arXiv subjects

Nassif Ghoussoub

Publications and source records attributed to Nassif Ghoussoub.

At least 19 recordsLinked to original sources

arXiv2026

A General Aubry-Mather Theory

This paper reproduces the front matter --- preface, overview and table of contents --- of a monograph by the author, submitted for publication under the title {\it Skew Linear Entropies and Kantorovich Operators: A General Aubry-Mather Theory}. The book isolates a class of non-linear operators, which we call {\it Kantorovich operators}, that are ubiquitous in analysis, probability, dynamical systems, mathematical economics and finance. We develop aspects of their ergodic theory in a way that extends classical ones involving Markov operators, free-energy transfers, or the Hopf--Lax--Oleinik semi-group. Having no adjoint, the duality between such an operator and measures is carried instead via a convex functional on {\it pairs} of probability distributions --- a source and a target --- which we call a {\it skew-linear entropy}, and which is a general form of optimal mass transport. The extensive overview reproduced here describes the resulting ergodic theory, in which minimal measures, a Mather constant, weak KAM solutions and an Aubry set are attached to an arbitrary Kantorovich operator, extending Aubry--Mather theory well beyond its origins in Hamiltonian dynamics.

math.AP↗

arXiv2023

Ergodic properties of Kantorovich operators

Kantorovich operators are non-linear extensions of Markov operators and are omnipresent in several branches of mathematical analysis. The asymptotic behaviour of their iterates plays an important role even in classical ergodic, potential and probability theories, which are normally concerned with linear Markovian operators, semi-groups, and resolvents. The Kantorovich operators that appear implicitly in these cases, though non-linear, are all positively 1-homogenous. General Kantorovich operators amount to assigning "a cost" to most operations on measures and functions normally conducted "for free" in these classical settings. Motivated by extensions of the Monge-Kantorovich duality in mass transport, the stochastic counterpart of Aubry-Mather theory for Lagrangian systems, weak KAM theory à la Fathi-Mather, and ergodic optimization of dynamical systems, we study the asymptotic properties of general Kantorovich operators.

math.AP↗

arXiv2023

Linear transfers as minimal costs of dilations of measures in balayage order

Linear transfers between probability distributions were introduced in [5,6] in order to extend the theory of optimal mass transportation while preserving the important duality established by Kantorovich. It is shown here that $\{0, +\infty\}$-valued linear transfers can be characterized by balayage of measures with respect to suitable cones of functions à la Choquet, while general linear transfers extend balayage theory by requiring the "sweeping out" of measures to optimize certain cost functionals. We study the dual class of Kantorovich operators, which are natural and manageable extensions of Markov operators. It is also an important subclass of capacities, and could be called "convex functional Choquet capacities," since they play for non-linear maps the same role that convex envelopes do for arbitrary numerical functions. A forthcoming paper [7] will study their ergodic properties and their applications.

math.AP↗

arXiv2021

The Hardy--Schrödinger Operator on the Poincaré Ball: Compactness and Multiplicity

