doi: 10.1685/journal.caim.531

### Stochastic processes related to time-fractional diffusion-wave equation

#### 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.

$$

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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