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.


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