Ricerc@Sapienza

Given a (finitely additive) full conditional probability space and a conditional measurable space , a multivalued mapping Γ from X to Y induces a class of full conditional probabilities on . A closed form expression for the lower and upper envelopes ⁎ and ⁎ of such class is provided: the envelopes can be expressed through a generalized Bayesian conditioning rule, relying on two linearly ordered classes of (possibly unbounded) inner and outer measures.

Gapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of |x| 2 diverges. Intermediate regimes are forbidden.

In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem (P)−Δu=|u|[Formula presented]u, in Ω,u=0, on ∂Ω on the annulus Ω:=x∈RN:a

Ci siamo resi conto di un buco nella dimostrazione del Teorema 2 (iii) e 6 (ii) di [1].

We study a (elliptic measurable coefficients) diffusion in the classical snowflake domain in the situation when there are
diffusion and drift terms not only in the interior but also on the fractal boundary, which is a union of three copies of
the classical Koch curve. In this example we can combine the fractal membrane analysis, the vector analysis for local
Dirichlet forms and quasilinear PDE and SPDE on fractals, non-symmetric Dirichlet forms, and analysis of Lipschitz

In the present paper we perform the homogenization of the semilinear elliptic problem
\begin{equation*}
\begin{cases}
u^\eps \geq 0 & \mbox{in} \; \oeps,\\
\displaystyle - div \,A(x) D u^\eps = F(x,u^\eps) & \mbox{in} \; \oeps,\\
u^\eps = 0 & \mbox{on} \; \partial \oeps.\\
\end{cases}
\end{equation*}

We consider an ensemble of Ornstein-Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable corresponding to this ensemble is statistically equivalent to a process driven by a non-autonomous stochastic differential equation with time-dependent drift and a white noise. In particular, the time scaling and the density function of such variable are driven by the population of timescales and of noise amplitudes, respectively.

We consider a system of differential equations of Monge–Kantorovich type which describes the equilibrium configurations of granular material poured by a constant source on a network. Relying on the definition of viscosity solution for Hamilton–Jacobi equations on networks introduced in [P.-L. Lions and P. E. Souganidis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), pp. 535–545], we prove existence and uniqueness of the solution of the system and we discuss its numerical approximation. Some numerical experiments are carried out.

We prove a representation formula of Hopf-Lax type for solutions to Hamilton-Jacobi equation involving a Caputo time-fractional derivative. Equations of this type are associated with optimal control problems where the controlled dynamics is given by a time-changed stochastic process describing the trajectory of a particle subject to random trapping effects.