arXiv subjects

Léonard Monsaingeon

Publications and source records attributed to Léonard Monsaingeon.

At least 19 recordsLinked to original sources

arXiv2026

Entropy and Fisher information in non-convex domains: one chain to rule them all

We prove that the (square root) Fisher information functional is a strong Wasserstein upper gradient of the entropy on non-convex Riemannian domains. This fills a gap in the literature by allowing one to completely dispense from $λ$-displacement convexity arguments. Along the way we establish a novel quantitative short-time control of the Fisher information along the Neumann heat flow, and establish an exact chain rule under stronger $AC_2$ assumptions typically satisfied by curves of measures obtained as limits of JKO schemes.

math.AP↗

arXiv2025

Construction of 2-bubbles for the energy critical bi-harmonic Schrödinger equation

We construct a blowing-up solution for the energy critical focusing biharmonic nonlinear Schrödinger equation in infinite time in dimension $N\geq 13$. Our solution is radially symmetric and converges asymptotically to the sum of two bubbles. The scale of one of the bubble is of order $1$ whereas the other one is of order $|t|^{-\frac{2}{N-12}}$. Moreover, the phase between the two bubbles form a right angle.

math.AP↗

arXiv2025

A gradient flow that is none: Heat flow with Wentzell boundary condition

We establish a representation of the heat flow with Wentzell boundary conditions on smooth domains as gradient descent dynamics for the entropy in a suitably extended Otto manifold of probability measures with additional boundary parts. Yet it is shown that for weak boundary diffusion, the associated Fokker-Planck dynamics cannot be recovered from any entropy-driven metric JKO-Wasserstein scheme, at least if the underlying point metric satisfies certain natural regularity assumptions. This discrepancy is illustrated in competing large-deviation heuristics in the Sanov and Schilder regimes.

math.AP↗

arXiv2025

Sticky-reflecting diffusion as a Wasserstein gradient flow

In this paper we identify the Fokker-Planck equation for (reflected) Sticky Brownian Motion as a Wasserstein gradient flow in the space of probability measures. The driving functional is the relative entropy with respect to a non-standard reference measure, the sum of an absolutely continuous interior part plus a singular part supported on the boundary. Taking the small time-step limit in a minimizing movement (JKO scheme) we prove existence of weak solutions for the coupled system of PDEs satisfying in addition an Energy Dissipation Inequality.

math.AP↗

arXiv2024

Discretizing the Fokker-Planck equation with second-order accuracy: a dissipation driven approach

We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time discretization is novel and relies on a tailored nonlinear mid-point rule, designed to accurately capture the dissipative structure of the model. We establish well-posedness for the scheme, positivity of the solutions, as well as a fully discrete energy-dissipation inequality mimicking the continuous one. We then prove the rigorous convergence of the scheme under mildly restrictive conditions on the unstructured grids, which can be easily satisfied in practice. Numerical simulations show that our scheme is second order accurate both in time and space, and that one can solve the discrete nonlinear systems arising at each time step using Newton's method with low computational cost.

math.AP↗

arXiv2023

Trivariate distribution of sticky Brownian motion

In this short note we derive a closed form for the trivariate distribution (position, local time at the origin, and positive occupation time) of the one-dimensional sticky Brownian motion, thereby filling some gaps and fixing some mistakes in the literature.

math.PR↗

arXiv2023

The dynamical Schrödinger problem in abstract metric spaces

In this paper we introduce the dynamical Schrödinger problem on abstract metric spaces, defined for a wide class of entropy and Fisher information functionals. Under very mild assumptions we prove a generic Gamma-convergence result towards the geodesic problem as the noise parameter $\varepsilon\downarrow 0$. We also investigate the connection with geodesic convexity of the driving entropy, and study the dependence of the entropic cost on the parameter $\varepsilon$. Some examples and applications are discussed.

