### ON THE MARTINGALE PROBLEM FOR PSEUDO-DIFFERENTIAL OPERATORS OF VARIABLE ORDER

**TAKASHI KOMATSU**

*Theory of Stochastic Processes*

*Vol. 14 (30), no. 2, 2008, pp. 42–51*

Consider parabolic pseudo-differential operators $L = partial_t – p(x,D_x)$ of variable order $alpha(x) leq 2$. The function $alpha(x)$ is assumed to be smooth, but the symbol $p(x,xi)$ is not always differentiable in $x$. We shall show the uniqueness of Markov processes with the generator $L$. The essential point in our study is to obtain the $Lˆp -$estimate for resolvent operators associated with solutions to the martingale problem for $L$. We shall show that by making use of the theory of pseudo-differential operators and a generalized Calderon – Zygmund inequality for singular integrals. As a consequence of our study, the Markov process with the generator $L$ is constructed and characterized, which may be called a stable-like process with perturbation.

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