Let $Ω$ be a compact smooth domain containing zero in the Poincaré ball model of the Hyperbolic space $\mathbb{B}^{n}$ ($n \geq 3$) and let $-Δ_{\mathbb{B}^{n}}$ be the Laplace-Beltrami operator on $\mathbb{B}^{n}$, associated with the metric $g_{\mathbb{B}^{n}}= \frac{4}{(1-|x|^{2})^2}g_{_{\hbox{Eucl}}}$. We consider issues of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, \begin{eqnarray*} (E)~ \left\{ \begin{array}{lll} -Δ_{\mathbb{B}^{n}}u-γ{V_2}u -λu&=V_{2^\star(s)}|u|^{2^\star(s)-2}u &\hbox{ in }Ω\\ \hfill u &=0 & \hbox{ on } \partial Ω, \end{array} \right. \end{eqnarray*} where $0\leq γ\leq \frac{(n-2)^2}{4}$, $0< s <2$, ${2^\star(s)}:=\frac{2(n-s)}{n-2}$ is the corresponding critical Sobolev exponent, $V_{2}$ (resp., $V_{2^\star(s)}$) is a Hardy-type potential (resp., Hardy-Sobolev weight) that is invariant under hyperbolic scaling and which behaves like $\frac{1}{r^{2}}$ (resp., $\frac{1}{r^{s}}$) at the origin. The bulk of this paper is a sharp blow-up analysis on approximate solutions of $(E)$ with bounded but arbitrary high energies. Our analysis leads to existence of positive ground state solutions for $(E)$, whenever $n \geq 4$, $0 \leq γ\leq \frac{(n-2)^2}{4}-1$ and $ λ> 0$. The latter result also holds true for $n\geq 3$ and $γ> \frac{(n-2)^2}{4}-1$ provided the domain has a positive "hyperbolic mass". On the other hand, the same analysis yields that if $γ> \frac{(n-2)^2}{4}-1$ and the mass is non vanishing, then there is a surprising stability of regimes where no variational positive solution exists. As for higher energy solutions to $(E)$, we show that there are infinitely many of them provided $n\geq 5$, $0\leq γ<\frac{(n-2)^2}{4}-4$ and $ λ> \frac{n-2}{n-4} \left(\frac{n(n-4)}{4}-γ\right)$.

math.AP↗

arXiv2020

Optimal Stopping of Stochastic Transport Minimizing Submartingale Costs

Given a stochastic state process $(X_t)_t$ and a real-valued submartingale cost process $(S_t)_t$, we characterize optimal stopping times $τ$ that minimize the expectation of $S_τ$ while realizing given initial and target distributions $μ$ and $ν$, i.e., $X_0\sim μ$ and $X_τ\sim ν$. A dual optimization problem is considered and shown to be attained under suitable conditions. The optimal solution of the dual problem then provides a contact set, which characterizes the location where optimal stopping can occur. The optimal stopping time is uniquely determined as the first hitting time of this contact set provided we assume a natural structural assumption on the pair $(X_t, S_t)_t$, which generalizes the twist condition on the cost in optimal transport theory. This paper extends the Brownian motion settings studied in [15, 16] and deals with more general costs.

math.PR↗

arXiv2020

Hidden convexity in a problem of nonlinear elasticity

We study compressible and incompressible nonlinear elasticity variational problems in a general context. Our main result gives a sufficient condition for an equilibrium to be a global energy minimizer, in terms of convexity properties of the pressure in the deformed configuration. We also provide a convex relaxation of the problem together with its dual formulation, based on measure-valued mappings, which coincides with the original problem under our condition.

math.AP↗

arXiv2020

A solution to the Monge transport problem for Brownian martingales

We provide a solution to the problem of optimal transport by Brownian martingales in general dimensions whenever the transport cost satisfies certain subharmonic properties in the target variable, as well as a stochastic version of the standard "twist condition" frequently used in deterministic Monge transport theory. This setting includes in particular the case of the distance cost $c(x,y)=|x-y|$. We prove existence and uniqueness of the solution and characterize it as the first time Brownian motion hits a barrier that is determined by solutions to a corresponding dual problem.

math.AP↗

arXiv2020

Multiplicity and stability of the Pohozaev obstruction for Hardy-Schrödinger equations with boundary singularity

