Iterated elastic Brownian motions and fractional diffusion equations

Fractional diffusion equations of order [nu][set membership, variant](0,2) are examined and solved under different types of boundary conditions. In particular, for the fractional equation on the half-line [0,+[infinity]) and with an elastic boundary condition at x=0, we are able to provide the general solution in terms of the density of the elastic Brownian motion. This permits us, for equations of order , to write the solution as the density of the process obtained by composing the elastic Brownian motion with the (n-1)-times iterated Brownian motion. Also the limiting case for n-->[infinity] is investigated and the explicit form of the solution is expressed in terms of exponentials. Moreover, the fractional diffusion equations on the half-lines [0,+[infinity]) and (-[infinity],a] with additional first-order space derivatives are analyzed also under reflecting or absorbing conditions. The solutions in this case lead to composed processes with general form , where only the driving process is affected by drift, while the role of time is played by iterated Brownian motion In-1.