arXiv subjects

Benjamin Gess

Publications and source records attributed to Benjamin Gess.

At least 19 recordsLinked to original sources

arXiv2026

Weak synchronisation for McKean--Vlasov SDEs

Synchronisation by noise for McKean--Vlasov stochastic differential equations is investigated. A transfer principle is introduced by which synchronisation by noise and diagonal mixing can be transferred from an associated limiting frozen-diffusion SDE to a genuinely law-dependent McKean--Vlasov SDE. The usefulness of this principle is demonstrated through an application to ensemble Kalman sampling, thereby providing its first application in a sampling context. Synchronisation for SDEs with multiplicative noise is then revisited from the perspective of sampling. Existing general frameworks providing sufficient conditions for synchronisation by noise are refined and extended to a control-oriented setting on noncompact state spaces, and coefficient-level conditions are derived by which these criteria can be verified. Motivated by extrapolation schemes in sampling, the freedom in the construction of couplings for a fixed sampler is emphasised, together with the advantage of choosing couplings that exhibit favourable synchronisation properties.

math.PR↗

arXiv2026

The Advective Fisher-Rao Geometry of Deterministic Measure Transport

A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin--Wentzell large deviation rate functional, and as the Hessian of the Benamou--Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss--Newton method yields optimal fitting of velocity fields.

math.OC↗

arXiv2026

The Incompressible Navier--Stokes--Fourier System with Thermal Noise

We establish a solution theory for the incompressible Navier--Stokes--Fourier system with thermal noise, posed on the three-dimensional torus. While in the incompressible deterministic setting the equation for the velocity can be solved independently of the temperature, the inclusion of the effects of thermal fluctuations by means of the GENERIC framework leads to a nonlinear gradient noise term, which couples the dynamics of both variables. Therefore, the analysis poses new challenges, which are absent in the deterministic incompressible Navier--Stokes--Fourier equations. In particular, the a priori estimates used in the deterministic setting are not readily generalizable, the noise introduces strongly nonlinear gradient terms and the total energy lacks convexity. These challenges are overcome in the present work by a novel variable transformation, and novel entropy dissipation estimates. Thereby, the existence of global-in-time weak solutions for $L_x^2$ initial data, the existence of local-in-time strong solutions for regular initial data, and weak-strong uniqueness are obtained.

math.PR↗

arXiv2026

Large Spikes in Stochastic Gradient Descent: A Large-Deviations View

Large loss spikes in stochastic gradient descent are studied through a rigorous large-deviations analysis for a shallow, fully connected network in the NTK scaling. In contrast to full-batch gradient descent, the catapult phase is shown to split into inflationary and deflationary regimes, determined by an explicit log-drift criterion. In both cases, large spikes are shown to be at least polynomially likely. In addition, these spikes are shown to be the dominant mechanism by which sharp minima are escaped and curvature is reduced, thereby favouring flatter solutions. Corresponding results are also obtained for certain ReLU networks, and implications for curriculum learning are derived.

cs.LG↗

arXiv2026

The Porous Medium Equation: Multiscale Integrability in Large Deviations

We consider a zero-range process $\eta^N_t(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation $\partial_t u=\frac12\Delta u^\alpha, \alpha>1$. As a main result we obtain a large deviation principle in any scaling regime of vanishing particle size $\chi_N\to 0$. The key challenge is to develop uniform integrability estimate on the nonlinearity $(\eta^N(x))^\alpha$ in a situation where neither pathwise regularity nor Dirichlet-form based regularity is readily available. We resolve this by introducing a novel multiscale argument exploiting the appearance of pathwise regularity across scales.

math.PR↗

arXiv2026

Ergodicity for SPDEs driven by divergence-free transport noise

We study the ergodic behaviour of the McKean-Vlasov equations driven by common, divergence-free transport noise. In particular, we show that in dimension $d\geq 2$, if the noise is mixing and sufficiently strong it can enforce the uniqueness of invariant probability measures, even if the deterministic part of equation has multiple steady states.

