doi: 10.1685/journal.caim.531

Stochastic processes related to time-fractional diffusion-wave equation

Rudolf Gorenflo

Abstract


It is known that the solution to the Cauchy problem:
$$
D^\beta_* u(x,t)= R^\alpha u(x,t) \,, \quad u(x,0)=\delta(x) \,,
\quad \frac{\partial}{\partial x}u(x,t=0) \equiv 0 \,,
\quad -\infty < x < \infty \,, \quad t > 0 \,,
$$
is a probability density if
$$ 1 < \beta \le \alpha \le 2 $$
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 $$R^\alpha=\displaystyle{\frac{\partial^2}{\partial x^2}}$$ Moreover, if \alpha=2 we can replace $$D^\beta_*$$ by an operator of distributed orders with a non-negative (generalized) weight function b(\beta):
$$
\displaystyle{\int_{(1,2]} \!\!\! b(\beta) \, D^\beta_* \dots d\beta}
$$
For this case u(x,t) is a probability density.

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Communications in Applied and Industrial Mathematics
ISSN: 2038-0909