Special Session 11: Stochastic Partial Differential Equations

On spectral gaps for Stochastic Wave equations.

Nikolay Barashkov Max Planck Institute for Mathematics in the Sciences Germany Co-Author(s):    Giacomo di Gesu, Petri Laarne

Abstract: We will discuss Spectral Gap of Stochastic Wave equations with additive noise, and its relation to the heat equation case. Our methods combine techiques from Hypocoercivity in an infinite dimesional setting, with techniques from singuarl SPDEs.

Concentration results for time-dependent SPDEs with Gaussian and fractional noise

Nils Berglund IDP, University of Orleans France Co-Author(s):    Alexandra Blessing, Rita Nader

Abstract: We consider slowly time-dependent parabolic SPDEs on the torus of dimension $1$ or $2$, driven by either Gaussian, or by fractional space-time white noise with Hurst parameter strictly above $1/4$. We obtain concentration estimates for solutions near stable critical curves of these equations. As an application, we prove a quantitative result on stochastic resonance for periodically forced SPDEs. References: 1. NB, Rita Nader, Stochastic resonance in stochastic PDEs, Stochastics and Partial Differential Equations: Analysis and Computations, 11:348-387 (2023). 2. NB, Rita Nader, Concentration estimates for slowly time-dependent singular SPDEs on the two-dimensional torus, Electronic J. Probability 29:1-35 (2024). 3. NB, Alexandra Blessing (Neamtu), Concentration estimates for SPDEs driven by fractional Brownian motion Electron. Commun. Probab. 30:1-13 (2025).

Renormalization destroys a finite time bifurcation in the $\Phi^4_2$ equation

Alexandra Blessing University of Konstanz Germany Co-Author(s):    Nicolas Perkowski, Chara Zhu

Abstract: We study the singular $\Phi^4_2$ equation at a pitchfork bifurcation of the underlying deterministic dynamics. To this aim, we linearize the SPDE along its stationary solution and show that the support of its finite-time Lyapunov exponents (FTLEs) is the real line, regardless of the bifurcation parameter and in sharp contrast to the non-singular $\Phi^4_1$ equation. The proof relies on a support theorem for the stationary solution and its renormalized square.

On the approximation of finite-time Lyapunov exponents for the stochastic Burgers equation

Dirk Bl\"omker Universit\"at Augsburg Germany Co-Author(s):    Alexandra Blessing

Abstract: We analyze stochastic partial differential equations (SPDEs) with quadratic nonlinearities close to a change of stability. To this aim we compute finite-time Lyapunov exponents (FTLEs), observing a change of sign based on the interplay between the distance towards the bifurcation and the noise intensity. We reduce the infinite dimensional equation to an SDE on the dominant modes and carry over results for FTLE from the finite to the infinite dimensional setting. A technical challenge is to provide a suitable control of the nonlinear terms coupling the dominant and stable modes of the SPDE and of the corresponding linearization. In order to illustrate our results we apply them to the stochastic Burgers equation.

Uniqueness for stochastic differential equations in Hilbert spaces with irregular drift

Oleg Butkovsky Weierstrass Institute (WIAS) Germany Co-Author(s):

Abstract: Joint work with Lukas Anzeletti, M\`at\`e Gerencs\`er, Alexander Shaposhnikov. We obtain strong uniqueness for SDEs in Hilbert spaces with irregular drift: $$ dX_t= (A X_t + b(X_t))dt +(-A)^{-\gamma/2}dW_t, X_0=x\in H, $$ where $H$ is a separable Hilbert space, $A$ is a self-adjoint negative definite operator, $W$ is a cylindrical Wiener process, $b$ is an $\alpha$-H\older continuous function $H\to H$, $\gamma\ge0$. We show that this equation has a unique strong solution provided that $\alpha > \alpha^*(\gamma)$, with an explicit function $\alpha^*$ that takes values in $(0,1)$ for all $\gamma\in[0,3)$. This substantially extends the seminal work of Da Prato and Flandoli (2010), as no structural assumption on $b$ is imposed. To obtain this result, we do not use infinite-dimensional Kolmogorov equations but instead develop a new technique combining L\^e`s theory of stochastic sewing in Hilbert spaces, Gaussian analysis, and the method of Lasry and Lions for approximation in Hilbert spaces.

