# Benjamin GessMax Planck Institute for Mathematics in the Sciences & Universität Bielefeld

Benjamin Gess

Prof. Dr.

## About

85

Publications

7,227

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

1,444

Citations

Citations since 2017

Introduction

Additional affiliations

October 2015 - present

February 2015 - October 2015

November 2013 - February 2015

## Publications

Publications (85)

We study a rescaling of the zero-range process with homogenous jump rates $g(k)=k^\alpha$ with arbitrary $\alpha\ge 1$. With a simultaneous rescaling of space, time and particle size, the rescaled empirical density is shown to converge to the porous medium equation with homogeneity $\alpha$. We further identify the dynamical large deviations from t...

We propose new limiting dynamics for stochastic gradient descent in the small learning rate regime called stochastic modified flows. These SDEs are driven by a cylindrical Brownian motion and improve the so-called stochastic modified equations by having regular diffusion coefficients and by matching the multi-point statistics. As a second contribut...

We consider the momentum stochastic gradient descent scheme (MSGD) and its continuous-in-time counterpart in the context of non-convex optimization. We show almost sure exponential convergence of the objective function value for target functions that are Lipschitz continuous and satisfy the Polyak-Lojasiewicz inequality on the relevant domain, and...

The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them are (i) a regularization by noise phenomenon for the mean curvature flow with homogeneous noise which establishes that the inclusion...

We prove optimal regularity estimates in Sobolev spaces in time and space for solutions to stochastic porous medium equations. The noise term considered here is multiplicative, white in time and coloured in space. The coefficients are assumed to be Hölder continuous and the cases of smooth coefficients of at most linear growth as well as √ u are co...

Quantitative estimates for the top Lyapunov exponents for systems of stochastic reaction-diffusion equations are proven. The treatment includes reaction potentials with degenerate minima. The proof relies on an asymptotic expansion of the invariant measure, with careful control on the resulting error terms. As a consequence of these estimates, sync...

The convergence of stochastic interacting particle systems in the mean-field limit to solutions to conservative stochastic partial differential equations is shown, with optimal rate of convergence. As a second main result, a quantitative central limit theorem for such SPDEs is derived, again with optimal rate of convergence. The results apply in pa...

The dynamics of the solutions to a class of conservative SPDEs are analysed from two perspectives: Firstly, a probabilistic construction of a corresponding random dynamical system is given for the first time. Secondly, the existence and uniqueness of invariant measures, as well as mixing for the associated Markov process is shown.

A new mechanism leading to a random version of Burgers’ equation is introduced: it is shown that the Totally Asymmetric Exclusion Process in discrete time (TASEP) can be understood as an intrinsically stochastic, non-entropic weak solution of Burgers’ equation on [Formula: see text]. In this interpretation, the appearance of randomness in the Burge...

The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but, to the best of our knowledge, none of them prove strong or weak convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with s...

We prove optimal regularity estimates in Sobolev spaces in time and space for solutions to stochastic porous medium equations. The noise term considered here is multiplicative, white in time and coloured in space. The coefficients are assumed to be H\"older continuous and the cases of smooth coefficients of at most linear growth as well as $\sqrt{u...

We prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-soli...

In micro-fluidics not only does capillarity dominate but also thermal fluctuations become important. On the level of the lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for the film height $h$ driven by space-time white noise. The gradient flow structure of its deterministic counterpart, the thin-film equatio...

In this paper we prove the well-posedness of the generalized Dean-Kawasaki equation driven by noise that is white in time and colored in space. The results treat diffusion coefficients that are only locally 1/2-H\"older continuous, including the square root. This solves several open problems, including the well-posedness of the Dean-Kawasaki equati...

Scaling limits for the weakly driven Zhang and the Bak-Tang-Wiesenfeld (BTW) model for self-organized criticality are considered. It is shown that the weakly driven Zhang model converges to a stochastic PDE with singular-degenerate diffusion. In addition, the deterministic BTW model is proved to converge to a singular-degenerate PDE. Alternatively,...

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...

We prove the path-by-path well-posedness of stochastic porous media and fast diffusion equations driven by linear, multiplicative noise. As a consequence, we obtain the existence of a random dynamical system. This solves an open problem raised in [Barbu, Röckner; JDE, 2011], [Barbu, Röckner; JDE, 2018], and [Gess; AoP, 2014].

We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) which include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully discrete numerical approximation of the considered SPDEs based on the very weak formulation. By exploiting the...

We introduce a new particle model, which we dub Active Bi-Directional Flow, conjugated to the Totally Asymmetric Exclusion Process in discrete time. We then associate to our model intrinsically stochastic, non-entropic weak solutions of Burgers' equation on R, thus linking the latter to the KPZ universality class.