Let $Ω$ be a smooth bounded domain in $\mathbb{R}^n$ ($n\geq 3$) such that $0\in\partial Ω$. In this memoir, we consider issues of non-existence, existence, and multiplicity of variational solutions in $H_{1,0}^2(Ω)$ for the borderline Dirichlet problem, $-Δu-γ\frac{u}{|x|^2}- h(x) u = \frac{|u|^{{2^\star(s)}-2}u}{|x|^s}$ in $Ω$, where $0<s<2$, ${2^\star(s)}:=\frac{2(n-s)}{n-2}$, $γ\in\mathbb{R}$ and $h\in C^0(\overlineΩ)$. We use sharp blow-up analysis on --possibly high energy-- solutions of corresponding subcritical problems to establish, for example, that if $γ<\frac{n^2}{4}-1$ and the principal curvatures of $\partialΩ$ at $0$ are non-positive but not all of them vanishing, then the above equation has an infinite number of (possibly sign-changing) solutions in ${H_{1,0}^2(Ω)}$. This complements results of the first and third authors, who had previously shown that if $γ\leq \frac{n^2}{4}-\frac{1}{4}$ and the mean curvature of $\partialΩ$ at $0$ is negative, then the equation has a positive solution. On the other hand, the sharp blow-up analysis also allows us to prove that if the mean curvature at $0$ is non-zero and if the mass (when defined) does not vanish, then there is a surprising stability under $C^1$-perturbations of the potential $h$ of those regimes where no variational positive solutions exist. In particular, and in sharp contrast with the non-singular case (i.e., when $γ=s=0$), we show non-existence of such solutions for (E) in any dimension, whenever $Ω$ is star-shaped and $h$ is close to $0$, which include situations not covered by the classical Pohozaev obstruction.

math.AP↗

arXiv2019

Stochastic optimal transport with free end time

We consider a stochastic transportation problem between two prescribed probability distributions (a source and a target) over processes with general drift dependence and with free end times. First, and in order to establish a dual principle, we associate two equivalent formulations of the primal problem in order to guarantee its convexity and lower semi-continuity with respect to the source and target distributions. We exhibit an equivalent Eulerian formulation, whose dual variational principle is given by Hamilton-Jacobi-Bellman type variational inequalities. In the case where the dependence on the drift is bounded, regularity results on the minimizers of the Eulerian problem then enable us to prove attainment in the corresponding dual problem. We also address attainment when the drift component of the cost defining Lagrangian $L$ is superlinear $L \approx |u|^p$ with $1<p<2$, in which case the setting is reminiscent of our approach -- in a previous work -- on deterministic controlled transport problems with free end time. We finally address criteria under which the optimal drift and stopping time are unique, namely strict convexity in the drift component and monotonicity in time of the Lagrangian.

math.OC↗

arXiv2019

Optimal Brownian stopping when the source and target are radially symmetric distributions

Given two probability measures $μ, ν$ on $\mathbb{R}^d$, in subharmonic order, we describe optimal stopping times $τ$ that maximize/minimize the cost functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Brownian motion with initial law $μ$ and with final distribution --once stopped at $τ$-- equal to $ν$. Under the assumption of radial symmetry on $μ$ and $ν$, we show that in dimension $d \geq 3$ and $α\neq 2$, there exists a unique optimal solution given by a non-randomized stopping time characterized as the hitting time to a suitably symmetric barrier. We also relate this problem to the optimal transportation problem for subharmonic martingales, and establish a duality result. This paper is an expanded version of a previously posted but not published work by the authors.

math.AP↗

arXiv2019

Mather Measures and Ergodic Properties of Kantorovich Operators associated to General Mass Transfers

We introduce and study the class of linear transfers between probability distributions and the dual class of Kantorovich operators between function spaces. Linear transfers can be seen as an extension of convex lower semi-continuous energies on Wasserstein space, of cost minimizing mass transports, as well as many other couplings between probability measures to which Monge-Kantorovich theory does not readily apply. Basic examples include balayage of measures, martingale transports, optimal Skorokhod embeddings, and the weak mass transports of Talagrand, Marton, Gozlan and others. The class also includes various stochastic mass transports such as the Schrödinger bridge associated to a reversible Markov process, and the Arnold-Brenier variational principle for the incompressible Euler equations. We associate to most linear transfers, a critical constant, a corresponding effective linear transfer and additive eigenfunctions to their dual Kantorovich operators, that extend Mané's critical value, Aubry-Mather invariant tori, and Fathi's weak KAM solutions for Hamiltonian systems. This amounts to studying the asymptotic properties of the nonlinear Kantorovich operators as opposed to classical ergodic theory, which deals with linear Markov operators. This allows for the extension of Mather theory to other settings such as its stochastic counterpart. We also introduce the class of convex transfers, which includes $p$-powers ($ p \geq 1$) of linear transfers, the logarithmic entropy, the Donsker-Varadhan information, optimal mean field plans, and certain free energies as functions of two probability measures, i.e., where the reference measure is also a variable. Duality formulae for general transfer inequalities follow in a very natural way. This paper is an expanded version of a previously posted but not published work by the authors.

