Some properties of path measures
We call any measure on a path space, a path measure. Some notions about path measures which appear naturally when solving the Schrödinger problem are presented and worked out in detail.
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arXiv subjects
Christian Léonard
Publications and source records attributed to Christian Léonard.
Stochastic derivatives and generalized h-transforms of Markov processes
Let $R$ be a continuous-time Markov process on the time interval $[0,1]$ with values in some state space $X$. We transform this reference process $R$ into $P:=f(X_0)\exp (-\int_0^1 V_t(X_t) dt) g(X_1)\,R$ where $f,g$ are nonnegative measurable functions on X and V is some measurable function on $[0,1]\times X$. It is easily seen that $P$ is also Markov. The aim of this paper is to identify the Markov generator of $P$ in terms of the Markov generator of $R$ and of the additional ingredients: $f,g$ and $V$ in absence of regularity assumptions on $f,g$ and $V.$ As a first step, we show that the extended generator of a Markov process is essentially its stochastic derivative. Then, we compute the stochastic derivative of $P$ to identify its generator, under a finite entropy condition. The abstract results are illustrated with continuous diffusion processes on $\mathbb{R}^d$ and Metropolis algorithms on a discrete space.
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Girsanov theory under a finite entropy condition
This paper is about Girsanov's theory. It (almost) doesn't contain new results but it is based on a simplified new approach which takes advantage of the (weak) extra requirement that some relative entropy is finite. Under this assumption, we present and prove all the standard results pertaining to the absolute continuity of two continuous-time processes with or without jumps. We have tried to give as much as possible a self-contained presentation. The main advantage of the finite entropy strategy is that it allows us to replace martingale representation results by the simpler Riesz representations of the dual of a Hilbert space (in the continuous case) or of an Orlicz function space (in the jump case).
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From the Schrödinger problem to the Monge-Kantorovich problem
The aim of this article is to show that the Monge-Kantorovich problem is the limit of a sequence of entropy minimization problems when a fluctuation parameter tends down to zero. We prove the convergence of the entropic values to the optimal transport cost as the fluctuations decrease to zero, and we also show that the limit points of the entropic minimizers are optimal transport plans. We investigate the dynamic versions of these problems by considering random paths and describe the connections between the dynamic and static problems. The proofs are essentially based on convex and functional analysis. We also need specific properties of Gamma-convergence which we didn't find in the literature. Hence we prove these Gamma-convergence results which are interesting in their own right.
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On the Duality Theory for the Monge-Kantorovich Transport Problem
The paper is accompanying "A general Duality Theorem for the Monge-Kantorovich Transport Problem". We explain the methods used in this article in an elementary setting and present two examples complementing the results obtained therein.
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Entropic Projections and Dominating Points
Generalized entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugate duality and functional analysis, criteria are derived for their existence. Representations of the generalized entropic projections are obtained: they are the ``measure component" of some extended entropy minimization problem.
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A generalized dual maximizer for the Monge--Kantorovich transport problem
The dual attainment of the Monge--Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be polish and equipped with Borel probability measures $μ$ and $ν$. The transport cost function $c:\XY \to [0,\infty]$ is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic and rely on Fenchel's perturbation technique.
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Transport Inequalities. A Survey
This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory.
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A large deviation approach to optimal transport
A probabilistic method for solving the Monge-Kantorovich mass transport problem on $R^d$ is introduced. A system of empirical measures of independent particles is built in such a way that it obeys a doubly indexed large deviation principle with an optimal transport cost as its rate function. As a consequence, new approximation results for the optimal cost function and the optimal transport plans are derived. They follow from the Gamma-convergence of a sequence of normalized relative entropies toward the optimal transport cost. A wide class of cost functions including the standard power cost functions $|x-y|^p$ enter this framework.
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Convex minimization problems with weak constraint qualifications
One revisits the standard saddle-point method based on conjugate duality for solving convex minimization problems. Our aim is to reduce or remove unnecessary topological restrictions on the constraint set. Dual equalities and characterizations of the minimizers are obtained with weak or without constraint qualifications. The main idea is to work with intrinsic topologies which reflect some geometry of the objective function. The abstract results of this article are applied in other papers to the Monge-Kantorovich optimal transport problem and the minimization of entropy functionals.
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Minimization of entropy functionals
Entropy functionals (i.e. convex integral functionals) and extensions of these functionals are minimized on convex sets. This paper is aimed at reducing as much as possible the assumptions on the constraint set. Dual equalities and characterizations of the minimizers are obtained with weak constraint qualifications.
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Characterization of the optimal plans for the Monge-Kantorovich transport problem
We present a general method, based on conjugate duality, for solving a convex minimization problem without assuming unnecessary topological restrictions on the constraint set. It leads to dual equalities and characterizations of the minimizers without constraint qualification. As an example of application, the Monge-Kantorovich optimal transport problem is solved in great detail. In particular, the optimal transport plans are characterized without restriction. This characterization improves the already existing literature on the subject.
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A large deviation approach to some transportation cost inequalities
New transportation cost inequalities are derived by means of elementary large deviation reasonings. Their dual characterization is proved; this provides an extension of a well-known result of S. Bobkov and F. Götze. Their tensorization properties are investigated. Sufficient conditions (and necessary conditions too) for these inequalities are stated in terms of the integrability of the reference measure. Applying these results leads to new deviation results: concentration of measure and deviations of empirical processes.
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