This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time‐changed by an inverse stable subordinator whose index equals the order of the fractional time derivative. Some applications are given, to demonstrate how to specify a well‐posed Dirichlet problem for space‐time fractional diffusions in one or several variables. This solves an open problem in numerical analysis.

We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump Lévy processes.

Abstract. We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian ∆ k in R d . As applications we derive the Poisson-Jensen formula for ∆ k -subharmonic functions and Hardy-Stein identities for the Poisson integrals of ∆ k . We also obtain sharp estimates of the Newton potential kernel, Green function and Poisson kernel in the rank one case in R d . These estimates contrast sharply with the well-known results in the potential theory of the classical Laplacian.

20 pages.International audienceWe determine the Hausdorff dimension of the set of double points for a symmetric operator stable L\'evy process in terms of the eigenvalues of its stability exponent