math.AP↗

arXiv2019

PDE Methods For Optimal Skorokhod Embeddings

We consider cost minimizing stopping time solutions to Skorokhod embedding problems, which deal with transporting a source probability measure to a given target measure through a stopped Brownian process. PDEs and a free boundary problem approach are used to address the problem in general dimensions with space-time inhomogeneous costs given by Lagrangian integrals along the paths. We introduce an Eulerian---mass flow---formulation of the problem, whose dual is given by Hamilton-Jacobi-Bellman type variational inequalities. Our key result is the existence (in a Sobolev class) of optimizers for this new dual problem, which in turn determines a free boundary, where the optimal Skorokhod transport drops the mass in space-time. This complements and provides a constructive PDE alternative to recent results of Beiglböck, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.

math.AP↗

arXiv2019

Optimal Controlled Transports with Free End Times Subject to Import/Export Tariffs

We analyze controlled mass transportation plans with free end-time that minimize the transport cost induced by the generating function of a Lagrangian within a bounded domain, in addition to costs incurred as export and import tariffs at entry and exit points on the boundary. We exhibit a dual variational principle à la Kantorovich, that takes into consideration the additional tariffs. We then show that the primal optimal transport problem has an equivalent Eulerian formulation whose dual involves the resolution of a Hamilton-Jacobi-Bellman quasi-variational inequality with non-homogeneous boundary conditions. This allows us to prove existence and to describe the solutions for both the primal optimization problem and its Eulerian counterpart.

math.AP↗

arXiv2018

A Theory of Transfers: Duality and convolution

We introduce and study the permanence properties of the class of linear transfers between probability measures. This class contains all cost minimizing mass transports, but also martingale mass transports, the Schrodinger bridge associated to a reversible Markov process, and the weak mass transports of Tala- grand, Marton, Gozlan and others. The class also includes various stochastic mass transports to which Monge-Kantorovich theory does not apply. We also introduce the cone of convex transfers, which include any p-power (p > 1) of a linear transfer, but also the logarithmic entropy, the Donsker-Varadhan infor- mation and certain free energy functionals. This first paper is mostly about exhibiting examples that point to the pervasiveness of the concept in the important work on correlating probability distributions. Duality formulae for general transfer inequalities follow in a very natural way. We also study the infinite self-convolution of a linear transfer in order to establish the existence of generalized weak KAM solutions that could be applied to the stochastic counterpart of Fathi-Mather theory.

math.AP↗

arXiv2018

Optimal Transport with Controlled Dynamics and Free End Times

We consider optimal transport problems where the cost is optimized over controlled dynamics and the end time is free. Unlike the classical setting, the search for optimal transport plans also requires the identification of optimal "stopping plans," and the corresponding Monge-Kantorovich duality involves the resolution of a Hamilton-Jacobi-Bellman quasi-variational inequality. We discuss both Lagrangian and Eulerian formulations of the problem, and its natural connection to Pontryagin's maximum principle. We also exhibit a purely dynamic situation, where the optimal stopping plan is a hitting time of a barrier given by the free boundary problem associated to the dual variational inequality. This problem was motivated by its stochastic counterpart, which will be studied in a companion paper.

