DISTRIBUTED-ORDER SPACE-TIME FRACTIONAL DIFFUSIONS IN BOUNDED DOMAINS

We consider the distributed-order space-time fractional heat type equation D(')u(t, x) = -(-Delta)alpha/2u(t, x), t > 0, x is an element of D, where D(') is the distributed-order derivative based on the Caputo time-fractional derivative,-(-Delta)alpha/2 is the generator of an isotropic stable process for alpha is an element of (0, 2], D is a bounded domain in Rd and nu is a finite Borel measure, known as mixing measure. An important application of distributed order diffusions is to model ultraslow diffusions where a plume of particles spreads at a logarithmic rate as described in Sinai (Theory Probab. Appl., 27 (1982) 256-268). Using analytical tools, we provide an explicit classical solution and a stochastic analogue for this equation in bounded domains with zero exterior boundary conditions. We also show that our results still hold when the mixing measure in the distributed-order time-derivative is singular. Our results extend the results in the case of the time-fractional distributed-order diffusions obtained in Naber (Fractals 12( 2004) 23-32) and Meerschaert et al. (J. Math. Anal. Appl., 379(2011) 216-228).