Transposition Approach to Optimal Control of McKean-Vlasov SPDEs

Liangying Chen FU Berlin & TU Berlin Germany Co-Author(s):    Wilhelm Stannat

Abstract: We introduce the notion transposition solution for a class of McKean-Vlasov backward stochastic partial differential equations, providing a flexible framework to establish well-posedness under low regularity conditions via duality methods. This approach is particularly suited to infinite-dimensional stochastic systems where the given filtration is not natural. As an application, we study optimal control problems governed by Mckean-Vlasov stochastic partial differential equations, in which the coefficient depend on bothe the state and its probability distribution. Within this framework, we derive first-order necessary optimal conditions in the form of a stochastic maximum principle, where he corresponding adjoint equations are characterized as BSPDEs in the transposition sense.

Large field problem in coercive SPDE via scaling

Ilya Chevyrev SISSA (International School for Advanced Studies) Italy Co-Author(s):    Massimiliano Gubinelli

Abstract: In this talk, I will show an approach to deriving a priori bounds for coercive SPDEs based on scaling. The basic idea is to first show bounds for the equation with a small noise and then rescale the bounds to a global scale. While many equations that the approach can handle have been treated recently with other methods, its advantages are that it is simple and allows one to state a single result that is applicable to a variety of equations, such as rough differential equations and parabolic/elliptic SPDEs.

It\^o perspective on variance renormalisation

Konstantinos Dareiotis University of Leeds England Co-Author(s):    M\`at\`e Gerencs\`er

Abstract: In this talk we will see that the It\^o solutions of the nonlinear stochastic heat equation $$ \partial_t u^\varepsilon- \Delta u^\varepsilon =\varepsilon^{3/4} g (u^\varepsilon) \nabla \xi_\varepsilon, $$ where $ \xi_\varepsilon$ denotes the mollification in space at scale $\varepsilon>0$ of a space-time white noise $\xi$, converge in law, as $\varepsilon \to 0$, to the solution of the stochastic heat equation with right-hand side $cg`g(u)\xi$ with a constant $c>0$. Since the noise $\nabla\xi$ is supercritical, the small prefactor is not unexpected to obtain a limit, but the exponent $3/4$ is not predicted by naive scaling arguments. The case $g(u)=u$, modulo a Cole-Hopf transform, corresponds to the result of Hairer (2025) for the KPZ equation. Our argument is relatively short and relies solely on stochastic analytic techniques.

Large deviations and weak solution concepts in a stochastic Galerkin-Euler system

Lucio Galeati University of L`Aquila Italy Co-Author(s):    Daniel Heydecker

Abstract: Motivated by fluctuating hydrodynamics, modelling of turbulent fluids, and previous analysis of the Landau-Lifschitz-Navier-Stokes equations, we study a dynamical large deviation principle for solutions to a Galerkin approximation of the stochastic 3D Euler equations; here the noise is divergence free, of Stratonovich transport type, and we consider a scaling regime where the noise intensity $\varepsilon$ and the inverse of the Fourier truncation parameter $N$ go to zero simultaneously. We first show the validity of both a restricted lower bound and an upper bound on compact sets, for a natural choice of rate function, concerning $L^2$-valued admissible weak solutions of the associated skeleton Euler equations. Exploiting the dynamical reversibility of the system, we then show the existence of energy inadmissible solutions, which must be seen by any extension of the rate function which results into a global LDP lower bound. Finally, we extend the LDP upper bound to a global one, by considering a newly introduced concept of H-measure-valued solutions to skeleton Euler. Based on joint work with Daniel Heydecker (Oslo).

