arXiv subjects

Katharina Eichinger

Publications and source records attributed to Katharina Eichinger.

10 recordsLinked to original sources

arXiv2026

Exponential Convergence of the Sinkhorn Algorithm for the Schrödinger Bridge with Regime Switching

This paper studies the convergence of the Sinkhorn algorithm for the Schrödinger bridge problem with regime switching, as introduced in Zlotchevski and Chen (2025). We consider a class of regime-switching stochastic systems on the hybrid state space $E=\mathbb{R}^d\times\{1,\ldots,m\}$, and construct the Sinkhorn iteration through the associated equivalent entropic optimal transport formulation. The main result of this paper is the exponential convergence of the Sinkhorn algorithm in relative entropy under compactness assumptions. Our proofs are inspired by the arguments recently developed for proving exponential convergence of the classical Schrödinger problem in Chiarini, Conforti, Greco and Tamanini (2024), Eckstein (2025). We perform a similar analysis for the partially observed terminal setting, where only the marginal distribution of the continuous component is prescribed at the terminal time, while the discrete regime is unobserved.

math.PR↗

arXiv2026

Near-Lipschitz stability of the Kim--Milman flow map

We prove that the Kim--Milman flow map enjoys favorable stability properties with respect to variations in the target measure, provided that one of the target measures is sufficiently regular. Our results include stability in relative entropy, and more notably, Lipschitz stability in the $2$-Wasserstein distance up to a logarithmic factor. We complement our results with a general existence theorem for these maps for any target measure with finite second moment.

math.PR↗

arXiv2025

Propagation of weak log-concavity along generalised heat flows via Hamilton-Jacobi equations

A well-known consequence of the Pr{é}kopa-Leindler inequality is the preservation of logconcavity by the heat semigroup. Unfortunately, this property does not hold for more general semigroups. In this paper, we exhibit a slightly weaker notion of log-concavity that can be propagated along generalised heat semigroups. As a consequence, we obtain logsemiconcavity properties for the ground state of Schr{ö}dinger operators for non-convex potentials, as well as propagation of functional inequalities along generalised heat flows. We then investigate the preservation of weak log-concavity by conditioning and marginalisation, following the seminal works of Brascamp and Lieb. To our knowledge, our results are the first of this type in non log-concave settings. We eventually study generation of log-concavity by parabolic regularisation and prove novel two-sided log-Hessian estimates for the fundamental solution of parabolic equations with unbounded coefficients, which can be made uniform in time. These properties are obtained as a consequence of new propagation of weak convexity results for quadratic Hamilton-Jacobi-Bellman (HJB) equations. The proofs rely on a stochastic control interpretation combined with a second order analysis of reflection coupling along HJB characteristics.

math.AP↗

arXiv2025

Monge-Kantorovich interpolation with constraints and application to a parking problem

We consider optimal transport problems where the cost for transporting a given probability measure $μ_0$ to another one $μ_1$ consists of two parts: the first one measures the transportation from $μ_0$ to an intermediate (pivot) measure $μ$ to be determined (and subject to various constraints), and the second one measures the transportation from $μ$ to $μ_1$. This leads to Monge-Kantorovich interpolation problems under constraints for which we establish various properties of the optimal pivot measures $μ$. Considering the more general situation where only some part of the mass uses the intermediate stop leads to a mathematical model for the optimal location of a parking region around a city. Numerical simulations, based on entropic regularization, are presented both for the optimal parking regions and for Monge-Kantorovich constrained interpolation problems.

math.OC↗

arXiv2025

A coupling approach to Lipschitz transport maps

In this note, we propose a probabilistic approach to bound the (dimension-free) Lipschitz constant of the Langevin flow map on $\mathbb{R}^d$ introduced by Kim and Milman (2012). As example of application, we construct Lipschitz maps from a uniformly $\log$-concave probability measure to $\log$-Lipschitz perturbations as in Fathi, Mikulincer, Shenfeld (2024). Our proof is based on coupling techniques applied to the stochastic representation of the family of vector fields inducing the transport map. This method is robust enough to relax the uniform convexity to a weak asymptotic convexity condition and to remove the bound on the third derivative of the potential of the source measure.

math.PR↗

arXiv2025

Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced by Bigot, Cazelles and Papadakis (2019) as a regularization of Wasserstein barycenters first presented by Agueh and Carlier (2011). After characterizing these barycenters in terms of a system of Monge-Ampère equations, we prove some global moment and Sobolev bounds as well as higher regularity properties. We finally establish a central limit theorem for entropic-Wasserstein barycenters.

math.AP↗

arXiv2025

Wasserstein medians: robustness, PDE characterization and numerics

We investigate the notion of Wasserstein median as an alternative to the Wasserstein barycenter, which has become popular but may be sensitive to outliers. In terms of robustness to corrupted data, we indeed show that Wasserstein medians have a breakdown point of approximately $\frac{1}{2}$. We give explicit constructions of Wasserstein medians in dimension one which enable us to obtain $L^p$ estimates (which do not hold in higher dimensions). We also address dual and multimarginal reformulations. In convex subsets of $\mathbb{R}^d$, we connect Wasserstein medians to a minimal (multi) flow problem à la Beckmann and a system of PDEs of Monge-Kantorovich-type, for which we propose a $p$-Laplacian approximation. Our analysis eventually leads to a new numerical method to compute Wasserstein medians, which is based on a Douglas-Rachford scheme applied to the minimal flow formulation of the problem.

math.OC↗

arXiv2024

The exponential turnpike phenomenon for mean field game systems: weakly monotone drifts and small interactions

This article aims at quantifying the long time behavior of solutions of mean field PDE systems arising in the theory of Mean Field Games and McKean-Vlasov control. Our main contribution is to show well-posedness of the ergodic problem and the exponential turnpike property of dynamic optimizers, which implies exponential convergence to equilibrium for both optimal states and controls to their ergodic counterparts. In contrast with previous works that require some version of the Lasry-Lions monotonicity condition, our main assumption is a weak form of asymptotic monotonicity on the drift of the controlled dynamics and some basic regularity and smallness conditions on the interaction terms. Our proof strategy is probabilistic and based on the construction of contractive couplings between controlled processes and forward-backward stochastic differential equations. The flexibility of the coupling approach allows us to cover several interesting situations. For example, we do not need to restrict ourselves to compact domains and can work on the whole space $\mathbb{R}^d$, we can cover the case of non-constant diffusion coefficients and we can sometimes show turnpike estimates for the hessians of solutions to the backward equation.

math.PR↗

arXiv2021

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations

We investigate the stability of traveling-pulse solutions to the stochastic FitzHughNagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Krüger and Stannat for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications.

math.AP↗

arXiv2019

Sample Paths Estimates for Stochastic Fast-Slow Systems driven by Fractional Brownian Motion

We analyze the effect of additive fractional noise with Hurst parameter $H > \frac{1}{2}$ on fast-slow systems. Our strategy is based on sample paths estimates, similar to the approach by Berglund and Gentz in the Brownian motion case. Yet, the setting of fractional Brownian motion does not allow us to use the martingale methods from fast-slow systems with Brownian motion. We thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold. In this setting, we provide a neighborhood, tailored to the fast-slow structure of the system, that contains the process with high probability. We prove this assertion by providing exponential error estimates on the probability that the system leaves this neighborhood. We also illustrate our results in an example arising in climate modeling, where time-correlated noise processes have become of greater relevance recently.

math.PR↗