## Stochastic solutions for fractional Cauchy problems.(English)Zbl 1057.35102

Let $$\{T_t\}$$ be a Feller semigroup associated to some infinitely divisible law on $${\mathbb R}^d$$, let $$\beta \in ]0,1[$$ and let $$\{T_t^\beta\}$$ be the subordinated semigroup of $$\{T_t\}$$ by means of the stable subordinator of order $$\beta$$. The authors prove that $$(t,x)\rightarrow T_t^\beta f(x)$$ solves the associated fractional Cauchy problem.

### MSC:

 35R60 PDEs with randomness, stochastic partial differential equations 26A33 Fractional derivatives and integrals 47D07 Markov semigroups and applications to diffusion processes 60E07 Infinitely divisible distributions; stable distributions 60J35 Transition functions, generators and resolvents