Special Session 168: Stochastic Analysis and Large Scale Interacting Systems

Numerical Approximation of the stochastic Cahn-Hilliard equation with singular potential

Lubomir Banas Bielefeld University Germany Co-Author(s):    Stefan Metzger

Abstract: We discuss the numerical approximation of the stochastic Cahn-Hilliard equation with a singular double-obstacle potential and multiplicative conservative noise. We propose a regularized fully discrete finite element approximation scheme for the problem and show that is satisfies stability estimates which are uniform with respect to the discretisation parameters. We show convergence of the approximation for vanishing discretisation parameters towards a regularised version of the singular stochastic Cahn-Hilliard equation by monotonicity arguments. Owing to a uniform $H^1$-estimate for the regularised problem we then establish convergence of the regularised solution to the probabilistically strong solution of the stochastic Cahn-Hilliard equation with double-obstacle potential. We also present numerical simulations where we compare the regularised numerical approximation to its unregularised counterpart and illustrate the effect of the conservative noise.

Existence and Uniqueness of a 3D Nonlinear Stochastic Thermolastic Model with Nonlinear Damping

Hakima Bessaih Florida Internaational University USA Co-Author(s):    Mogtaba Mohammed

Abstract: We study a three-dimensional stochastic thermoelastic system with nonlinear damping and forcing. The model couples a nonlinear elasticity equation with a heat equation driven by additive noise. Using compactness, variational, and monotonicity methods, we prove the existence of weak martingale solutions. Under additional structural assumptions on the damping and noise, we establish pathwise uniqueness. These results contribute to the analysis of stochastic thermoelastic systems with nonlinear dissipation

Homogenisation of a Passive Scalar Transported by Locally Supported White Noise

Federico Butori Scuola Normale Superiore, Pisa Italy Co-Author(s):    Mayorcas, Avi and Morlacchi, Silvia

Abstract: Transport-type stochastic perturbations are a common way of representing turbulent effects in fluid dynamics models. A recent research thread on the topic is the so-called \emph{It\^o-Stratonovich diffusion limit}. By selecting Stratonovich transport noise with carefully arranged coefficients, one can show that the solution of certain SPDEs are close, in an appropriate topology, to a deterministic equation with an effective second order elliptic operator, linked to the Ito-Stratonovich corrector. In this talk, we deal with viscous a passive scalar model. Starting from [Flandoli \emph{et al.}, 2022, \emph{Philos. Trans. Roy. Soc. A}, 380(2219)], we consider a transport noise made by a sum of independent and compactly supported vector fields. Due to the anisotropic nature of the noise, the identification of the limit equation is not straightforward as in other examples, as the Ito-Stratonovich corrector is a generic second order elliptic operator with non-constant coefficients. Using tools from Homogenisation theory, we obtain a representation for the limiting effective diffusivity matrix. Exploiting this representation, we study both analytically and numerically asymptotics, in the zero-viscosity regime, of the effective diffusivity across a number of vector field regimes parametrised by the radius of their support.

Invariant Measures for Stochastic Fluid Flows in Bounded Domains with Permeable BoundariesAbstract

Fernanda Cipriano NOVA University Lisbon Portugal Co-Author(s):

Abstract: This talk is devoted to the study of the well-posedness and the existence of invariant measures for a class of incompressible viscous fluids filling a 2D bounded domain with permeable boundaries, under the action of random forces.

Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line

Robert Dalang Ecole Polytechnique Federale de Lausanne (EPFL) Switzerland Co-Author(s):    Fei Pu (Beijing) & David Nualart (Kansas)

Abstract: For Gaussian random fields with values in $\R^d$, sharp upper and lower bounds on the probability of hitting a fixed set have been available for many years. These apply in particular to the solutions of systems of linear SPDEs. For non-Gaussian random fields, the available bounds are less sharp. For systems of stochastic heat equations, a sharp lower bound was obtained in [R.C. Dalang and F. Pu, Optimal lower bounds on hitting probabilities for stochastic heat equations in spatial dimension k \geq 1. Electron. J. Probab. 25 (2020), Paper No. 40, 31 pp]. Here, we obtain the corresponding sharp upper bound.

Stochastic diffuse interface models driven by conservative noise

Andrea Di Primio Scuola Normale Superiore Italy Co-Author(s):

Abstract: Diffuse interface models have attracted significant interest starting with the pioneering work of Cahn and Hilliard in 1958. In this talk, I will introduce and discuss examples of systems of Allen--Cahn and Cahn--Hilliard type perturbed by conservative noise, i.e., keeping the key mass conservation property, enjoyed by the deterministic counterparts of the systems, pathwise. The systems are endowed with a singular potential, as prescribed by the thermodynamical derivation of the model. In particular, results on existence and uniqueness of martingale and/or probabilistically-strong solutions are shown. The talk is based on joint works with Maurizio Grasselli (Politecnico di Milano), Andrea Papini (Chalmers University of Technology), Luca Scarpa (Politecnico di Milano) and Margherita Zanella (Politecnico di Milano).