We prove the existence of nonnegative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid...

In this paper, we provide a continuum model for the fluctuations of the symmetric simple exclusion process about its hydrodynamic limit. The model is based on an approximating sequence of stochastic PDEs with nonlinear, conservative noise. In the small-noise limit, we show that the fluctuations of the solutions are to first-order the same as the fl...

We prove uniform synchronization by noise with rates for the stochastic quantization equation in dimensions two and three. The proof relies on a combination of coming down from infinity estimates and the framework of order-preserving Markov semigroups derived in [O. Butkovsky and M. Scheutzow, Couplings via comparison principle and exponential ergo...

We consider the stochastic thin-film equation with colored Gaussian Stratonovich noise in one space dimension and establish the existence of nonnegative weak (martingale) solutions. The construction is based on a Trotter-Kato-type decomposition into a deterministic and a stochastic evolution, which yields an easy to implement numerical algorithm. C...

This article establishes optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise. Thereby, this work proves the optimality of the strong convergence rates for certain full-discrete approximations of stochastic Allen–Cahn equations with space-time whit...

We consider finite dimensional rough differential equations driven by centered Gaussian processes. Combining Malliavin calculus, rough paths techniques and interpolation inequalities, we establish upper bounds on the density of the corresponding solution for any fixed time $t>0$. In addition, we provide Varadhan estimates for the asymptotic behavio...

We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-α...

The existence of martingale solutions for stochastic porous media equations driven by nonlinear multiplicative space-time white noise is established in spatial dimension one. The Stroock-Varopoulos inequality is identified as a key tool in the derivation of the corresponding estimates.

We show that the initial value problem for Hamilton–Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the “skeleton” of the path, that is a piecewise linear path ob...

We identify the large deviations rate function for nonlinear diffusion equations with conservative, nonlinear white noise by proving the $\Gamma$-convergence of rate functions to approximating stochastic PDE. The limiting rate function is shown to coincide with the rate function describing the large deviations of the zero range process from its hyd...

We prove uniform synchronisation by noise with rates for the stochastic quantisation equation in dimensions two and three. The proof relies on a combination of coming down from infinity estimates and the framework of order-preserving Markov semigroups derived in [Butkovsky, Scheutzow; 2019]. In particular, it is shown that this framework can be app...

The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker–Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with common noise. The uniqueness of solutions is obtained without any higher moment assumption on the solution by means...

We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive L\'{e}vy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Lera...

The long time behaviour of solutions to stochastic porous media equations on smooth bounded domains with Dirichlet boundary data is studied. Based on weighted $L^{1}$-estimates the existence and uniqueness of invariant measures with optimal bounds on the rate of mixing are proved. Along the way the existence and uniqueness of entropy solutions is s...

We prove the pathwise well-posedness of stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. As a consequence, the generation of a random dynamical system is obtained. This extends results of the second author and Souganidis, who considered analogous spatially homogeneous and first-order equations, and earli...

We consider the stochastic thin-film equation with colored Gaussian Stratonovich noise and establish the existence of nonnegative weak (martingale) solutions. The construction is based on a Trotter-Kato-type decomposition into a deterministic and a stochastic evolution, which yields an easy to implement numerical algorithm. Compared to previous wor...

We consider the stochastic thin-film equation with colored Gaussian Stratonovich noise and establish the existence of nonnegative weak (martingale) solutions. The construction is based on a Trotter-Kato-type decomposition into a deterministic and a stochastic evolution, which yields an easy to implement numerical algorithm. Compared to previous wor...

The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker-Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with common noise. The uniqueness of solutions is obtained without any higher moment assumption on the solution by means...

We prove the local convergence to minima and estimates on the rate of convergence for the stochastic gradient descent method in the case of not necessarily globally convex nor contracting objective functions. In particular, the results are applicable to simple objective functions arising in machine learning.

We study regularizing effects of nonlinear stochastic perturbations for fully nonlinear PDE. More precisely, path-by-path L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen...

Regularity estimates in time and space for solutions to the porous medium equation are shown in the scale of Sobolev spaces. In addition, higher spatial regularity for powers of the solutions is obtained. Scaling arguments indicate that these estimates are optimal. In the linear limit, the proven regularity estimates are consistent with the optimal...

We prove the existence and uniqueness of entropy solutions for nonlinear diffusion equations with nonlinear conservative gradient noise. As particular applications our results include stochastic porous media equations, as well as the one-dimensional stochastic mean curvature flow in graph form.

Optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise are obtained. In particular, we establish the optimality of strong convergence rates for full-discrete approximations of stochastic Allen-Cahn equations with space-time white noise which have rec...

