous function in \(\xi\) with index \(\alpha\) such that Re \(\psi^{(\alpha)}(x,\xi)<0\) for \(| \xi | =1\), and \(\phi^{(\alpha)}(x,\xi)=o(| \xi |^{\alpha})\) for large \(| \xi |.\) A Markov process is associated with L in the sense that its resolvent \(\{R_{\lambda}\}\) satisfies the equality \(R_{\lambda}(\lambda -L)f=f\) for any test function f. It is proved that existence and uniqueness hold for Markov processes associated with operators in a class of non- degenerate pseudo-differential operators. The uniqueness of solutions of the martingale problem for L is discussed to prove the uniqueness of Markov processes.
scientific article; zbMATH DE number 3858126