Lp{L^p}-Liouville Theorems for Invariant Partial Differential Operators in Rn{\mathbb{R}^n}

We prove some $L^p$-Liouville theorems for hypoelliptic second order Partial Differential Operators left translation invariant with respect to a Lie group composition law in $\mathbb{R}^n$. Results for both solutions and subsolutions are given

Wiener-Landis criterion for Kolmogorov-type operators

We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepy's Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials

ON THE HARMONIC CHARACTERIZATION OF DOMAINS VIA MEAN VALUE FORMULAS

The Euclidean ball have the following harmonic characterization, via Gauss-mean value property: Let D be an open set with finite Lebesgue measure and let x(0) be a point of D. If u(x(0)) = 1/|D| int_{D} u(y)dy for every nonnegative harmonic function u in D, then D is a Euclidean ball centered at x(0). On the other hand, on every sufficiently smooth domain D and for every point x(0) in D there exist Radon measures mu such that u(x(0)) = int_{D} u(y)d mu(y)for every nonnegative harmonic function u in D. In this paper we give sufficient conditions so that this last mean value property characterizes the domain D

STOCHASTIC PERTURBATION OF A CUBIC ANHARMONIC OSCILLATOR

We perturb with an additive noise the Hamiltonian system associated to a cubic anharmonic oscillator. This gives rise to a system of stochastic differential equations with quadratic drift and degenerate diffusion matrix. Firstly, we show that such systems possess explosive solutions for certain initial conditions. Then, we carry a small noise expansion's analysis of the stochastic system which is assumed to start from initial conditions that guarantee the existence of a periodic solution for the unperturbed equation. We then investigate the probabilistic properties of the sequence of coefficients which turn out to be the unique strong solutions of stochastic perturbations of the well-known Lamé's equation. We also obtain explicit expressions of these in terms of Jacobi elliptic functions. Furthermore, we prove, in the case of Brownian noise, a lower bound for the probability that the truncated expansion stays close to the solution of the deterministic problem. Lastly, when the noise is bounded, we provide conditions for the almost sure convergence of the global expansion

Asymptotic average solutions to linear second order semi-elliptic PDEs: a Pizzetti-type Theorem

By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of {\it asymptotic average solutions} making pointwise solvable every Poisson equation $\mathcal{L} u(x)=-f(x)$ with continuous data $f$, where $\mathcal{L}$ is a hypoelliptic linear partial differential operator with positive semidefinite characteristic form

On the Perron solution of the caloric Dirichlet problem: an elementary approach

By an easy trick taken from caloric polynomial theory we construct a family $\mathscr{B}$ of $almost\ regular$ domains for the caloric Dirichlet problem. $\mathscr{B}$ is a basis of the Euclidean topology. This allows to build, with a basically elementary procedure, the Perron solution to the caloric Dirichlet problem on every bounded domain

Una base di insiemi risolutivi per l'equazione del calore: una costruzione elementare

By an easy “trick” taken from the caloric polynomial theory, we prove the existence of a basis of the Euclidean topology whose elements are resolutive sets of the heat equation. This result can be used to construct, with a very elementary approach, the Perron solution of the caloric Dirichlet problem on arbitrary bounded open subsets of the Euclidean space-time.Con un semplice espediente preso dalla teoria dei polinomi calorici, dimostriamo l'esistenza di una base della topologia euclidea i cui elementi sono insiemi risolutivi per l'equazione del calore. Questo risultato può essere utilizzato per costruire, con un approccio elementare, la soluzione di Perron del problema di Dirichlet calorico su arbitrari insiemi aperti limitati dello spazio-tempo euclideo

On the Perron solution of the caloric Dirichlet problem: an elementary approach

By an easy trick taken from caloric polynomial theory, we construct a family ℬ of almost regular domains for the caloric Dirichlet problem. ℬ is a basis of the Euclidean topology. This allows to build, with a basically elementary procedure, the Perron solution to the caloric Dirichlet problem on every bounded domain