We prove the path-by-path well-posedness of stochastic porous media and fast diffusion equations driven by linear, multiplicative noise. As a consequence, we obtain the existence of a random dynamical system. This solves an open problem raised in [Barbu, R\"ockner; JDE, 2011], [Barbu, R\"ockner; JDE, 2018+], and [Gess, AoP, 2014].

We give a brief introduction and overview of the topic of regularization and well-posedness by noise for ordinary and partial differential equations. The article is an attempt to outline in a concise fashion different directions of research in this field that have attracted attention in recent years. We close the article with a look on more recent...

We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results obtained in [Gess, Souganidis; Comm. Pure Appl. Math. (2017)]. In addition, we introduce a new path-by-path s...

We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the "skeleton" of the path, that is a piecewise linear path ob...

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well - posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\Delta (|u|^{m-1}u)$ for all $m\in(1,\infty)$, and H\"older continuous diffusion nonlin...

We prove a priori estimates in $L_\infty$ for a class of quasilinear stochastic partial differential equations. The estimates are obtained independently of the ellipticity constant $\varepsilon$ and thus imply analogous estimates for degenerate quasilinear stochastic partial differential equations, such as the stochastic porous medium equation.

We prove the pathwise well-posedness of stochastic porous media equations driven by nonlinear, conservative noise. As a consequence, the generation of a random dynamical system is obtained. The noise is assumed to be regular in space and to be given by a multi-dimensional rough path in time. This extends results of the second author and Souganidis,...

The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the existence and uniqueness of entropy solutions under hypotheses on the velocity field which are weaker than those require...

We establish improved velocity averaging Lemmata with applications to non-isotropic parabolic-hyperbolic PDE. In particular, this leads to improved spatial regularity estimates for solutions to porous media equations with a force in fractional Sobolev spaces. Scaling arguments indicate that the obtained regularity is optimal and it is consistent wi...

We provide sufficient conditions for synchronization by noise, i.e. under
these conditions we prove that weak random attractors for random dynamical
systems consist of single random points. In the case of SDE with additive
noise, these conditions are also essentially necessary. In addition, we provide
sufficient conditions for the existence of a mi...

We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control...

We consider stochastic scalar conservation laws with spatially inhomogeneous flux. The regularity of the flux function with respect to its spatial variable is assumed to be low, so that entropy solutions are not necessarily unique in the corresponding deterministic scalar conservation law. We prove that perturbing the system by noise leads to well-...

We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^1$ sett...

We introduce the notion of pathwise entropy solutions for a class of degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity and fluxes with rough time dependence and prove their well-posedness. In the case of Brownian noise and periodic boundary conditions, we prove that the pathwise entropy solutions converge to their spatial av...

Ergodicity for local and nonlocal stochastic singular p-Laplace equations is proven, without restriction on the spatial dimension and for all p∈[1,2). This generalizes previous results from [Gess, Tölle; J. Math. Pures Appl., 2014], [Liu, Tölle; Electron. Commun. Probab., 2011], [Liu; J. Evol. Equations, 2009]. In particular, the results include th...

We develop a semi-discretization approximation for scalar conservation laws
with multiple rough time dependence in inhomogeneous fluxes. The method is
based on Brenier's transport-collapse algorithm and uses characteristics
defined in the setting of rough paths. We prove strong $L^1$-convergence for
inhomogeneous fluxes and provide a rate of conver...

We study Brownian flows on manifolds for which the associated Markov process
is strongly mixing with respect to an invariant probability measure and for
which the distance process for each pair of trajectories is a diffusion $r$. We
provide a sufficient condition on the boundary behavior of $r$ at $0$ which
guarantees that the statistical equilibri...

We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46–57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance o...

We provide a general framework for the stability of solutions to stochastic
partial differential equations with respect to perturbations of the drift. More
precisely, we consider stochastic partial differential equations with drift
given as the subdifferential of a convex function and prove continuous
dependence of the solutions with regard to rand...

We provide sufficient conditions for weak synchronization by noise for
order-preserving random dynamical systems on Polish spaces. That is, under
these conditions we prove the existence of a weak point attractor consisting of
a single random point. This generalizes previous results in two directions:
First, we do not restrict to Banach spaces and s...

We consider singular-degenerate, multivalued stochastic fast diffusion
equations with multiplicative Lipschitz continuous noise. In particular, this
includes the stochastic sign fast diffusion equation arising from the
Bak-Tang-Wiesenfeld model for self-organized criticality. A well-posedness
framework based on stochastic variational inequalities (...

We study the long-time behavior and the regularity of pathwise entropy
solutions to stochastic scalar conservation laws with random in time spatially
homogeneous fluxes and periodic initial data. We prove that the solutions
converge to their spatial average, which is the unique invariant measure, and
provide a rate of convergence, the latter being...

