arXiv subjects

Mattia Martini

Publications and source records attributed to Mattia Martini.

11 recordsLinked to original sources

arXiv2026

A McKean--Vlasov semigroup and its application to a martingale representation problem

We introduce a McKean--Vlasov semigroup in the presence of common noise and study its first-order differential structure. Given a terminal functional $G$, the semigroup is defined by propagating the initial law through a reference conditional McKean--Vlasov flow. Using a finite-particle approximation and a diagonal/off-diagonal decomposition of the tangent processes, we identify the Lions derivative of the propagated functional and derive a backward representation along arbitrary admissible conditional law flows. This formula involves only first-order derivatives in the measure variable and yields both a Feynman--Kac representation for linear equations on the Wasserstein space and, in the driftless case, a martingale representation problem whose unique solution is given by the McKean--Vlasov heat semigroup and its Lions derivative. We also discuss mixed finite-dimensional/Wasserstein representations and discrete path-dependent extensions.

math.PR↗

arXiv2026

Mean convergence rates for Gaussian-smoothed Wasserstein distances and classical Wasserstein distances

We establish upper bounds for the expected $p$-th power of the Gaussian-smoothed $p$-Wasserstein distance between a probability measure $μ$ and the corresponding empirical measure $μ_N$, whenever $μ$ has finite $q$-th moment for some $q>p$. This generalizes recent results that were valid only for $q>2p+2d$. We provide two distinct proofs of such a result. We also investigate the optimality of these bounds by establishing a lower bound of order $N^{-1/2-\varepsilon}$ for a probability measure possessing finite moments of all orders. Finally, we exploit a third upper bound for the Gaussian-smoothed $p$-Wasserstein distance to derive new convergence rates for the classical $p$-Wasserstein distance in the critical regime where $μ$ has finite $p$-th moment but infinite moments of order $q > p$, covering for instance the case of Zygmund classes $L^p(\log L)^α$.

math.PR↗

arXiv2025

HJB equations driven by the Dirichlet-Ferguson Laplacian in Wasserstein-Sobolev spaces

We study linear and nonlinear PDEs defined on the space of $\mathcal{P}(\mathbb{T}^d)$ over the flat torus $\mathbb{T}^d$, equipped with the Dirichlet-Ferguson measure $\mathcal{D}$. We first develop an analytic framework based on the Wasserstein-Sobolev space $H^{1,2}(\mathcal{P}(\mathbb{T}^d), W_2, \mathcal{D})$ associated with the Dirichlet form induced by the infinite-dimensional Laplacian acting on functions of measures. Within this setting, we establish existence and uniqueness results for transport-diffusion and Hamilton-Jacobi equations in the Wasserstein space. Our analysis connects the PDE approach with a corresponding interacting particles system providing a probabilistic (Kolmogorov-type) representation of strong solutions. Finally, we extend the theory to semilinear equations and mean-field optimal control problems, together with consistent finite-dimensional approximations.

math.OC↗

arXiv2025

Quantitative convergence for mean field control with common noise and degenerate idiosyncratic noise

We consider the convergence problem in the setting of mean field control with common noise and degenerate idiosyncratic noise. Our main results establish a rate of convergence of the finite-dimensional value functions $V^N$ towards the mean field value function $U$. In the case that the idiosyncratic noise is constant (but possibly degenerate), we obtain the rate $N^{-1/(d+7)}$, which is close to the conjectured optimal rate $N^{-1/d}$, and improves on the existing literature even in the non-degenerate setting. In the case that the idiosyncratic noise can be both non-constant and degenerate, the argument is more complicated, and we instead find the rate $N^{-1/(3d + 19)}$. Our proof strategy builds on the one initiated in [Daudin, Delarue, Jackson - JFA, 2024] in the case of non-degenerate idiosyncratic noise and zero common noise, which consists of approximating $U$ by more regular functions which are almost subsolutions of the infinite-dimensional Hamilton-Jacobi equation solved by $U$. Because of the different noise structure, several new steps are necessary in order to produce an appropriate mollification scheme. In addition to our main convergence results, we investigate the case of zero idiosyncratic noise, and show that sharper results can be obtained there by purely control-theoretic arguments. We also provide examples to demonstrate that the value function is sensitive to the choice of admissible controls in the zero noise setting.

math.OC↗

arXiv2024

Fourier Galerkin approximation of mean field control problems

The purpose of this work is to provide a finite dimensional approximation of the solution to a mean field optimal control problem set on the $d$-dimensional torus. The approximation is obtained by means of a Fourier-Galerkin method, the main principle of which is to convolve probability measures on the torus by the Dirichlet kernel or, equivalently, to truncate the Fourier expansion of probability measures on the torus. However, this operation has the main feature not to leave the space of probability measures invariant, which drawback is know as \textit{Gibbs}' phenomenon. In spite of this, we manage to prove that, for initial conditions in the `interior' of the space of probability measures and for sufficiently large levels of truncation, the Fourier-Galerkin method induces a new finite dimensional control problem whose trajectories take values in the space of probability measures with a finite number of Fourier coefficients. Our main result asserts that, whenever the cost functionals are smooth and convex, the distance between the optimal trajectories of the original and approximating control problems decreases at a polynomial rate as the index of truncation in the Fourier-Galerkin method tends to $\infty$. A similar result holds for the distance between the corresponding value functions. From a practical point of view, our approach provides an efficient strategy to approximate mean field control optimizers by finite dimensional parameters and opens new perspectives for the numerical analysis of mean field control problems. It may be also applied to discretize more general mean field game systems.