Pathwise Solvability and Bubbling in 2D Stochastic Landau-Lifshitz-Gilbert Equations

Ben Goldys Sydney University Australia Co-Author(s):    Chunxi Jiao, Christof Melcher

Abstract: We investigate the stochastic Landau-Lifshitz-Gilbert (LLG) equation on a periodic 2D domain, driven by infinite-dimensional Gaussian noise in a Sobolev class. We establish strong local well-posedness in the energy space and characterize blow-up at random times in terms of energy concentration at small scales (bubbling). By iteration, we construct pathwise global weak solutions, with energy evolving as a c\`adl\`ag process, and prove uniqueness within this class. These results offer a stochastic counterpart to the deterministic concept of Struwe solutions. The approach relies on a transformation that leads to a magnetic Landau-Lifshitz-Gilbert equation with random gauge coefficients.

The Leibenson process as a strong solution to its associated McKean--Vlasov SDE

Sebastian Grube Bielefeld University Germany Co-Author(s):    Viorel Barbu, Marco Rehmeier, Michael Roeckner

Abstract: This talk continues the presentation by Marco Rehmeier. The Leibenson process, a nonlinear Markov process, is constructed from the path laws of probabilistically weak solutions to a class of McKean--Vlasov SDEs associated to the Leibenson equation. Despite the low regularity of the coefficients of these highly degenerate McKean--Vlasov SDEs, we can show that the weak solutions constituting the Leibenson process are, in fact, strong solutions. As a special case, we show that the $p$-Brownian motion is a strong solution to its associated McKean--Vlasov SDE.

The Porous Medium Equation: Multiscale Integrability in Large Deviations

Daniel Heydecker University of Oslo Norway Co-Author(s):    Benjamin Gess

Abstract: We consider the large deviations of a particle system $\eta^N$ with degenerate and superlinear diffusivity. 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. This is resolved introducing a novel multiscale argument exploiting the appearance of pathwise regularity across scales.

Milstein-type Schemes for Hyperbolic SPDEs

Katharina Klioba TU Delft Netherlands Co-Author(s):

Abstract: Discretisations of SPDEs with multiplicative noise in time that rely solely on the Wiener increments $W(t_{j+1})-W(t_j)$ are known to converge at most at rate $1/2$. To overcome this limitation, one needs to consider higher-order terms from an Ito-Taylor expansion, leading to iterated stochastic integrals. The resulting Milstein scheme is well-studied for both SDEs and parabolic SPDEs. In this talk, convergence rates of the Milstein scheme are presented for hyperbolic SPDEs, where no smoothing effect over time can be leveraged. Optimal convergence rates are derived for the pathwise uniform strong error $$E_h^\infty:= (\mathbb{E}\max_{1\le j \le M}\|U(t_j)-u_j\|_X^p)^{1/p}$$ on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $u_j$ its Milstein approximation at time $t_j=jh$ with step size $h>0$ and final time $T=Mh>0$. For sufficiently regular nonlinearity and noise, we establish strong convergence of order one, with the error satisfying $E_h^\infty\le h\sqrt{\log(T/h)}$ for rational Milstein schemes and $E_h^\infty \le h$ for exponential Milstein schemes. This extends previous results from parabolic to hyperbolic SPDEs, from exponential to rational Milstein schemes, and from root-mean-square error estimates to pathwise uniform estimates. Applications include the stochastic Schroedinger and transport equations. This is joint work with Felix Kastner.

Variational Framework for SPDE: Old and New

Wei Liu Jiangsu Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk we mainly present some recent progress on the variational framework for SPDE, especially on Mckean-Vlasov SPDE and regularization effect by noise.