math.FA↗

arXiv2022

Hidden dissipation and convexity for Kimura equations

In this paper we establish a rigorous gradient flow structure for one-dimensional Kimura equations with respect to some Wasserstein-Shahshahani optimal transport geometry. This is achieved by first conditioning the underlying stochastic process to non-fixation in order to get rid of singularities on the boundaries, and then studying the conditioned $Q$-process from a more traditional and variational point of view. In doing so we complete the work initiated in [Chalub et Al., Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective. Acta App Math., 171(1), 1-50], where the gradient flow was identified only formally. The approach is based on the Energy Dissipation Inequality and Evolution Variational Inequality notions of metric gradient flows. Building up on some convexity of the driving entropy functional, we obtain new contraction estimates and quantitative long-time convergence towards the stationary distribution.

math.AP↗

arXiv2022

Schrödinger encounters Fisher and Rao: a survey

In this short note we review the dynamical Schrödinger problem on the non-commutative Fisher-Rao space of positive semi-definite matrix-valued measures. The presentation is meant to be self-contained, and we discuss in particular connections with Gaussian optimal transport, entropy, and quantum Fisher information.

math.ST↗

arXiv2021

Invariant measures and global well-posedness for a fractional Schrödinger equation with Moser-Trudinger type nonlinearity

In this paper, we construct invariant measures and global-in-time solutions for a fractional Schr\" odinger equation with a Moser-Trudinger type nonlinearity $$ i\partial_t u= (-Δ)^αu+ 2βu e^{β|u|^2},\qquad\mbox{for}\qquad(x,t)\in \ M\times \mathbb{R} $$ on a compact Riemannian manifold $M$ without boundary of dimension $d\geq 2$. To do so, we use the so-called Inviscid-Infinite-dimensional limits introduced by Sy ('19) and Sy and Yu ('21). More precisely, we show that if $s>d/2$ or if $s\leq d/2$ and $s\leq 1+α$, there exists an invariant measure $μ^{s}$ and a set $Σ^s \subset H^s$ containing arbitrarily large data such that $μ^{s}(Σ^s ) =1$ and that the fractional NLS is globally well-posed on $Σ^{s}$. For strong regularities $s>d/2$ we also obtain a logarithmic upper bound on the growth of the $H^r$-norm of our solutions for $r<s$. This gives new examples of invariant measures supported in highly regular spaces in comparison with the Gibbs measure constructed by Robert ('21) for the same equation.

math.AP↗

arXiv2021

Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps

If $u : Ω\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in a metric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convex function, in the sense that it generates an ${\rm EVI}_0$-gradient flow, we prove that the pullback ${\sf E} \circ u : Ω\to \mathbb{R}$ is subharmonic. This property was known in the smooth Riemannian manifold setting or with curvature restrictions on ${\rm X}$, while we prove it here in full generality. In addition, we establish generalized maximum principles, in the sense that the $L^q$ norm of ${\sf E} \circ u$ on $\partial Ω$ controls the $L^p$ norm of ${\sf E} \circ u$ in $Ω$ for some well-chosen exponents $p \geq q$, including the case $p=q=+\infty$. In particular, our results apply when ${\sf E}$ is a geodesically convex entropy over the Wasserstein space, and thus settle some conjectures of Y. Brenier, "Extended Monge-Kantorovich theory" in Optimal transportation and applications (Martina Franca, 2001), volume 1813 of Lecture Notes in Math., pages 91-121. Springer, Berlin, 2003.

math.MG↗

arXiv2021

A degenerate elliptic-parabolic system arising in competitive contaminant transport

In this work we investigate a coupled system of degenerate and nonlinear partial differential equations governing the transport of reactive solutes in groundwater. We show that the system admits a unique weak solution provided the nonlinear adsorption isotherm associated with the reaction process satisfies certain physically reasonable structural conditions. We conclude, moreover, that the solute concentrations stay non-negative if the source term is componentwise non-negative and investigate numerically the finite speed of propagation of compactly supported initial concentrations, in a two-component test case.

