Iterated elastic Brownian motions and fractional diffusion equations

Fractional diffusion equations of order ν ∈ ( 0 , 2 ) are examined and solved under different types of boundary conditions. In particular, for the fractional equation on the half-line [ 0 , + ∞ ) 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 ν = 1 2 n , 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 → ∞ is investigated and the explicit form of the solution is expressed in terms of exponentials. er, the fractional diffusion equations on the half-lines [ 0 , + ∞ ) and ( − ∞ , 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 X ( | I n − 1 ( t ) | ) , where only the driving process X is affected by drift, while the role of time is played by iterated Brownian motion I n − 1 .