math.PR↗

arXiv2026

Probabilistically Strong Solutions to Stochastic Euler Equations

In this paper, we establish the existence of probabilistically strong, measure-valued solutions for the stochastic incompressible Navier--Stokes equations and prove their convergence, in the vanishing viscosity limit, to probabilistically strong solutions for the stochastic incompressible Euler equations. In particular, this solves the open problem of constructing probabilistically strong solutions for the stochastic Euler equations that satisfy the energy inequality for general $L^2$ initial data. We introduce the concept of energy-variational solutions in the stochastic context in order to treat the nonlinearities without changing the probability space. Furthermore, we extend these results to fluids driven by transport noise.

math.AP↗

arXiv2025

THINNs: Thermodynamically Informed Neural Networks

Physics-Informed Neural Networks (PINNs) are a class of deep learning models aiming to approximate solutions of PDEs by training neural networks to minimize the residual of the equation. Focusing on non-equilibrium fluctuating systems, we propose a physically informed choice of penalization that is consistent with the underlying fluctuation structure, as characterized by a large deviations principle. This approach yields a novel formulation of PINNs in which the penalty term is chosen to penalize improbable deviations, rather than being selected heuristically. The resulting thermodynamically consistent extension of PINNs, termed THINNs, is subsequently analyzed by establishing analytical a posteriori estimates, and providing empirical comparisons to established penalization strategies.

cs.LG↗

arXiv2025

Matching Large Deviation Bounds of the Zero-Range Process in the whole space

We consider the large deviations of the hydrodynamic rescaling of the zero-range process on $\mathbb{Z}^d$ in any dimension $d\ge 1$. Under mild and canonical hypotheses on the local jump rate, we obtain matching upper and lower bounds, thus resolving the problem opened by \cite{KL99}. On the probabilistic side, we extend the superexponential estimate to any dimension, and prove the superexponential concentration on paths with finite entropy dissipation. In addition, we extend the theory of the parabolic-hyperbolic skeleton equation to the whole space, and remove global convexity/concavity assumptions on the nonlinearity.

math.PR↗

arXiv2025

Random dynamical systems for McKean--Vlasov SDEs via rough path theory

The existence of random dynamical systems for McKean--Vlasov SDEs is established. This is approached by considering the joint dynamics of the corresponding nonlinear Fokker-Planck equation governing the law of the system and the underlying stochastic differential equation (SDE) as a dynamical system on the product space $\RR^d \times \mathcal{P}(\RR^d)$. The proof relies on two main ingredients: At the level of the SDE, a pathwise rough path-based solution theory for SDEs with time-dependent coefficients is implemented, while at the level of the PDE a well-posedness theory is developed, for measurable solutions and allowing for degenerate diffusion coefficients. The results apply in particular to the so-called ensemble Kalman sampler (EKS), proving the existence of an associated RDS under some assumptions on the posterior, as well as to the Lagrangian formulation of the Landau equation with Maxwell molecules. As a by-product of the main results, the uniqueness of solutions non-linear Fokker--Planck equations associated to the EKS is shown.

math.PR↗

arXiv2025

Existence of martingale solutions to a stochastic kinetic model of chemotaxis

We show the existence of local and global in time weak martingale solutions for a stochastic version of the Othmer-Dunbar-Alt kinetic model of chemotaxis under suitable assumptions on the turning kernel and stochastic drift coefficients, using dispersion and stochastic Strichartz estimates. The analysis is based on new Strichartz estimates for stochastic kinetic transport. The derivation of these estimates involves a local in time dispersion analysis using properties of stochastic flows, and a time-splitting argument to extend the local in time results to arbitrary time intervals.

math.AP↗

arXiv2024

Conservative stochastic PDEs on the whole space

The purpose of this paper is to establish a well-posedness theory for conservative stochastic partial differential equations on the whole space. This class of stochastic PDEs arises in fluctuating hydrodynamics, and includes the Dean--Kawasaki equation with correlated noise. In combination with the analysis of the authors and Heydecker [35], the connection between fluctuating hydrodynamics and macroscopic fluctuation theory in the context of the zero range particle process is made rigorous.

