I will discuss recent results on stochastic homogenization for linear elliptic equations with random coefficients. The main focus will be on the degenerate elliptic setting, where ellipticity is controlled by suitable (p,q)-moment conditions rather than uniform bounds. I will present several results, including large-scale regularity estimates. If time permits, I will also discuss recent results on boundary correctors and their role in extending homogenization results up to the boundary.
The talk is based on joint works with Mathias Schäffner, Michael Kniely, Julian Fischer, Claudia Raithel, and Marc Josien.
Rough paths and regularity structures study pathwise S(P)DEs using local expansions. The associated remainders are usually estimated via analytic methods but one can also write them explicitly, and in fact there are many possible ways of expressing them. With Lorenzo Agabiti and Lorenzo Zambotti we found a particularly nice form of the remainders, which allows to establish a priori bounds on the solution of rough ODEs under optimal assumptions on the regularity of the coefficients. The formula is however surprisingly difficult and involves several non-trivial combinatorial tools. In the first part of this bipartite talk, I will present the main hurdles that one has to face.
In this talk, I will describe a construction of a gauge field potential for the 2D Yang-Mills measure with optimal regularity. The method is inspired by classical Uhlenbeck compactness and involves solving for the Coulomb gauge. In the low regularity setting that we consider, this boils down to analysing a singular, geometric (group-valued), elliptic PDE. We develop a somewhat new solution theory for this equation based on implicit function theorem and regularity structures, which we believe could be of interest for other geometric SPDEs. Joint work with Tom Klose and Abdulwahab Mohamed.
I will address the construction of fundamental solutions and Hadamard states for the Klein-Gordon field on half-Minkowski spacetime with Robin boundary conditions in spacetime dimension d≥2. First, building on a higher-dimensional generalization of the Robin-to-Dirichlet map used by Bondurant and Fulling (J. Phys. A: Math. Theor. 38:7, 2005) in dimension two, I will show that the advanced and retarded Green operators can be represented as convolutions with the kernel of the inverse Robin-to-Dirichlet map. These operators are unique and satisfy the expected causal support properties.
Second, I will discuss a local representation of the Hadamard parametrix adapted to this setting. This provides the appropriate local formulation of the Hadamard condition in d≥2 dimensions and accounts for the additional “reflected” singularities produced by the spacetime boundary. I will then show that the fundamental solutions constructed above are compatible with this local parametrix representation.
Finally, in the same framework, I will prove the equivalence between the local and global Hadamard conditions, where the relevant wavefront-set characterization is formulated in terms of generalized broken bicharacteristics. This yields a Radzikowski-type theorem for half-Minkowski spacetime with Robin boundary conditions.
Based on joint work with Claudio Dappiaggi, Benito A. Juárez-Aubry, and Raman Deep Singh, https://doi.org/10.1007/s00023-026-01702-2.
In this talk, we consider the random conductance model with long-range jump. This model is defined by assigning to each pair of vertices x , y in Z^d, a conductance c(x,y) = a(x,y) / |x - y|^(d + a) where a(x, y) are i.i.d. and uniformly elliptic, and by considering a random walk starting from 0 and jumping from x to y with rate c(x,y). The large scale behaviour of the random walk depends on the value of the range exponent a. When a < 2, it converges to a stable process, after a superdiffusive rescaling t^{1/a}. When a > 2, it converges to a Brownian motion, after a diffusive scaling t^{1/2}. In this talk, we will be interested in the critical case a = 2, and show that the random walk converges to a Brownian motion, after a superdiffusive rescaling (t ln t)^{1/2}. This is joint work with A. Bou-Rabee.
In this talk, we address the existence and uniqueness of invariant and reversible measures for a class of stochastic partial differential equations (SPDEs) on the full space R, and more generally on R^n. In this setting, standard approaches to proving the uniqueness and ergodicity of invariant measures (typically based on establishing the strong Feller or asymptotic strong Feller property of the associated Markov semigroup) fail. To overcome this difficulty, we propose a different strategy. We show that any reversible measure for a (sufficiently regular) SPDE on R is a Gibbs measure satisfying suitable Dobrushin–Lanford–Ruelle (DLR) equations, and vice versa. Exploiting this connection, we prove an analogue of Fernique's theorem for Gibbs measures, extend certain uniqueness results for them, and show that whenever the DLR equations admit a unique solution, the corresponding SPDE admits a unique reversible invariant measure, which is ergodic. This talk is based on joint work with Davide Bignamini, Carlo Orrieri, and Carlos Villanueva Mariz.
