# zbMATH — the first resource for mathematics

Limit theorems for certain functionals associated to stable processes in a Hölder space. (Théorèmes limites pour certaines fonctionnelles associées aux processus stables sur l’espace de Hölder.) (French) Zbl 0995.60037
Let $$\{L_t^x : t\geq 0, x\in\mathbb R\}$$ be the local time of an $$\alpha$$-Levy motion $$(Z_t)_{t\geq 0}$$ where $$1<\alpha\leq 2$$. The aim of the present paper is to investigate $$L_t^x$$ as function of the two variables $$t$$ and $$x$$ jointly. For example, a Hölder type estimate is proved for $$(t,x)\to L_t^x$$. Furthermore, the authors verify Hölder conditions for certain fractional derivatives of $$L_t^x$$ (here the derivative is taken with respect to $$x\in\mathbb R$$). As a consequence, limits of some special functionals of the motion $$(Z_t)_{t\geq 0}$$ may be described by means of fractional derivatives of $$L_t^x$$.

##### MSC:
 60F25 $$L^p$$-limit theorems 60G52 Stable stochastic processes
##### Keywords:
stable processes; local times
Full Text: