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

Baeumer, Boris; Meerschaert, Mark M.

Fract. Calc. Appl. Anal. 4, No. 4, 481-500 (2001).

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.

Reviewer: Mohamed Hmissi (Tunis)

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

Keywords:

fractional derivative; Cauchy problem; stable law; Feller semigroup