arXiv subjects

Antonello Pesce

Publications and source records attributed to Antonello Pesce.

7 recordsLinked to original sources

arXiv2025

Fractional Sobolev spaces related to an ultraparabolic operator

We propose a functional framework of fractional Sobolev spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak H\"ormander condition. We characterize these spaces as real interpolation of natural order intrinic Sobolev spaces recently introduced in [27], and prove continuous embeddings into $L^p$ and intrinsic H\"older spaces from [24]. These embeddings naturally extend the standard Euclidean ones, coherently with the homogeneous structure of the associated Kolmogorov group. Our approach to interpolation is based on approximation of intrinsically regular functions, the latter heavily relying on integral estimates of the intrinsic Taylor remainder. The embeddings exploit the aforementioned interpolation property and the corresponding embeddings of natural order intrinsic spaces.

math.AP↗

arXiv2024

Nash equilibrium in a singular stochastic game between two renewable power producers with price impact

We study the singular stochastic game, formulated in Awerkin and Vargiolu (Decis. Econ. Finance 44(2), 2021), between two agents aiming at maximizing their profits by installing photovoltaic panels and selling the produced electricity, net of installation costs, in the case that their cumulative installations have an impact on power prices. We first solve explicitly the static, one-step, version of the game, and find that Nash equilibria divide the state-space into four regions: one where both players are idle, two where only one player installs new panels, and one where both players install. In some particular regimes, we find that the latter may not be uniquely distinguished from the previous two. We then consider the dynamic, continuous-time, problem. Led by the intuition garnered in the static case, we assume a free-boundary structure similar to that arising in the one-step game and provide a rigorous verification theorem for the corresponding system of free-boundary HJB equations, also taking into account the lack of smoothness of the value functions near the free boundaries. Finally, for each regular solution of the HJB system, we show that there exists a unique equilibrium strategy, which is obtained as the solution to the Skorokhod-type problem associated with the free boundary.

math.OC↗

arXiv2024

Sobolev embeddings for kinetic Fokker-Planck equations

We introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hörmander condition. We prove continuous embeddings into Lorentz and intrinsic Hölder spaces. We also prove approximation and interpolation inequalities by means of an intrinsic Taylor expansion, extending analogous results for Hölder spaces. The embedding at first order is proved by adapting a method by Luc Tartar which only exploits scaling properties of the intrinsic quasi-norm, while for higher orders we use uniform kernel estimates.

math.AP↗

arXiv2024

Approximation and Interpolation in Kolmogorov-type groups

We prove a real interpolation characterization for some non Euclidean Hölder spaces, built on the Lie structure induced by a class of ultra-parabolic Kolmogorov-type operators satisfying the Hörmander condition. As a by-product we also obtain an approximation property for intrinsically regular functions on the whole space.

math.AP↗

arXiv2021

Backward and forward filtering under the weak Hörmander condition

We derive the forward and backward filtering equations for a class of degenerate partially observable diffusions, satisfying the weak Hörmander condition. Our approach is based on the Hölder theory for degenerate SPDEs that allows to pursue the direct approaches proposed by N. V. Krylov and A. Zatezalo, and A. Yu. Veretennikov, avoiding the use of general results from filtering theory. As a by-product we also provide existence, regularity and estimates for the filtering density.

math.PR↗

arXiv2019

The parametrix method for parabolic SPDEs

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Ito-Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.

math.PR↗

arXiv2019

On stochastic Langevin and Fokker-Planck equations: the two-dimensional case

We prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution of the stochastic Langevin equation. This degenerate SPDE satisfies the weak Hörmander condition. We use a Wentzell's transform to reduce the SPDE to a PDE with random coefficients; then we apply a new method, based on the parametrix technique, to construct a fundamental solution. This approach avoids the use of the Duhamel's principle for the SPDE and the related measurability issues that appear in the stochastic framework. Our results are new even for the deterministic equation.

math.PR↗