math.OC↗

arXiv2018

Dynamic and Stochastic Propagation of Brenier's Optimal Mass Transport

We investigate how mass transports that optimize the inner product cost -considered by Y. Brenier- propagate in time along a given Lagrangian. In the deterministic case, we consider transports that maximize and minimize the following "ballistic" cost functional on phase space $M^*\times M$, \[ b_T(v, x):=\inf\{\langle v, γ(0)\rangle +\int_0^TL(t, γ(t), {\dot γ}(t))\, dt; γ\in C^1([0, T), M); γ(T)=x\}, \] where $M=\mathbb{R}^d$, $T>0$, and $L:M\times M \to \mathbb{R}$ is a suitable Lagrangian. We also consider the stochastic counterpart: \begin{align*}%\tag{$\star$} \underline{B}_T^s(μ,ν):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle +\int_0^T L(t, X,β(t,X))\,dt\right]; X\in \mathcal{A}, V\simμ,X_T\sim ν\right\} \end{align*} where $\mathcal{A}$ is the set of stochastic processes satisfying $dX=β_X(t,X)\,dt+ dW_t,$ for some drift $β_X(t,X)$, and where $W_t$ is $σ(X_s:0\le s\le t)$-Brownian motion. While inf-convolution allows us to easily obtain Hopf-Lax formulas on Wasserstein space for cost minimizing transports, this is not the case for total cost maximizing transports, which actually are sup-inf problems. However, in the case where the Lagrangian $L$ is jointly convex on phase space, Bolza-type dualities --well known in the deterministic case but novel in the stochastic case--transform sup-inf problems to sup-sup settings. Hopf-Lax formulas relate optimal ballistic transports to those associated with dynamic fixed-end transports studied by Bernard-Buffoni and Fathi-Figalli in the deterministic case, and by Mikami-Thieullen in the stochastic setting. We also write Eulerian formulations and point to links with the theory of mean field games.

math.AP↗

arXiv2018

Mass and Extremals Associated with the Hardy-Schrödinger Operator on Hyperbolic Space

We consider the Hardy-Schrödinger operator $ -Δ_{\mathbb{B}^n}-γ{V_2}$ on the Poincaré ball model of the Hyperbolic space ${\mathbb{B}^n}$ ($n \geq 3$). Here $V_2$ is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at $0$, i.e., $V_2(r)\sim \frac{1}{r^2}$. Just like in the Euclidean setting, the operator $ -Δ_{\mathbb{B}^n}-γ{V_2}$ is positive definite whenever $γ<\frac{(n-2)^2}{4}$, in which case we exhibit explicit solutions for the equation $$-Δ_{\mathbb{B}^n}u-γ{V_2}u=V_{2^*(s)}u^{2^*(s)-1}\quad{\text{ in }}\mathbb{B}^n,$$ where $0\leq s <2$, $2^*(s)=\frac{2(n-s)}{n-2}$, and $V_{2^*(s)}$ is a weight that behaves like $\frac{1}{r^s}$ around $0$. The same equation, on bounded domains $Ω$ of ${\mathbb{B}^n}$ containing $0$ but not touching the hyperbolic boundary, has positive solutions if $0 < γ\leq \frac{(n-2)^{2}}{4}-\frac{1}{4}$. However, if $\frac{(n-2)^{2}}{4}-\frac{1}{4}< γ< \frac{(n-2)^{2}}{4}$, the existence of solutions requires the positivity of the "hyperbolic Hardy mass" $m_{_{\mathbb{B}^n}}(Ω)$ of the domain, a notion that we introduce and analyse therein.

math.AP↗

arXiv2018

A Self-dual Variational Approach to Stochastic Partial Differential Equations

Unlike many deterministic PDEs, stochastic equations are not amenable to the classical variational theory of Euler-Lagrange. In this paper, we show how self-dual variational calculus leads to solutions of various stochastic partial differential equations driven by monotone vector fields. We construct weak solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We deal with both additive and non-additive noise. The equations considered in this paper have already been resolved by other methods, starting with the celebrated thesis of Pardoux, and many other subsequent works. This paper is about presenting a new variational approach to this type of problems, hoping it will lead to progress on other still unresolved situations.

math.AP↗