Stochastic processes related to time-fractional diffusion-wave equation

It is known that the solution to the Cauchy problem: D-*(beta) u(x, t) = R(alpha)u(x,t), u(x, 0) = delta(x), partial derivative/partial derivative x u(x, t = 0) equivalent to 0, -infinity < x < infinity, t > 0, is a probability density if 1 < beta <= alpha <= 2, where D-*(beta) is the time fractional Caputo derivative of order beta whereas R-alpha denotes the spatial Riesz fractional pseudo-differential operator. In the present paper it is considered the question if u(x, t) can be interpreted in a natural way as the sojourn probability density (in point x, evolving in time t) of a randomly wandering particle starting in the origin x = 0 at instant t = 0. We show that this indeed can be done in the extreme case alpha = 2, that is R-alpha = partial derivative(2)/partial derivative x(2). Moreover, if alpha = 2 we can replace D-*(beta) by an operator of distributed orders with a non-negative (generalized) weight function b(beta): integral((1,2])b(beta) D-*(beta) ... d beta. For this case u(x, t) is a probability density.