The vanishing latent heat limit of a stochastic Stefan problem : An error estimate

Perla El Kettani University of Toulon France Co-Author(s):    Ioana Ciotir, Dan Goreac, Danielle Hilhorst.

Abstract: We extend an article by Hilhorst, Mimura and Schatzle about the limit as the latent heat coefficient tends to zero of a two-phase Stefan problem arising in biology. We introduce a one-dimensional additive white noise in time, and search for the limit of the solution of the corresponding stochastic Stefan problem as the latent heat coefficient vanishes. We first prove the existence and uniqueness of the weak solution of this problem, and then study the limit of the solution as the latent heat coefficient tends to zero. Our method of proof is based upon an error estimate between the solution of the Stefan problem with positive latent heat and the one of the Stefan problem with zero latent heat, which seems to be novel even in the deterministic case when no noise is added.

Optimal Regularity and Stability for Numerical Schemes of SPDEs via functional calculus

Foivos F Evangelopoulos-Ntemiris TU DELFT Netherlands Co-Author(s):

Abstract: The $H^\infty$-calculus provides a powerful framework for establishing well-posedness and optimal regularity results for stochastic partial differential equations (SPDEs). In particular, it is the cornerstone for proving stochastic maximal regularity, a crucial tool for handling non-linear SPDEs via linearization techniques. While this continuous theory is well-established, its applications to numerical analysis---specifically for deriving sharp stability and convergence rates for numerical schemes---have only recently begun to emerge. In this talk, I will present a framework for establishing a priori regularity estimates for semi-discrete numerical schemes of linear SPDEs, which subsequently enable the convergence analysis of non-linear problems. The core of this approach relies on proving the uniform boundedness of the $H^\infty$-calculus for the finite element discretizations $\{A_h \colon h>0\}$ of a second-order elliptic operator $A$.

Kolmogorov equations for stochastic Volterra processes with singular kernels

Ioannis Gasteratos TU Berlin Germany Co-Author(s):    Alexandre Pannier

Abstract: We associate backward and forward Kolmogorov equations to a class of fully nonlinear Stochastic Volterra Equations (SVEs) with convolution kernels $K$ that are singular at the origin. Working on a carefully chosen Hilbert space $\mathcal{H}_1$, we rigorously establish a link between solutions of SVEs and Markovian mild solutions of a Stochastic Partial Differential Equation (SPDE) of transport-type. Then, we obtain two novel It\^o formulae for functionals of mild solutions and, as a byproduct, show that their laws solve corresponding Fokker-Planck equations. Finally, we introduce a natural notion of singular directional derivatives along $K$ and prove that (conditional) expectations of SVE solutions can be expressed in terms of the unique solution to a backward Kolmogorov equation on $\mathcal{H}_1$. Our analysis relies on stochastic calculus in Hilbert spaces, the reproducing kernel property of the state space $\mathcal{H}_1$, as well as crucial invariance and smoothing properties that are specific to the SPDEs of interest. In the special case of singular power-law kernels, our conditions guarantee well-posedness of the backward equation either for all values of the Hurst parameter $H$, when the noise is additive, or for all $H>1/4$ when the noise is multiplicative.

Large Deviations for the Porous Medium Equation via Multiscale Integrability

Benjamin Gess TU Berlin, MPI MiS Leipzig Germany Co-Author(s):

Abstract: This talk investigates a large deviation principle for a class of zero-range particle systems whose macroscopic behavior is governed by the porous medium equation. The result applies in the full small-particle regime and extends earlier work that required stronger restrictions on the scaling. The main difficulty is that the model combines degenerate diffusion with superlinear growth, so the standard methods used in hydrodynamic large deviations, like two-block estimates, do not apply. The key new idea is a multiscale approach that recovers enough effective regularity at intermediate scales to obtain the necessary superexponential and uniform integrability estimates. In this talk, I will describe the particle model, explain where the usual replacement arguments break down, and show how coarse-graining and multiscale integrability make it possible to obtain the large deviation principle in complete generality.