math.PR↗

arXiv2024

A quantitative central limit theorem for the simple symmetric exclusion process

A quantitative central limit theorem for the simple symmetric exclusion process (SSEP) on a $d$-dimensional discrete torus is proven. The argument is based on a comparison of the generators of the density fluctuation field of the SSEP and the generalized Ornstein-Uhlenbeck process, as well as on an infinite-dimensional Berry-Essen bound for the initial particle fluctuations. The obtained rate of convergence is optimal.

math.PR↗

arXiv2024

Higher Order Fluctuation Expansions for Nonlinear Stochastic Heat Equations in Singular Limits

Higher order fluctuation expansions for stochastic heat equations (SHE) with nonlinear, non-conservative and conservative noise are obtained. These Edgeworth-type expansions describe the asymptotic behavior of solutions in suitable joint scaling regimes of small noise intensity and diverging singularity. The results include both the case of the SHE with regular and irregular diffusion coefficients. In particular, this includes the correlated Dawson-Watanabe and Dean-Kawasaki SPDEs, as well as SPDEs corresponding to the Fleming-Viot and symmetric simple exclusion processes.

math.PR↗

arXiv2024

Low temperature expansion for the Euclidean $\Phi^4_2$-measure

We study asymptotic expansions of the Euclidean $\Phi^4_2$-measure in the low-temperature regime. In particular, this extends the asymptotic expansions of Gaussian function space integrals developed in Schilder (1966) and Ellis and Rosen (1982) to the singular setting, where the field is no longer a function, but just a distribution. As a consequence, we deduce limit theorems, specifically the law of large numbers and the central limit theorem for the $\Phi^4_2$-measure in the low-temperature limit.

math.PR↗

arXiv2024

SVI solutions to stochastic nonlinear diffusion equations on general measure spaces

We establish a framework for the existence and uniqueness of solutions to stochastic nonlinear (possibly multi-valued) diffusion equations driven by multiplicative noise, with the drift operator $L$ being the generator of a transient Dirichlet form on a finite measure space $(E,\mathcal{B},\mu)$ and the initial value in $\mathcal{F}_e^*$, which is the dual space of an extended transient Dirichlet space. $L$ and $\mathcal{F}_e^*$ replace the Laplace operator $\Delta$ and $H^{-1}$, respectively, in the classical case. This framework includes stochastic fast diffusion equations, stochastic fractional fast diffusion equations, the Zhang model, and apply to cases with $E$ being a manifold, a fractal or a graph. In addition, our results apply to operators $-f(-L)$, where $f$ is a Bernstein function, e.g. $f(\lambda)=\lambda^\alpha$ or $f(\lambda)=(\lambda+1)^\alpha-1$, $0<\alpha<1$.

math.PR↗

arXiv2024

Stochastic Modified Flows for Riemannian Stochastic Gradient Descent

We give quantitative estimates for the rate of convergence of Riemannian stochastic gradient descent (RSGD) to Riemannian gradient flow and to a diffusion process, the so-called Riemannian stochastic modified flow (RSMF). Using tools from stochastic differential geometry we show that, in the small learning rate regime, RSGD can be approximated by the solution to the RSMF driven by an infinite-dimensional Wiener process. The RSMF accounts for the random fluctuations of RSGD and, thereby, increases the order of approximation compared to the deterministic Riemannian gradient flow. The RSGD is build using the concept of a retraction map, that is, a cost efficient approximation of the exponential map, and we prove quantitative bounds for the weak error of the diffusion approximation under assumptions on the retraction map, the geometry of the manifold, and the random estimators of the gradient.

cs.LG↗

arXiv2023

Optimal Regularity in Time and Space for Nonlocal Porous Medium Type Equations

A broad class of possibly non-unique generalized kinetic solutions to hyperbolic-parabolic PDEs is introduced. Optimal regularity estimates in time and space for such solutions to nonlocal, and spatially inhomogeneous variants of the porous medium equation are shown in the scale of Sobolev spaces. The optimality of these results is shown by comparison to the nonlocal Barenblatt solution. The regularity results are used in order to obtain existence of generalized kinetic solutions.

math.AP↗