A certain class of distribution-valued additive functionals. II: For the case of stable process. (Q1411704)

In part I of the paper [ibid. 40, 293--314 (2000; Zbl 0981.60076)], the author has given the properties of the distribution valued additive functionals of the form \(A_T(a:t,\omega)=\int_0^t T(B_s-a)ds\) for a \(d\)-dimensional Brownian motion \(B_t\) and a distribution \(T\). In the present paper, he proves similar properties for 1-dimensional stable process or \(d\)-dimensional symmetric stable process \((X_t)\) of index \(\alpha\). The distribution \(T\) is assumed to be an element of \(H^\beta_p =\{T \in {\mathcal S}':\hat{T}(\lambda)(1+| \lambda| ^2)^{\beta/2} \in L^p\}\), where \(\hat{T}\) is the Fourier transform of \(T\). By using the estimates of the Fourier transform, for \(T \in H^\beta_p\) with suitable choices of \(\beta\) and \(p\), it is proved that \(A_T(a:t,\omega)\) is approximated by locally Hölder continuous additive functionals. Furthermore, \(A_T\) becomes a continuous additive functional of zero energy. In the case of 1-dimensional stable process, a representation theorem of \(\int_0^t f(X_s)ds\) for \(f \in {\mathcal S}\) by using the Hilbert transform is also given.