Iterated elastic Brownian motions and fractional diffusion equations

Fractional diffusion equations of order nu is an element of (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 ofthe elastic Brownian motion. This permits us, for equations of order nu = 1/2(n), to write the Solution its 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 ill this case lead to composed processes with general form chi(vertical bar I(n-1)(t)vertical bar), where only the driving process chi is affected by drift, while the role of time is played by iterated Brownian motion I(n-1). (C) 2008 Elsevier B.V. All rights reserved