math.OC↗

arXiv2024

Latent Abstractions in Generative Diffusion Models

In this work we study how diffusion-based generative models produce high-dimensional data, such as an image, by implicitly relying on a manifestation of a low-dimensional set of latent abstractions, that guide the generative process. We present a novel theoretical framework that extends NLF, and that offers a unique perspective on SDE-based generative models. The development of our theory relies on a novel formulation of the joint (state and measurement) dynamics, and an information-theoretic measure of the influence of the system state on the measurement process. According to our theory, diffusion models can be cast as a system of SDE, describing a non-linear filter in which the evolution of unobservable latent abstractions steers the dynamics of an observable measurement process (corresponding to the generative pathways). In addition, we present an empirical study to validate our theory and previous empirical results on the emergence of latent abstractions at different stages of the generative process.

cs.LG↗

arXiv2023

On smooth approximations in the Wasserstein space

In this paper we investigate the approximation of continuous functions on the Wasserstein space by smooth functions, with smoothness meant in the sense of Lions differentiability. In particular, in the case of a Lipschitz function we are able to construct a sequence of infinitely differentiable functions having the same Lipschitz constant as the original function. This solves an open problem raised in [11]. For (resp. twice) continuously differentiable function, we show that our approximation also holds for the first-order derivative (resp. second-order derivatives), therefore solving another open problem raised in [11].

math.PR↗

arXiv2023

Time-inconsistent mean-field optimal stopping: A limit approach

We provide a characterization of an optimal stopping time for a class of finite horizon time-inconsistent optimal stopping problems (OSPs) of mean-field type, adapted to the Brownian filtration, including those related to mean-field diffusion processes and recursive utility functions. Despite the time-inconsistency of the OSP, we show that it is optimal to stop when the value-process hits the reward process for the first time, as is the case for the standard time-consistent OSP. We solve the problem by approximating the corresponding value-process with a sequence of Snell envelopes of processes, for which a sequence of optimal stopping times is constituted of the hitting times of each of the reward processes by the associated value-process. Then, under mild assumptions, we show that this sequence of hitting times converges in probability to the hitting time for the mean-field OSP and that the limit is optimal.

math.PR↗

arXiv2023

Kolmogorov equations on spaces of measures associated to nonlinear filtering processes

We introduce and study some backward Kolmogorov equations associated to stochastic filtering problems. Measure-valued processed arise naturally in the context of stochastic filtering and one can formulate two stochastic differential equations, called Zakai and Kushner-Stratonovitch equation, that are satisfied by a positive measure and a probability measure-valued process respectively. The associated Kolmogorov equations have been intensively studied, mainly assuming that the measure-valued processes admit a density and then by exploiting stochastic calculus techniques in Hilbert spaces. Our approach differs from this since we do not assume the existence of a density and we work directly in the context of measures. We first formulate two Kolmogorov equations of parabolic type, one on a space of positive measures and one on a space of probability measures, and then we prove existence and uniqueness of classical solutions. In order to do that, we prove some intermediate results of independent interest. In particular, we prove Itô formulas for the composition of measure-valued filtering processes and real-valued functions. Moreover we study the regularity of the solution to the filtering equations with respect to the initial datum. In order to achieve these results, proper notions of derivatives on space of positive measures have been introduced and discussed.

math.PR↗

arXiv2022

Kolmogorov equations on the space of probability measures associated to the nonlinear filtering equation: the viscosity approach

We study the backward Kolmogorov equation on the space of probability measures associated to the Kushner-Stratonovich equation of nonlinear filtering. We prove existence and uniqueness in the viscosity sense and, in particular, we provide a comparison theorem. In the context of stochastic filtering it is natural to consider measure-valued processes that satisfy stochastic differential equations. In the literature, a classical way to address this problem is by assuming that these measure-valued processes admit a density. Our approach is different and we work directly with measures. Thus, the backward Kolmogorov equation we study is a second-order partial differential equation of parabolic type on the space of probability measures with compact support. In the literature only few results are available on viscosity solutions for this kind of problems and in particular the uniqueness is a very challenging issue. Here we find a viscosity solution and then we prove that it is unique via comparison theorem.

math.PR↗

arXiv2019

Numerical Methods for a System of Coupled Cahn-Hilliard Equations

In this work, we study a system of coupled Cahn-Hilliard equations describing the phase separation of a copolymer and a homopolymer blend. The numerical methods we propose are based on suitable combinations of existing schemes for the single Cahn-Hilliard equation. As a verification for our approach, we present some tests and a detailed description of the numerical solutions' behaviour obtained by varying the values of the parameters.

math.NA↗