Ricerc@Sapienza

Instantaneous frequency (IF) is a fundamental feature in multicomponent signals analysis and its estimation is required in many practical applications. This goal can be successfully reached for well separated components, while it still is an open problem in case of interfering modes. Most of the methods addressing this issue are parametric, that is, they apply to a specific IF class. Alternative approaches consist of non-parametric time filtering-based procedures, which do not show robustness to destructive interference---the most critical scenario in crossing modes.

We study discounted Hamilton Jacobi equations on networks, without putting any restriction on their geometry. Assuming the Hamiltonians continuous and coercive, we establish a comparison principle and provide representation formulae for solutions. We follow the approach introduced in 11, namely we associate to the differential problem on the network, a discrete functional equation on an abstract underlying graph.

We study a one--parameter family of Eikonal Hamilton-Jacobi
equations on an embedded network, and prove that there exists a
unique critical value for which the corresponding equation admits
global solutions, in a suitable viscosity sense. Such a solution is
identified, via an Hopf--Lax type formula, once an admissible trace
is assigned on an {it intrinsic boundary}. The salient point of
our method is to associate to the network an {it abstract graph},
encoding all of the information on the complexity of the network,

We propose a probabilistic construction for the solution of a general class of fractional high-order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with space fractional derivatives of any order, and later we show the extension to equations with time fractional derivative (in the sense of Caputo derivative) of order α∈ (0 , 1).