Feynman-kac Penalisations of Symmetric Stable Pro- Cesses

In [9], [10], B. Roynette, P. Vallois and M. Yor have studied limit theorems for Wiener processes normalized by some weight processes. In [16], K. Yano, Y. Yano and M. Yor studied the limit theorems for the one-dimensional symmetric stable process normalized by non-negative functions of the local times or by negative (killing) Feynman-Kac functionals. They call the limit theorems for Markov processes normalized by Feynman-Kac functionals the Feynman-Kac penalisations. Our aim is to extend their results on Feynman-Kac penalisations to positive Feynman-Kac functionals of multi-dimensional symmetric α-stable processes. Let M = (Ω,F ,Ft ,Px , X t) be the symmetric α-stable process on Rd with 0 < α ≤ 2, that is, the Markov process generated by −(1/2)(−∆)α/2, and (E ,D(E )) the Dirichlet form of M (see (2.1),(2.2)). Let μ be a positive Radon measure in the class K∞ of Green-tight Kato measures (Definition 2.1). We denote by Aμt the positive continuous additive functional (PCAF in abbreviation) in the Revuz correspondence to μ: for a positive Borel function f and γ-excessive function g,