math.AP↗

arXiv2021

A new transportation distance with bulk/interface interactions and flux penalization

We introduce and study a new optimal transport problem on a bounded domain $\barΩ\subset \mathbb R^d$, defined via a dynamical Benamou-Brenier formulation. The model handles differently the motion in the interior and on the boundary, and penalizes the transfer of mass between the two. The resulting distance interpolates between classical optimal transport on $\barΩ$ on the one hand, and on the other hand between two independent optimal transport problems set on $Ω$ and $\partial Ω$.

math.AP↗

arXiv2021

The Schrödinger problem on the non-commutative Fisher-Rao space

We present a self-contained and comprehensive study of the Fisher-Rao space of matrix-valued non-commutative probability measures, and of the related Hellinger space. Our non-commutative Fisher-Rao space is a natural generalization of the classical commutative Fisher-Rao space of probability measures and of the Bures-Wasserstein space of Hermitian positive-definite matrices. We introduce and justify a canonical entropy on the non-commutative Fisher-Rao space, which differs from the von Neumann entropy. We consequently derive the analogues of the heat flow, of the Fisher information, and of the dynamical Schrödinger problem. We show the $Γ$-convergence of the $ε$-Schrödinger problem towards the geodesic problem for the Fisher-Rao space, and, as a byproduct, the strict geodesic convexity of the entropy.

math.FA↗

arXiv2020

Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective

We consider three classical models of biological evolution: (i) the Moran process, an example of a reducible Markov Chain; (ii) the Kimura Equation, a particular case of a degenerated Fokker-Planck Diffusion; (iii) the Replicator Equation, a paradigm in Evolutionary Game Theory. While these approaches are not completely equivalent, they are intimately connected, since (ii) is the diffusion approximation of (i), and (iii) is obtained from (ii) in an appropriate limit. It is well known that the Replicator Dynamics for two strategies is a gradient flow with respect to the celebrated Shahshahani distance. We reformulate the Moran process and the Kimura Equation as gradient flows and in the sequel we discuss conditions such that the associated gradient structures converge: (i) to (ii) and (ii) to (iii). This provides a geometric characterisation of these evolutionary processes and provides a reformulation of the above examples as time minimization of free energy functionals.

q-bio.PE↗

arXiv2020

Computing discrete equivariant harmonic maps

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explicit convergence rate. We also examine center of mass methods, after showing a generalized mean value property for harmonic maps. We feature a concrete illustration of these methods with Harmony, a computer software that we developed in C++, whose main functionality is to numerically compute and display equivariant harmonic maps.

math.GT↗

arXiv2020

Computing harmonic maps between Riemannian manifolds

In the previous paper [GLM2018], we showed that the theory of harmonic maps between Riemannian manifolds may be discretized by introducing triangulations with vertex and edge weights on the domain manifold. In the present paper, we study convergence of the discrete theory to the smooth theory when taking finer and finer triangulations. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps. Our computer software Harmony implements these methods to computes equivariant harmonic maps in the hyperbolic plane.

math.DG↗

arXiv2019

Small noise limit and convexity for generalized incompressible flows, Schrödinger problems, and optimal transport

This paper is concerned with six variational problems and their mutual connections: The quadratic Monge-Kantorovich optimal transport, the Schrödinger problem, Brenier's relaxed model for incompressible fluids, the so-called Brödinger problem recently introduced by M. Arnaudon & al. [3], the multiphase Brenier model, and the multiphase Brödinger problem. All of them involve the minimization of a kinetic action and/or a relative entropy of some path measures with respect to the reversible Brownian motion. As the viscosity parameter $ν\to 0$ we establish Gamma-convergence relations between the corresponding problems, and prove the convergence of the associated pressures arising from the incompressibility constraints. We also present new results on the time-convexity of the entropy for some of the dynamical interpolations. Along the way we extend previous results by H. Lavenant [30] and J-D. Benamou & al. [10].

math.AP↗