Solitary Waves in a Stochastic Parametrically Forced Nonlinear Schr\odinger Equation

Manuel Gnann TU Delft Netherlands Co-Author(s):    Rik Westdorp and Joris van Winden

Abstract: We study a parametrically forced nonlinear Schr\odinger (PFNLS) equation, driven by multiplicative translation-invariant noise. We show that a solitary wave in the stochastic equation is orbitally stable on a timescale which is exponential in the inverse square of the noise strength. We give explicit expressions for the phase shift and fluctuations around the shifted wave which are accurate to second order in the noise strength. This is done by developing a new perspective on the phase-lag method introduced by Kr\uger and Stannat.

Stochastic Maximum Principle for Delay Equations: the Non-Convex Case

giuseppina guatteri Politecnico di Milano Italy Co-Author(s):    Federica Masiero

Abstract: This talk is based on joint work with Federica Masiero. We discuss a stochastic maximum principle for controlled stochastic differential equations with delay in the state, control-dependent noise, and non-convex control sets. The cost functional may depend on both the present state and its weighted past through general finite measures. In the regular case, where the delay measures admit square-integrable densities, the problem can be reformulated as an infinite-dimensional stochastic evolution equation in a Hilbert space, and necessary optimality conditions are derived through spike variation and first- and second-order adjoint equations. For general finite measures, including pointwise delays, the analysis combines anticipated backward stochastic differential equations with an approximation procedure for the second-order term. This yields a stochastic maximum principle beyond the convex setting and clarifies the role of infinite-dimensional methods in delay control problems.

Nonlinear stochastic PDEs in biochemical systems

Erika Hausenblas Technical University of Leoben Austria Co-Author(s):    Deboprija Mukherjee

Abstract: Nonlinear partial differential equations (PDEs) arise naturally in many biological and chemical systems, particularly in the context of cross-diffusion systems such as chemotaxis. Additionally, random fluctuations are prevalent in the real world, and this randomness can lead to various new phenomena that significantly impact the behavior of the solutions. The introduction of a stochastic term (or noise) in the model often results in qualitatively new behaviors, enhancing our understanding of real processes and often making them more realistic. The interplay between noise and nonlinearity can give rise to effects such as noise-induced transitions, stochastic resonance, metastability, or even noise-induced chaos. In the case of simple Keller-Segel models, a dichotomy exists based on the initial mass; depending on this mass, a blow-up phenomenon may occur. However, it is possible to suppress this blow-up by introducing an additional term. When incorporating some multiplicative noise, the mass may change over time. In this scenario, it is also possible to demonstrate the existence of a global solution if a proliferation term or a porous media term is included. In the talk, we will introduce some Keller-Segel systems for which a global solution exists even in the two-dimensional case.

Fernique-type bounds for BPHZ models and their applications

Masato Hoshino Institute of Science Tokyo Japan Co-Author(s):    Ismael Bailleul and Ryoji Takano

Abstract: For general locally subcritical parabolic singular SPDEs without variance blow-up, we prove that the BPHZ model satisfies a Fernique-type theorem, namely exponential square integrability, whenever the driving noise is stationary and satisfies a suitable Poincare inequality. As applications, we obtain two consequences. First, if the SPDE admits appropriate a priori estimates, then its solution satisfies a corresponding concentration inequality. Second, we show that the BPHZ model satisfies a Schilder-type large deviation principle, and that the solution to the SPDE satisfies a Freidlin-Wentzell-type large deviation principle, extending the result of Hairer and Weber (2015).

Fractional Diffusion Bridges

Yuzuru Inahama Kyushu University Japan Co-Author(s):

Abstract: Consider ``stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/4, 1)$. Their solutions are sometimes called fractional diffusion processes. The main purpose of this talk is conditioning these processes to reach a given terminal point. We call the conditioned processes fractional diffusion bridges. Our main tool for mathematically rigorous conditioning is quasi-sure analysis, which is a potential theoretic part of Malliavin calculus. We also prove a small-noise large deviation principle of Freidlin-Wentzell type for scaled fractional diffusion bridges under a mild ellipticity assumption on the coefficient vector fields.

Chengcheng Ling University of Augsburg Germany Co-Author(s):

Abstract:

On the short memory limit in a stochastic Coleman-Gurtin model of heat conduction

Vincent R Martinez CUNY Hunter College & Graduate Center USA Co-Author(s):    Nathan Glatt-Holtz, Vincent R. Martinez, Hung D. Nguyen

Abstract: This talk presents recent results on a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials and sufficiently regular noise the system always admits invariant probability measures, defined on the extended phase space, that possess higher regularity properties dictated by the structure of the nonlinearities in the equation. Furthermore, we investigate the singular limit as the memory kernel collapses to a Dirac function. Specifically, provided sufficiently many directions in the phase space are stochastically forced, we show that there is a unique stationary measure to which the system converges, in a suitable Wasserstein distance, at exponential rates independent of the decay of the memory kernel. We then prove the convergence of the statistically steady states to the unique invariant probability of the classical stochastic reaction-diffusion equation in the desired singular limit. As a consequence, we establish the validity of the small memory approximation for solutions on the infinite time horizon.

