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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
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