We study pathwise entropy solutions for scalar conservation laws with
inhomogeneous fluxes and quasilinear multiplicative rough path dependence. This
extends the previous work of Lions, Perthame and Souganidis who considered
spatially independent and inhomogeneous fluxes with multiple paths and a single
driving singular path respectively. The appro...

We provide an abstract variational existence and uniqueness result for
multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert
spaces with general additive and Wiener multiplicative noise. As examples we
discuss certain singular diffusion equations such as the stochastic 1-Laplacian
evolution (total variation flow) in all sp...

The regularity and characterization of solutions to degenerate, quasilinear
SPDE is studied. Our results are two-fold: First, we prove regularity results
for solutions to certain degenerate, quasilinear SPDE driven by Lipschitz
continuous noise. In particular, this provides a characterization of solutions
to such SPDE in terms of (generalized) stro...

We prove the existence and uniqueness of solutions to a class of stochastic
scalar conservation laws with joint space-time transport noise and
affine-linear noise driven by a geometric p-rough path. In particular,
stability of the solutions with respect to the driving rough path is obtained,
leading to a robust approach to stochastic scalar conserv...

We prove finite time extinction for stochastic sign fast diffusion equations
driven by linear multiplicative space-time noise, corresponding to the
Bak-Tang-Wiesenfeld model for self-organized criticality. This solves a problem
posed and left open in several works: [Barbu, Da Prato, R\"ockner, CMP, 2009],
[Barbu, Da Prato, R\"ockner, CRMAS, 2009],...

The existence of random attractors for singular stochastic evolution equations (SEE) perturbed by general additive noise is proven. The drift is only assumed to satisfy the standard assumptions of the variational approach to SEE with compact embeddings in the Gelfand triple and singular coercivity. For ergodic, monotone, contractive random dynamica...

We discuss stochastic calculus for large classes of Gaussian processes, based
on rough path analysis. Our key condition is a covariance measure structure
combined with a classical criterion due to Jain-Monrad [Ann. Probab. (1983)].
This condition is verified in many examples, even in absence of explicit
expressions for the covariance or Volterra ke...

It is a well-known fact that finite rho-variation of the covariance (in 2D
sense) of a general Gaussian process implies finite rho-variation of
Cameron-Martin paths. In the special case of fractional Brownian motion (think:
2H=1/rho), in the rougher than Brownian regime, a sharper result holds thanks
to a Besov-type embedding [Friz-Victoir, JFA, 20...

We present sufficient conditions for finite controlled rho-variation of the
covariance of Gaussian processes with stationary increments, based on concavity
or convexity of their variance function. The motivation for this type of
conditions comes from recent work of Hairer [CPAM,2011]. Our results allow to
construct rough paths lifts of solutions to...

We prove finite speed of propagation for stochastic porous media equations
perturbed by linear multiplicative space-time rough signals. Explicit and
optimal estimates for the speed of propagation are given. The result applies to
any continuous driving signal, thus including fractional Brownian motion for
all Hurst parameters. The explicit estimates...

We prove the existence of random attractors for a large class of degenerate
stochastic partial differential equations (SPDE) perturbed by joint additive
Wiener noise and real, linear multiplicative Brownian noise, assuming only the
standard assumptions of the variational approach to SPDE with compact
embeddings in the associated Gelfand triple. Thi...

The existence of random attractors for singular stochastic partial
differential equations (SPDE) perturbed by general additive noise is proven.
The drift is assumed only to satisfy the standard assumptions of the
variational approach to SPDE with compact embeddings in the Gelfand triple and
singular coercivity. For ergodic, monotone, contractive ra...

Unique existence of solutions to porous media equations driven by continuous
linear multiplicative space-time rough signals is proven for initial data in
$L^1(\mathcal{O})$ on bounded domains $\mathcal{O}$. The generation of a
continuous, order-preserving random dynamical system (RDS) on
$L^1(\mathcal{O})$ and the existence of a random attractor fo...

The existence of random attractors for a large class of stochastic partial differential equations (SPDE) driven by general additive noise is established. The main results are applied to various types of SPDE, as e.g. stochastic reaction–diffusion equations, the stochastic p-Laplace equation and stochastic porous media equations. Besides classical B...

Unique existence of analytically strong solutions to stochastic partial
differential equations (SPDE) with drift given by the subdifferential of a
quasi-convex function and with general multiplicative noise is proven. The
proof applies a genuinely new method of weighted Galerkin approximations based
on the "distance" defined by the quasi-convex fun...

We prove new $L^2$-estimates and regularity results for generalized porous media equations "shifted by" a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of "$\zeta$-monotonicity" for the non-linear function in the equation. As a consequence we prove that stochastic porous medi...