Ricerc@Sapienza

A general class of probability density functions (1 (‖x‖) )γ β u(x, t) = Ct−αd − ctα, x ∈ Rd,t >0, + is considered, containing as particular case the Barenblatt solutions arising, for instance, in the study of nonlinear heat equations. Alternative probabilistic representations of the Barenblatt-type solutions u(x, t) are proposed. In the one-dimensional case, by means of this approach, u(x, t) can be connected with the wave propagation.

It is well-known that compositions of Markov processes with inverse subordinators are governed by integro-differential equations of generalized fractional type. This kind of processes are of wideinterest in statistical physics as they are connected to anomalous diffusions. In this paper we consider a generalization; more precisely we mean componentwise compositions of R^d -valued Markov processes

Transport equations with a nonlocal velocity field have been introduced as a continuum model for interacting particle systems arising in physics, chemistry and biology. Fractional time derivatives, given by convolution integrals of the time-derivative with power-law kernels, are typical for memory effects in complex systems. In this paper we consider a nonlinear transport equation with a fractional time-derivative.

Lévy walks (LWs) define a fundamental class of finite velocity stochastic processes that can be introduced as a special case of continuous time random walks. Alternatively, there is a hyperbolic representation of them in terms of partial probability density waves. Using the latter framework we explore the impact of aging on LWs, which can be viewed as a specific initial preparation of the particle ensemble with respect to an age distribution.