Peng`s Maximum Principle for Stochastic Delay Differential Equations of Mean-Field Type

Federica Masiero Milano-Bicocca University Italy Co-Author(s):    Giuseppina Guatteri, Lukas Wessels

Abstract: We extend Peng`s maximum principle to the case of stochastic delay differential equations of mean-field type. More precisely, the coefficients of our control problem depend on the state, on the past trajectory and on its expected value. Moreover, the control enters the noise coefficient and the control domain may be non-convex. Our approach is based on a lifting of the state equation to an infinite dimensional Hilbert space that removes the explicit delay in the state equation. The main ingredient in the proof of the maximum principle is a precise asymptotic for the expectation of the first order variational process, which allows us to neglect the corresponding second order terms in the expansion of the cost functional.

Interpretation of stochastic primitive equations with relaxed hydrostatic assumption as a higher order approximation of 3D stochastic Navier-Stokes

Antoine Moneyron Inria Rennes France Co-Author(s):    Arnaud Debussche, Etienne Memin, Antoine Moneyron

Abstract: We investigate the convergence of solutions of a stochastic representation of the three-dimensional Navier-Stokes equations to those of their primitive equations counterpart. Our analysis covers both weak and strong convergence regimes, corresponding respectively to rigid-lid and fully periodic boundary conditions. Furthermore, we explore the impact of relaxing the hydrostatic assumption in the stochastic primitive equations by retaining martingale terms as deviations from hydrostatic equilibrium. This modified model, obtained through a specific asymptotic scaling accessible only within the stochastic framework, captures non-hydrostatic effects while remaining within the primitive equations formalism. The resulting generalized hydrostatic model has been shown to be well-posed when the additional terms are regularized using a suitable filter for divergence-free noises under suitable assumptions. Within this setting, we demonstrate that the model provides a higher-order approximation of the 3D Navier-Stokes equations for appropriately scaled noises.

Stochastic Ericksen-Leslie System: Convergence of the Ginzburg-Landau Approximation

Paul Razafimandimby Dublin City University Ireland Co-Author(s):    Zdzislaw Brzezniak, Gabriel Deugoue, Erika Hausenblas

Abstract: The Ericksen-Leslie equations describe the hydrodynamics of nematic liquid crystals, coupling a fluid velocity with a director field constrained to the unit sphere. This non-convex constraint poses significant analytical challenges. The Ginzburg-Landau approximation relaxes the constraint via a penalization term, providing a tractable framework. In this talk, we consider the stochastic Ginzburg-Landau system and prove that, as the penalization vanishes, its solutions converge to a local, strong martingale solution of the original stochastic Ericksen-Leslie equations. The convergence is established for initial data in the energy space $\mathbb{H}^1 \times \mathbb{H}^2$, improving upon existing results that required higher regularity. Key ingredients include uniform estimates independent of the penalization, tightness arguments, and the Jakubowski-Skorokhod representation theorem. This work rigorously justifies the Ginzburg-Landau approximation in the stochastic setting and lays the foundation for studying noise effects in liquid crystal dynamics.

Invariant measures for the open KPZ equation

Tommaso Rosati University of Warwick England Co-Author(s):    Alex Dunlap and Yu Gu

Abstract: We provide an analytic proof of celebrated relative density formulas of the invariant measures of the open KPZ equation with respect to white noise. The proof relies on a Girsanov transform, a time reversal and a subtle use of the theory of regularity structures to reconstruct the force of the solution to the KPZ equation at the boundary of the domain. This is joint work with A. Dunlap and Y. Gu.

Mass generation for the O(N) Linear Sigma Model in the large N limit

Scott A Smith Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Matias Delgadino

Abstract: This talk will focus on the $O(N)$ Linear Sigma Model on $\R^{2}$ under a scaling dictated by the formal $1/N$ expansion. We show that in the large $N$ limit, correlations decay exponentially fast, where the acquired mass decays exponentially in the inverse temperature. In fact, each marginal converges to a massive Gaussian Free Field (GFF) on $\R^{2}$, quantified in the $2$-Wasserstein distance with a weighted $H^{1}(\R^{2})$ cost function. In contrast to prior work on the torus via parabolic stochastic quantization, our results hold without restrictions on the coupling constants, allowing us to also obtain a massive GFF in a suitable double scaling limit. Our proof combines the Feyel/\Ust\unel extension of Talagrand`s inequality with some classical tools in Euclidean Quantum Field Theory.

