arXiv subjects

Benjamin Gess

Publications and source records attributed to Benjamin Gess.

At least 19 recordsLinked to original sources

arXiv2026

Long-time behaviour and bifurcation analysis of a non-reciprocal two-species aggregation-diffusion system on the torus

A two-species aggregation-diffusion system on the one-dimensional torus with antisymmetric cross-interactions modeling pursuit-evasion dynamics is studied. The self-interaction strength is used as a bifurcation parameter, and the linear stability of the homogeneous state is characterized by Fourier analysis. It is shown that non-reciprocal coupling enlarges the stability region. In a symmetric diffusion setting, exponential convergence to the homogeneous state is established for sufficiently weak self-interactions by means of an entropy argument, with a decay condition independent of the cross-interaction strength. Under suitable nondegeneracy and spectral assumptions, the first loss of stability is shown to occur through a Hopf bifurcation associated with a single spatial Fourier mode. A branch of time-periodic patterns is thereby obtained, along which the two species oscillate with a fixed phase difference. An explicit formula for the cubic Hopf coefficient is derived, and an essential dependence of the direction and stability of the bifurcating branch on the second spatial harmonic is identified. In the case of matching diffusion coefficients and interaction kernels, sufficiently strong non-reciprocity is shown to transform a subcritical stationary instability of the corresponding reciprocal system into a supercritical Hopf bifurcation with stable periodic solutions. An exchange-of-stability result is proved, and the theory is illustrated for Kuramoto, two-mode cosine, and top-hat interaction kernels.

math.AP↗

arXiv2026

Dirichlet form approach to stochastic thin-film equations

The thin-film equation describes the evolution of a thin liquid film driven by capillarity and limited by viscosity. For sufficiently small film heights, thermal fluctuations may not be neglected. Since the thin-film equation as an overdamped evolution equation has a gradient flow structure, the informal implementation of the fluctuation-dissipation principle is straightforward. In this paper, we work out that the concept of Dirichlet forms and the related construction of Markov processes are tailor-made for this type of overdamped continuum evolution with thermal noise. However, the concept of Dirichlet forms relies on the availability of a Gibbsian equilibrium measure, which, in case of the thin-film equation, limits the treatment to the $(1+1)$-dimensional case. By means of Dirichlet form theory, we obtain the existence and uniqueness of the evolution of laws on configuration space. We show that the regularity of the realizations of this Markov process matches the heuristics. In a simplified situation with a Gaussian reference measure and a non-degenerate mobility function, we characterize it as what is called an energy solution. The latter provides a pathwise decomposition of the driving forces into a deterministic and a thermal one, which was the starting point. Applying the same techniques in the spatially discretized setting proposed in [Math. Comp. 92 (2023), 1931--1976], we obtain the same results uniformly in the discretization parameter. In particular, Galerkin as well as spectral approximations of the stochastic thin-film equation are shown to be uniformly tight in optimal topologies. In the simplified setting, we also prove Mosco convergence of the associated Dirichlet forms and thereby obtain convergence of these approximations.

math.PR↗

arXiv2026

Non-local to local limit for stochastic conservation laws

This work proves the convergence of stochastic nonlocal conservation laws, with spatially inhomogeneous noise, to stochastic conservation laws when the range of particle interaction vanishes. This provides rigorous justification for a class of stochastic conservation laws that has received considerable attention in the last decade. To handle technical challenges such as gradient-dependent noise, modified versions of the equations with specific cutoff boundaries are utilized and new uniform estimates to ensure stability during the limiting process are established. In addition, the work demonstrates that this convergence depends on the presence of a viscous term, as a counterexample shows that without it, the limiting solution may fail to be an entropy solution.

math.PR↗

arXiv2026

Higher-Order Gaussian Approximations on Path Space for the Simple Symmetric Exclusion Process

Quantitative Gaussian approximation estimates on path space are established for the nonequilibrium density fluctuations of the simple symmetric exclusion process (SSEP) on a $d$-dimensional discrete torus of mesh size $1/n$, started from a product measure with a smooth density profile. A novel $n$-dependent modified Ornstein-Uhlenbeck process is constructed that incorporates higher-order lattice information while remaining linear and Gaussian. It is shown that, for sufficiently high modification orders, this modification yields quantitative Gaussian approximation error bounds of order $n^{-\min\{d/2,2\}}$ for a class of smooth functionals of the entire fluctuation trajectory. In addition, it is shown that retaining an explicit microscopic correlation correction leaves a remainder of order $n^{-d/2}$, thereby distinguishing contributions to the bound due to correlations from those due to spatial approximation. For nonconstant initial profiles in dimensions $d\ge3$, a separate argument yields matching upper and lower bounds of order $n^{-2}$ for the direct weak error of suitable quadratic test functionals, provided that the modification orders are sufficiently high. The proof is based on a novel path-dependent Trotter formula. Its derivation builds on the introduction of a concept of horizontal absolute continuity for non-anticipative functionals and a corresponding martingale formula for Markov jump processes. This formula is subsequently combined with an integral backward Kolmogorov equation for Gaussian continuation functionals. A second ingredient of the proof is a refinement of the exchangeable-pair argument introduced by the authors in a previous work, thereby improving the central-limit rate $n^{-\min\{d/2,1\}}$ to $n^{-d/2}$ for the initial empirical fluctuations.

math.PR↗

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

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 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

The Porous Medium Equation: Multiscale Integrability in Large Deviations

We consider a zero-range process $η^N_t(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation $\partial_t u=\frac12Δu^α, α>1$. As a main result we obtain a large deviation principle in any scaling regime of vanishing particle size $χ_N\to 0$. The key challenge is to develop uniform integrability estimate on the nonlinearity $(η^N(x))^α$ 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↗

arXiv2025

Stabilization by transport noise and enhanced dissipation in the Kraichnan model

Stabilization and sufficient conditions for mixing by stochastic transport are shown. More precisely, given a second order linear operator with possibly unstable eigenvalues on a smooth compact Riemannian manifold, it is shown that the inclusion of transport noise can imply global asymptotic stability. Moreover, it is shown that an arbitrary large exponential rate of convergence can be reached, implying enhanced dissipation. The sufficient conditions are shown to be satisfied by the so-called Kraichnan model for stochastic transport of passive scalars in turbulent fluids. In addition, an example is given showing that it can be sufficient to force four modes in order to induce stabilization.

math.PR↗

arXiv2025

The Porous Medium Equation: Large Deviations and Gradient Flow with Degenerate and Unbounded Diffusion

The problem of deriving a gradient flow structure for the porous medium equation which is {\em thermodynamic}, in that it arises from the large deviations of some microscopic particle system, is studied. To this end, a rescaled zero-range process with jump rate $g(k)=k^α, α>1$ is considered, and its hydrodynamic limit and dynamical large deviations are shown in the presence of both degenerate and unbounded diffusion. The key superexponential estimate is obtained using pathwise discretised regularity estimates in the spirit of the Aubin-Lions-Simons lemma. This allows to exhibit the porous medium equation as the gradient flow of the entropy in a thermodynamic metric via the energy-dissipation inequality.

math.PR↗

arXiv2025

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↗