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Sergio Polidoro

Publications and source records attributed to Sergio Polidoro.

At least 19 recordsLinked to original sources
arXiv2025

The Dirichlet problem for a family of totally degenerate differential operators

In the framework of Potential Theory we prove existence or the Perron-Weiner-Brelot-Bauer solution to the Dirichlet problem related to a family of totally degenerate, in the sense of Bony, differential operators. We also state and prove a Wiener-type criterium and an exterior cone condition for the regularity of a boundary point. Our results apply to a wide family of strongly degenerate operators that includes the following example $\mathcal{L} = t^2Δ_x + \langle x, \nabla_y \rangle -\partial_t$, for $(x,y,t) \in \mathbb{R}^N \times \mathbb{R}^{N} \times \mathbb{R}$.

math.AP↗
arXiv2024

A study of the Kuramoto model for synchronization phenomena based on degenerate Kolmogorov-Fokker-Planck equations

We study a nonlinear partial differential equation that arises when introducing inertial effects in the Kuramoto model. Based on the known theory of degenerate Kolmogorov operators, we prove existence, uniqueness and a priori estimates of the solution to the relevant Cauchy problem. Moreover, a stable numerical operator, which is consistent with the degenerate Kolmogorov operator, is introduced in order to produce numerical solutions. Finally, numerical experiments show how the synchronization phenomena depend on the parameters of the Kuramoto model with inertia.

math.AP↗
arXiv2023

Intrinsic Hölder spaces for fractional kinetic operators

We introduce anisotropic Hölder spaces useful for the study of the regularity theory for non local kinetic operators $\mathcal{L}$ whose prototypal example is \begin{equation} \mathcal{L} u (t,x,v) = \int_{\mathbb{R}^d} \frac{C_{d,s}}{|v - v'|^{d+2s}} (u(t,x,v') - u(t,x,v)) d v' + \langle v , \nabla_x \rangle + \partial_t, \quad (t,x,v)\in\mathbb{R}\times\mathbb{R}^{2d}. \end{equation} The Hölder spaces are defined in terms of an anisotropic distance relevant to the Galilean geometric structure on $\mathbb{R}\times\mathbb{R}^{2d}$ the operator $\mathcal{L}$ is invariant with respect to. We prove an intrinsic Taylor-like formula, whose reminder is estimated in terms of the anisotropic distance of the Galilean structure. Our achievements naturally extend analogous known results for purely differential operators on Lie groups.

math.AP↗
arXiv2023

Mean value formulas for classical solutions to subelliptic evolution equations in stratified Lie groups

We prove mean value formulas for classical solutions to second order linear differential equations in the form $$ \partial_t u = \sum_{i,j=1}^m X_i (a_{ij} X_j u) + X_0 u + f, $$ where $A = (a_{ij})_{i,j=1, \dots,m}$ is a bounded, symmetric and uniformly positive matrix with $C^1$ coefficients under the assumption that the operator $\sum_{j=1}^m X_j^2 + X_0 - \partial_t$ is hypoelliptic and the vector fields $X_1, \dots, X_m$ and $X_{m+1} :=X_0 - \partial_t$ are invariant with respect to a suitable homogeneous Lie group. Our results apply e.g. to degenerate Kolmogorov operators and parabolic equations on Carnot groups $\partial_t u = \sum_{i,j=1}^m X_i (a_{ij} X_j u) + f$.

math.AP↗
arXiv2023

Hölder continuity and boundedness estimates for nonlinear fractional equations in the Heisenberg group

We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator on the Heisenberg-Weyl group $\mathbb{H}^n$. Amongst other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates.

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 Hörmander 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↗
arXiv2022

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relevant the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeters on stratified Lie groups.

math.AP↗
arXiv2021

Mean value formulas for classical solutions to uniformly parabolic equations in divergence form

We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in divergence form. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C^1 boundary, the other stated for sets with finite perimeter.

math.AP↗
arXiv2021

A Yosida's parametrix approach to Varadhan's estimates for a degenerate diffusion under the weak Hörmander condition

We adapt and extend Yosida's parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak Hörmander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain the Varadhan formula \begin{equation} \frac{-2 \log p(t,x;T,y) } { Ψ(t,x;T,y) } \to 1, \qquad \text{as } \quad T-t \to 0^+, \end{equation} where $p$ denotes the transition density and $Ψ$ denotes the optimal cost function of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function $Ψ$, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.

math.PR↗
arXiv2021

Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

We study the regularity properties of the second order linear operator in $\mathbb{R}^{N+1}$: \begin{equation*} \mathscr{L} u := \sum_{j,k= 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum_{j,k= 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} where $A = \left( a_{jk} \right)_{j,k= 1, \dots, m}, B= \left( b_{jk} \right)_{j,k= 1, \dots, N}$ are real valued matrices with constant coefficients, with $A$ symmetric and strictly positive. We prove that, if the operator $\mathscr{L}$ satisfies Hörmander's hypoellipticity condition, and $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $\mathscr{L} u = f$ are Dini continuous functions as well. We also consider the case of Dini continuous coefficients $a_{jk}$'s. A key step in our proof is a Taylor formula for classical solutions to $\mathscr{L} u = f$ that we establish under minimal regularity assumptions on $u$.

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↗
arXiv2020

Fundamental solutions for Kolmogorov-Fokker-Planck operators with time-depending measurable coefficients

We consider a Kolmogorov-Fokker-Planck operator of the kind studied by Lanconelli-Polidoro in [Rend. Sem. Mat. Univ. Politec. Torino 52 (1994)], where the leading coefficients $a_{ij}$, instead of being constant, are bounded measurable functions of t. We construct an explicit fundamental solution for this operator, study its property, show a comparison result between this function and the fundamental solution of some model operators with constant $a_{ij}$, and show the unique solvability of the Cauchy problem under various assumptions on the initial datum.

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↗
arXiv2018

Sharp Estimates for Geman-Yor Processes and applications to Arithmetic Average Asian options

We prove the existence and pointwise lower and upper bounds for the fundamental solution of the degenerate second order partial differential equation related to Geman-Yor stochastic processes, that arise in models for option pricing theory in finance. Lower bounds are obtained by using repeatedly an invariant Harnack inequality and by solving an associated optimal control problem with quadratic cost. Upper bounds are obtained by the fact that the optimal cost satisfies a specific Hamilton-Jacobi-Bellman equation.

math.AP↗
arXiv2018

A compactness result for the Sobolev embedding via potential theory

In this note we give a proof of the Sobolev and Morrey embedding theorems based on the representation of functions in terms of the fundamental solution of suitable partial differential operators. We also prove the compactness of the Sobolev embedding. We first describe this method in the classical setting, where the fundamental solution of the Laplace equation is used, to recover the classical Sobolev and Morrey theorems. We next consider degenerate Kolmogorov equations. In this case, the fundamental solution is invariant with respect to a non-Euclidean translation group and the usual convolution is replaced by an operation that is defined in accordance with this geometry. We recover some known embedding results and we prove the compactness of the Sobolev embedding. We finally apply our regularity results to a kinetic equation.

math.AP↗