We investigate the stability of traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a stochastic partial differential equation (SPDE) coupled to an ordinary differential equation (ODE). This approach has been employed by Krüger and Stannat [46] 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 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. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs that are not self-adjoint. This provides a major generalization as these systems are prevalent in many applications.

We analyze the effect of additive fractional noise with Hurst parameter H > 1 2 on fastslow 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.

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 Wasserstein 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 Wasserstein constrained interpolation problems.

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced in [5] as a regularization of Wasserstein barycenters [1]. 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.

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