Searcharxiv⌕ Search

arXiv subjects

Francesca Anceschi

Publications and source records attributed to Francesca Anceschi.

16 recordsLinked to original sources
arXiv2025

Solvability of Dirichlet boundary value problems governed by non-monotone differential operators

We prove existence results for Dirichlet boundary value problems for equations of the type \begin{align*} \left( \Phi(k(t) x'(t) ) \right)' = f(t, x(t) , x'(t) ) \qquad \text{for a.e. } t \in I:=[0,T] , \end{align*} where $\Phi : J \to \mathbb{R} $ is a generic possibly non-monotone differential operator defined in a open interval $J\subseteq \mathbb{R}$, $k:I \to \mathbb{R}$, $k$ is measurable with $k(t) >0$ for a.e. $t \in I$ and $f: \mathbb{R}^3 \to \mathbb{R}$ is a Carath\'eodory function. Under very mild assumptions, we prove the existence of solutions for suitably prescribed boundary conditions, and we also address the study of the existence of heteroclinic solutions on the half-line $[0,+\infty)$.

math.CA↗
arXiv2024

Optimal control problems driven by nonlinear degenerate Fokker-Planck equations

The well-posedness of a class of optimal control problems is analysed, where the state equation couples a nonlinear degenerate Fokker-Planck equation with a system of Ordinary Differential Equations (ODEs). Such problems naturally arise as mean-field limits of Stochastic Differential models for multipopulation dynamics, where a large number of agents (followers) is steered through parsimonious intervention on a selected class of leaders. The proposed approach combines stability estimates for measure solutions of nonlinear degenerate Fokker-Planck equations with a general framework of assumptions on the cost functional, ensuring compactness and lower semicontinuity properties. The Lie structure of the state equations allows one for considering non-Lipschitz nonlinearities, provided some suitable dissipativity assumptions are considered in addition to non-Euclidean H\"{o}lder and sublinearity conditions.

math.OC↗
arXiv2024

Well-posedness of Kolmogorov-Fokker-Planck equations with unbounded drift

We consider Kolmogorov-Fokker-Planck equations with unbounded drift terms which are only measurable in time and locally H\"older continuous in space. In particular, we extend the parametrix method to this setting and we prove existence and uniqueness of measure solutions to the associated Cauchy problem, as well as the equivalence with the corresponding stochastic formulation.

math.AP↗
arXiv2024

De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusions

We extend the De Giorgi-Nash-Moser theory to a class of nonlocal hypoelliptic equations arising naturally in kinetic theory, in which a first-order transport operator is coupled with an elliptic nonlocal operator involving fractional derivatives only in part of the variables. Under the sole assumption that the nonlocal tail in velocity of weak solutions is $p$-summable along the drift variables, we prove a local $L^2$-$L^\infty$ estimate for kinetic integral equations and a corresponding strong Harnack inequality. The tail condition is satisfied in standard kinetic regimes considered in the literature, for instance under the usual boundedness of the mass density in the Boltzmann equation without cut-off, and it is consistent with the recent counterexample by Kassmann and Weidner (Adv. Math. 2024). These estimates further lead to a geometric characterization of the Harnack inequality, in the spirit of the seminal work of Aronson and Serrin (Arch. Ration. Mech. Anal. 1967) for the local parabolic counterpart.

math.AP↗
arXiv2024

Poincar\'e inequality and quantitative De Giorgi method for hypoelliptic operators

We propose a systematic approach based on trajectories to prove a Poincar\'e inequality for weak non-negative sub-solutions to hypoelliptic equations with an arbitrary number of H\"ormander commutators, both in the local and in the non-local case. As a consequence, we deduce the weak Harnack inequality and H\"older regularity along the line of the De Giorgi method.

math.AP↗
arXiv2023

New perspectives on recent trends for Kolmogorov operators

After carrying out an overview on the non Euclidean geometrical setting suitable for the study of Kolmogorov operators with rough coefficients, we list some properties of the functional space $\mathcal{W}$, mirroring the classical $H^1$ theory for uniformly elliptic operators. Then we provide the reader with the proof of a new Sobolev embedding for functions in $\mathcal{W}$. Additionally, after reviewing recent results regarding weak regularity theory, we discuss some of their recent applications to real life problems arising both in Physics and in Economics. Finally, we conclude our analysis stating some recent results regarding the study of nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck operators.

