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Subordination in the sense of S. Bochner – An approach through pseudo differential operators. (English) Zbl 0923.47023
The authors investigate subordination which was considered by S. Bochner [“Harmonic analysis and the theory of probability” (1955; Zbl 0068.11702)] in accordance with pseudodifferential operator theory.
From the introduction: “We want to examine Bochner’s theory of subordination by using methods from the theory of pseudodifferential operators. Roughly speaking, our idea is to work on the level of the generator and to use a type of symbolic calculus in order to get approximations for \(A^f\) \[ A^fu= -(2\pi)^{-n/2} \int_{\mathbb{R}^n} e^{i(x,\xi)} f(\psi(\xi))\widehat u(\xi)d\xi\tag{1.8} \] and \(\{T^f_t\}_{t\geq 0}\) \[ T^f_t u= \int^\infty_0 T_su\rho_t(ds)\tag{1.7} \] which will also lead to an approximation on the level of the process.
In some sense, the paper is introductory for we only consider one rather simple type of generators, a strongly elliptic symmetric differential operator \[ A\equiv L(x,D)=- \sum^n_{k,\ell= 1} a_{k\ell}(x){\partial^2\over \partial x_k\partial x_\ell}+ c(x),\tag{1.11} \] imposing rather restrictive assumptions on the coefficients \(a_{k\ell}\) and \(c\).”
The authors consider several results on negative definite functions, Bernstein functions, and generators of semigroups derived from (1.11). Also, they consider several norm inequalities which are necessary for their theory.
Reviewer: S.Koshi (Sapporo)

MSC:
47D07 Markov semigroups and applications to diffusion processes
47G30 Pseudodifferential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators
60J35 Transition functions, generators and resolvents
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