Saturated Fronts in Crowds Dynamics

We consider a degenerate scalar parabolic equation, in one spatial dimension, of flux-saturated type. The equation also contains a convective term. We study the existence and regularity of traveling-wave solutions; in particular we show that they can be discontinuous. Uniqueness is recovered by requiring an entropy condition, and entropic solutions turn out to be the vanishing-diffusion limits of traveling-wave solutions to the equation with an additional non-degenerate diffusion. Applications to crowds dynamics, which motivated the present research, are also provided.

On the genealogy of branching populations and their diffusion limits

We consider the evolution of the ancestral structure of a classical branching process in space and its diffusion limit. We also indicate how the conditional structure of the past can be described asymptotically in terms of suitable uniform Brownian trees.