Strong uniqueness by Kraichnan transport noise for the inviscid 2D Boussinesq system

Dejun Luo Academy of Mathematics and Systems Science, Chinese Academy of Sciehces Peoples Rep of China Co-Author(s):

Abstract: We consider the inviscid 2D Boussinesq system driven by rough transport noise of Kraichnan type with regularity index $\alpha\in (0,1/2)$. For all $p>1$, we establish the existence and uniqueness of probabilistic strong solutions for all $L^p$ initial vorticity and $L^2$ initial temperature, under the parameter constraint $\alpha\in (0, 1-1/(p\wedge 2))$. The talk is based on a joint work with Dr. Shuaijie Jiao.

Ergodicity for SPDEs driven by divergence-free transport noise

Adrian Martini Technical University of Berlin Germany Co-Author(s):    Benjamin Gess, Rishabh S. Gvalani

Abstract: We study the ergodic behaviour of McKean--Vlasov equations driven by common, divergence-free transport noise. In particular, we show that in dimensions greater or equal to 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. This is joint work with Benjamin Gess and Rishabh S. Gvalani.

The FBSDE approach to EQFTs

Sarah-Jean SJM Meyer University of Oxford England Co-Author(s):    Massimiliano Gubinelli

Abstract: I will present recent progress on the forward-backward SDE (FBSDE) approach to Euclidean quantum field theories (EQFT), focusing on the example of the sine-Gordon model. The FBSDE describes the dynamics of transitioning between scales and can be viewed as a rigorous implementation of the renormalization group for the model, providing a powerful description of the interacting theory as a coupling with the Gaussian free field. Once established, the FBSDE is the starting point for a stochastic analysis of the EQFT to derive additional properties of the field, such as large deviations, decay of correlations, singularity, and a full verification of the Osterwalder-Schrader Axioms. I will outline the general construction of the stable reformulation and discuss ongoing work on the sine-Gordon model.

On modified stochastic Boussinesq-Benard equations in dimension 3

Annie A MILLET University Paris 1 Pantheon Sorbonne France Co-Author(s):    Hakima Bessaih

Abstract: We study the Boussinesq-B\`enard equations in dimension 3 subject to a multiplicative random perturbation; we have to add a Brinkman-Fochheimer smoothing term in the evolution equation for the velocity $u$. We prove that for $H^1$-initial velocity $u_0$ and temperature $\theta_0$ with proper moments, the system of SPDEs is a.s. globally well posed in $\big( C([0,T];H) \cap L^4(0,T;V)\big) \times \big(C([0,T]; L^2)\cap L^2(0,T;H^1)\big)$. We also prove the existence of higher moments of the solution $(u,\theta)$.

The Leibenson equation and its associated nonlinear Markov process

Marco Rehmeier TU Berlin Germany Co-Author(s):    Viorel Barbu, Sebastian Grube, Michael R\ockner

Abstract: We construct a probabilistic counterpart for the Barenblatt solutions of the doubly nonlinear Leibenson equation \begin{equation*} \partial_t u(t,x) = \Delta_p u^q(t,x),\quad (t,x) \in (0,\infty) \times \mathbb{R}^d, \end{equation*} where $\Delta_p f = \divv(|\nabla f|^{p-2}\nabla f)$ denotes the $p$-Laplace operator, $p>2 and $q>0$. This counterpart is a family of stochastic processes with one-dimensional time marginal densities given by the Barenblatt solutions. These processes are constructed as weak solutions to a McKean--Vlasov equation whose coefficients depend on the solution law pointwise via the gradient of its density. To this end, we first identify the Leibenson equation as a nonlinear Fokker--Planck equation. Moreover, we prove that these processes form a unique nonlinear Markov process with Barenblatt time marginals, which we call the Leibenson process. Joint work with Viorel Barbu, Sebastian Grube and Michael R\ockner.