math.AP↗
arXiv2023

On the obstacle problem associated to the Kolmogorov-Fokker-Planck operator with rough coefficients

This work is devoted to the study of the obstacle problem associated to the Kolmogorov-Fokker-Planck operator with rough coefficients through a variational approach. In particular, after the introduction of a proper anisotropic Sobolev space and related properties, we prove the existence and uniqueness of a weak solution for the obstacle problem by adapting a classical perturbation argument to the convex functional associated to the case of our interest. Finally, we conclude this work by providing a one-sided associated variational inequality, alongside with an overview on related open problems.

math.AP↗
arXiv2022

Harnack inequality and asymptotic lower bounds for the relativistic Fokker-Planck operator

We consider a class of second order degenerate kinetic operators $\mathscr{L}$ in the framework of special relativity. We first describe $\mathscr{L}$ as an H\"ormander operator which is invariant with respect to Lorentz transformations. Then we prove a Lorentz-invariant Harnack type inequality, and we derive accurate asymptotic lower bounds for positive solutions to $\mathscr{L} f = 0$. As a consequence we obtain a lower bound for the density of the relativistic stochastic process associated to $\mathscr{L}$.

math.AP↗
arXiv2021

Spatial regularity for a class of degenerate Kolmogorov equations

We establish spatial a priori estimates for the solution u to a class of dilation invariant Kolmogorov equation, where u is assumed to only have a certain amount of regularity in the diffusion's directions. The result is that u is also regular with respect to the remaining directions. The approach we propose is based on the commutators identities and allows us to obtain a Sobolev exponent that does not depend on the integrability assumption of the right-hand side. Lastly, we provide an alternative proof to that of Theorem 1.5 of [9] for the optimal spatial regularity.

math.AP↗
arXiv2021

A note on the weak regularity theory for degenerate Kolmogorov equations

The aim of this work is to prove a Harnack inequality and the H\"older continuity for weak solutions to the Kolmogorov equation $\mathscr{L} u = f$ with measurable coefficients, integrable lower order terms and nonzero source term. We introduce a functional space $\mathcal{W}$, suitable for the study of weak solutions to $\mathscr{L}u = f$, that allows us to prove a weak Poincar\'e inequality. More precisely, our goal is to prove a weak Harnack inequality for non-negative super-solutions by considering their Log-transform and following S. N. Kruzkov (1963). Then this functional inequality is combined with a classical covering argument (Ink-Spots Theorem) that we extend for the fist time to the case of ultraparabolic equations.

math.AP↗
arXiv2021

On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptotics

We investigate the Cauchy problem and the diffusion asymptotics for a spatially inhomogeneous kinetic model associated to a nonlinear Fokker-Planck operator. We derive the global well-posedness result with instantaneous smoothness effect, when the initial data lies below a Maxwellian. The proof relies on the hypoelliptic analog of classical parabolic theory, as well as a positivity-spreading result based on the Harnack inequality and barrier function methods. Moreover, the scaled equation leads to the fast diffusion flow under the low field limit. The relative phi-entropy method enables us to see the connection between the overdamped dynamics of the nonlinearly coupled kinetic model and the correlated fast diffusion. The global in time quantitative diffusion asymptotics is then derived by combining entropic hypocoercivity, relative phi-entropy and barrier function methods.

math.AP↗
arXiv2020

Existence of a Fundamental Solution of Partial Differential Equations associated to Asian Options

We prove the existence and uniqueness of the fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in the Black & Scholes setting for the pricing problem relevant to path dependent options. We improve previous results in that we provide a closed form expression for the solution of the Cauchy problem under weak regularity assumptions on the coefficients of the differential operator. Our method is based on a limiting procedure, whose convergence relies on some barrier arguments and uniform a priori estimates recently discovered.

math.AP↗
arXiv2019

A survey on the classical theory for Kolmogorov equation

We present a survey on the regularity theory for classic solutions to subelliptic degenerate Kolmogorov equations. In the last part of this note we present a detailed proof of a Harnack inequality and a strong maximum principle.

math.AP↗