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A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation. (English) Zbl 1341.60073

Summary: A stochastic solution with Gaussian stationary increments is established for the symmetric space-time fractional diffusion equation when \(0 < \beta < \alpha \leq 2\), where \(0 < \beta \leq 1\) and \(0 < \alpha \leq 2\) are the fractional derivation orders in time and space, respectively. This solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuous-time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental solution of the space-time fractional diffusion equation is derived first. It is then shown that, in the symmetric case, a stochastic solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variably distributed according to an arrangement of two extremal Lévy stable densities. This stochastic solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure.
Numerical simulations are carried out by choosing as Gaussian process, the fractional Brownian motion. Sample paths and probability density functions are shown to be in agreement with the fundamental solution of the symmetric space-time fractional diffusion equation.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
35R11 Fractional partial differential equations
60G15 Gaussian processes
60G22 Fractional processes, including fractional Brownian motion
60J60 Diffusion processes
60G10 Stationary stochastic processes
60G18 Self-similar stochastic processes
60G20 Generalized stochastic processes
26A33 Fractional derivatives and integrals
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics

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