Boltzmann processes

Barbara Ruediger University Wuppertal Germany Co-Author(s):    Barbara R\udiger , Padmanabhan Sundar

Abstract: The Boltzmann equation describes the dynamics of a density in position and velocity of a rarified gas expanding in vacuum. Ludwig Eduard Boltzmann (1844 -1906) derived the Boltzmann equation, by assuming any gas molecule of a rarified gas to travel straight in vacuum until an elastic collision occurs with another molecule of the same gas. In the Boltzmann equation, only binary centered collisions are considered. In this talk we present the Boltzmann process \cite{ARS}, i.e the process whose density evolves according to the Boltzmann equation. Using the Ito formula, we prove that this is a solution of a stochastic differential equation of McKean Vlasov type, for which we prove the existence \cite{RS} \ \begin{thebibliography}{8} \bibitem{ARS} Albeverio, S., R\udiger B., Sundar P.: On the construction and identifcation of Boltzmann processes. In: Brasesco S., Butta P., Cassandro M., Picco P., Vares M. E. (eds.) , ENSAIOS MATHEMATICOS, vol.38, pp. 1-22, . Papers in honor of Errico Presutti, Sociedade Brasileira de Mathematica (2023). \doi{10.21711/217504322023/em381} \bibitem{RS} R\udiger B., Sundar P.: .: Identification and existence of Boltzmann processes, arxiv 2301.08662v2 (2025). \doi{10.48550/arXiv.2301.08662} \end{thebibliography}

Mean-field games and Hamilton--Jacobi equations with nonlocal diffusions

Artur Rutkowski Wroc\l{}aw University of Science and Technology Poland Co-Author(s):    Espen Jakobsen, Robin Lien

Abstract: A mean-field game system is a system consisting of a backward Hamilton--Jacobi equation and a forward Fokker--Planck equation. It was proposed 20 years ago by Huang, Malham\`{e}, Caines, Lasry, and Lions, as a means to study large population games. We will discuss the well-posedness results for mean-field game systems and the classical solutions to associated master equations in the whole space $\mathbb{R}^d$, driven by individual L\`{e}vy noise of order $\alpha \in (1,2]$, without \textit{a priori} assuming the existence of any moments. We will also present a rather optimal way to express the order $\alpha$ of the noise in terms of the heat kernel of its generator. In this regime, we prove Schauder estimates for the Hamilton--Jacobi equations in the subcritical case $\alpha\in (1,2]$, where the noise dominates the gradient nonlinearity, and we give partial results for the critical case $\alpha=1$. Our results are purely deterministic, but extensions to the framework of SPDEs are possible and interesting. In this connection, we will briefly discuss mean-field games with common noise.

The incompressible Navier--Stokes--Fourier system with thermal noise

Max Sauerbrey MPI MiS, Leipzig Germany Co-Author(s):    Benjamin Gess, Zhengyan Wu

Abstract: We establish a solution theory (global weak existence, local strong existence and weak-strong uniqueness) 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 $u$ can be solved independently of the temperature $\vt$, 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 of the system for $(u,\vt)$ poses new challenges, which are absent in deterministic Navier--Sokes--Fourier equations.

A priori bounds for stochastic porous media equations

Markus Tempelmayr EPFL Switzerland Co-Author(s):

Abstract: We discuss a priori regularity estimates for solutions of stochastic porous media equations, in a regime where the equation is both singular and degenerate. For simplicity we consider only a mildly singular regime, which allows to prove modelledness of the solution with respect to the solution of the corresponding linear stochastic heat equation. If time permits we discuss some ingredients of the proof, including the kinetic formulation of the equation and a renormalized energy inequality.

Stochastic reaction diffusion equations with non-trace class noise

Mark Veraar TU Delft Netherlands Co-Author(s):    Antonio Agresti and Fabian Germ

Abstract: In this talk, I will give an overview of recent results on stochastic reaction-diffusion equations driven by non-trace class noise. Specifically, I will highlight new developments regarding: critical spaces of initial values, novel regularization phenomena, and improved linear estimates for stochastic convolutions