On the density of the supremum of nonlinear SPDEs

Alexandra Stavrianidi University of Munster Germany Co-Author(s):    G. Karali, K. Tzirakis, P. Zoubouloglou

Abstract: The existence and regularity of densities for the supremum of stochastic processes is a classical problem in probability theory. While such questions are well understood for several Gaussian processes, much less is known for nonlinear stochastic partial differential equations. In this talk, I will present joint work with G. Karali, K. Tzirakis, and P. Zoubouloglou establishing the existence of a density for the supremum of solutions to a class of nonlinear SPDEs in one spatial dimension driven by space time white noise. The class includes the nonlinear stochastic heat equation on a bounded domain with Dirichlet or Neumann boundary conditions. The proof relies on Malliavin calculus and a Bouleau Hirsch type criterion adapted to suprema of random fields.

Critical stationary fluctuations in reaction--diffusion processes

Kenkichi Tsunoda Kyushu University Japan Co-Author(s):    L. Cardoso, C. Landim, K. Tsunoda

Abstract: We study stationary fluctuations at criticality for a one-dimensional reaction--diffusion process combining symmetric simple exclusion dynamics with Glauber-type spin flips. The strength of the Glauber interaction is tuned to the critical regime in which the quadratic term in the effective potential vanishes. Focusing on the stationary distribution, we show that the total magnetization scaled by $n^{3/4}$ exhibits non-Gaussian fluctuations.

Kac`s program for the Landau equation

Zhenfu Wang Peking University Peoples Rep of China Co-Author(s):    Xuanrui Feng

Abstract: We study the derivation of the spatially homogeneous Landau equation from the mean-field limit of a conservative $N$-particle system, obtained by passing to the grazing limit on Kac`s walk in his program for the Boltzmann equation. Our result covers the full range of interaction potentials, including the physically important Coulomb case. This provides the first resolution of propagation of chaos for a many-particle system approximating the Landau equation with Coulomb interactions, and the first extension of Kac`s program to the Landau equation in the soft potential regime. The convergence is established in weak, Wasserstein, and entropic senses, together with strong $L^1$ convergence. To handle the singularity of soft potentials, we extend the duality approach of Bresch-Duerinckx-Jabin and establish key functional inequalities, including an extended commutator estimate and a new second-order Fisher information estimate.

Convergence rate of lattice approximations for reflected stochastic partial differential equation

BIN XIE Shinshu University Japan Co-Author(s):

Abstract: We investigate the convergence rate of a lattice approximation scheme for a reflected stochastic partial differential equation (SPDE) driven by Gaussian space--time white noise, which is viewed as an infinite-dimensional Skorohod problem. We establish the rate of $L^p(\Omega)$-convergence for the approximations under Lipschitz continuous condition. Our approach is based on a detailed analysis of the corresponding deterministic parabolic obstacle problem via penalization methods.

Remarks on the three-dimensional Navier-Stokes equations with Lions` exponent forced by space-time white noise

Kazuo Yamazaki University of Nebraska-Lincoln USA Co-Author(s):

Abstract: Navier-Stokes equations forced by space-time white becomes singular in a way that in all dimensions equal to or higher than two, the product of the nonlinear term becomes ill-defined. In case dimension is two, Da Prato-Debussche proved its global solution theory using explicit knowledge of invariant measure. More recently, Hairer-Rosati developed a new way to prove its global solution theory without relying on invariant measure. We apply the latter approach to the three-dimensional Navier-Stokes equations forced by space-time white noise and energy-critical diffusion; despite the absence of any knowledge of its invariant measure, we prove its global solution theory.

From particles via fluctuating hydrodynamics to gradient flows: Rigorous error estimates

Johannes Zimmer TU Munich Germany Co-Author(s):    Nicolas Dirr and Zhengyan Wu

Abstract: The aim is to learn gradient flow evolution from underlying particle models. Specifically, we will discuss how the mobility of the thermodynamic evolution operator of suitable diffusive processes can be learned from particle data. Results will consider the case where the evolution is of Wasserstein gradient flow type. The central tool is a stochastic partial differential equation of fluctuating hydrodynamics type, which will be introduced in the talk. As rigorous result, error estimates for the mobility associated with the simple exclusion process are presented. Methodologically, this approach relies on the fact that fluctuating hydrodynamics can be seen as thermodynamically correct stochastic perturbation of a deterministic gradient flow.