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Characteristic functions and symbols in the theory of Feller processes. (English) Zbl 0908.60041
Let \(X=((X_t)_{t\geq 0}, P^x)_{x\in R^n}\) be a Feller process on \(R^n\) with the semigroup \((T_t; t\geq 0)\) and the infinitesimal generator \(A\). The generator \(A\) satisfies the positive maximum principle, and therefore, by a result of P. Courrège [Sémin. Théorie Potentiel M. Brelot, G. Choquet et J. Deny 10 (1965/66), No. 3, 48 p. (1967; Zbl 0155.17403)], has on \(C_0^{\infty}(R^n)\) the representation \[ Au(x)=-(2\pi)^{-n/2}\int_{R^n} e^{ix\xi} q(x,\xi) \widehat{u}(\xi) d\xi, \tag{\(*\)} \] where \(\widehat{u}\) denotes the Fourier transform of \(u\), and \(q:R^n\times R^n \to C\) is the symbol of \(A\). The goal of this paper is to give a probabilistic interpretation of \(q\). Let \({\lambda}_t(x,\xi)=E^x[e^{-i(X_t-x)\xi}]\). The author proves that under certain regularity assumptions, the following two statements are valid:
(i) \(T_tu(x) = (2\pi)^{-n/2} \int_{R^n} e^{ix\xi}{\lambda}_t(x,\xi) \hat{u} (\xi) d\xi\), for any \(u\in {\mathcal S}(R^n)\), and
(ii) formula \((*)\) is valid with the symbol \( -q(x,\xi)= {d\over{dt}} {\lambda}_t(x,\xi)|_{t=0}\).

60G99 Stochastic processes
60J35 Transition functions, generators and resolvents
47D07 Markov semigroups and applications to diffusion processes
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