Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations

We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. We prove logarithmic Sobolev and Poincar\'e inequalities with respect to an associated evolution system of measures $\{\mu_t: t \in I\}$, and we deduce hypercontractivity and asymptotic behaviour results for the evolution operator $G(t,s)$

Asymptotic behavior in time periodic parabolic problems with unbounded coefficients

We study asymptotic behavior in a class of non-autonomous second order parabolic equations with time periodic unbounded coefficients in $\mathbb R\times \mathbb R^d$. Our results generalize and improve asymptotic behavior results for Markov semigroups having an invariant measure. We also study spectral properties of the realization of the parabolic operator $u\mapsto {\cal A}(t) u - u_t$ in suitable $L^p$ spaces

On coupled systems of Kolmogorov equations with applications to stochastic differential games

We prove that a family of linear bounded evolution operators $({\bf G}(t,s))_{t\ge s\in I}$ can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators $\bm{\mathcal A}$ with unbounded coefficients defined in I\times \Rd (where $I$ is a right-halfline or $I=\R$) all having the same principal part. We establish some continuity and representation properties of $({\bf G}(t,s))_{t \ge s\in I}$ and a sufficient condition for the evolution operator to be compact in C_b(\Rd;\R^m). We prove also a uniform weighted gradient estimate and some of its more relevant consequence

Kernel estimates for nonautonomous Kolmogorov equations

Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients

A Five Dimensional Perspective on Many Particles in the Snyder basis of Double Special Relativity

After a brief summary of Double Special Relativity (DSR), we concentrate on a five dimensional procedure, which consistently introduce coordinates and momenta in the corresponding four-dimensional phase space, via a Hamiltonian approach. For the one particle case, the starting point is a de Sitter momentum space in five dimensions, with an additional constraint selected to recover the mass shell condition in four dimensions. Different basis of DSR can be recovered by selecting specific gauges to define the reduced four dimensional degrees of freedom. This is shown for the Snyder basis in the one particle case. We generalize the method to the many particles case and apply it again to this basis. We show that the energy and momentum of the system, given by the dynamical variables that are generators of translations in space and time and which close the Poincar\'e algebra, are additive magnitudes. From this it results that the rest energy (mass) of a composite object does not have an upper limit, as opposed to a single component particle which does.Comment: 12 pages, no figures, AIP Conf. Pro