In this talk, we discuss the classical and quantum KMS conditions in the context of spin lattice systems. Using the Berezin quantization of a 2-sphere-valued spin system on a discrete lattice, we lift a classical dynamics on the algebra of classical observables to a corresponding quantum dynamics on the algebra of quantum observables. This framework allows us to compare the notions of classical and quantum thermal equilibrium by proving that every weak* limit point of a family of quantum KMS states satisfies the classical KMS condition. Consequently, the semiclassical limit of quantum thermal equilibrium states describes classical thermal equilibrium, thereby providing further justification for the physical interpretation of the classical KMS condition.
In the second part of the talk, we present new sufficient conditions for the uniqueness of both classical and quantum KMS states in the high-temperature (subcritical) regime. These conditions substantially enlarge the class of admissible interactions while also improving the lower bounds on the critical inverse temperature. In particular, they identify a common subcritical regime in which both the classical and the quantum KMS state are unique.
This is joint work with L. Pettinari and C. J. F. van de Ven.
In the 60's Kraichnan proposed a synthetic model for passive scalar turbulence, consisting of a scalar advected by a random Gaussian velocity field, white in time and α-Holder continuous in space. Despite its simplicity, this SPDE displays anomalous dissipation of energy, spontaneous stochasticity and intermittency, which are also expected for more realistic turbulent fluids. At the same time, solutions to the inviscid SPDE are unique and can be recovered by vanishing viscosity and mollification schemes.
In this talk I will present some recent further understandings on this model: i) solutions to the transport equation with L² initial data display anomalous regularization and almost gain Sobolev regularity H^(1−α) , but not better; ii) solutions to the continuity equation starting from Dirac deltas instantaneously gain Lebesgue integrability, due to the diffusive behaviour of Lagrangian particle splitting, and their variance at small times grows like t^(1/(1−α)) .
Time permitting I will also shortly discuss ongoing investigations concerning nonlinear, active scalar SPDEs in the presence of such a rough, transport noise.
Based on joint works with M. Maurelli, F. Grotto, U. Pappalettera and T. Drivas.
We present the recent construction of equilibrium states for a gas of weakly interacting non-relativistic bosons, focusing on the case of a non-trivial background field in infinitely extended space. The construction is based on a Hubbard-Stratonovich transformation for the interaction and on the convergence of the loop vertex expansion for the state in the infrared regime. Notably, the result holds in the Gross-Pitaevskii scaling for the interaction.
This is based on collaborations with Nicola Pinamonti
The Asymmetric Simple Exclusion Process (ASEP) is a paradigmatic interacting particle system in nonequilibrium statistical mechanics. While its large-scale behaviour is by now well understood in dimension d=1 (polynomially superdiffusive, with fluctuations governed by the KPZ fixed point) and in dimensions d≥3 (diffusive, with Gaussian fluctuations), the critical dimension d=2 remains much less understood.
In this talk, I will present ongoing work on the two-dimensional ASEP, showing that its diffusivity grows like log(t)^{2/3} (improving the previously known log(t)^{2/3+o(1)} bound due to Yau). Moreover, under the corresponding superdiffusive rescaling, we obtain a central limit theorem for the second-class particle and for the fluctuation process. The key ingredient is a connection to the two-dimensional Stochastic Burgers Equation (SBE), for which an analogous result was recently proved in joint work arXiv:2501.00344 with Giuseppe Cannizzaro and Fabio Toninelli.
The semiclassical approximation of the Einstein–Klein–Gordon system is a framework where gravity is treated as the curvature of a Lorentzian manifold, while matter is modeled by a quantum field. The backreaction of matter on the geometry is implemented by equating the Einstein tensor with the expectation value of the quantum stress-energy tensor in a suitable quantum state. In this talk, we focus on the linearized problem around Minkowski spacetime. Using the Møller operator, the system decouples into two distinct Cauchy problems, where the metric perturbations are governed by a higher-order, nonlocal hyperbolic PDE. By relegating the nonlocal contributions to subleading order, we establish the well-posedness of this Cauchy problem. Furthermore, we provide a rigorous asymptotic analysis for physically admissible choices of the renormalization constants, demonstrating late-time exponential growth. This instability ultimately drives a clear transition from a Minkowski to a de Sitter geometry. This is joint work with S. Galanda, P. Meda, N. Pinamonti and G. Schmid.
In this talk I will present a joint work with P. Duch and M.Gubinelli concerning the stochastic quantization of the fractional \Phi^4_3 model in the full subcritical regime using flow equations techniques.
Gaussian Quantum Markov Semigroups (GQMSs) constitute a nat- ural class of quantum dynamical semigroups describing the evolution of bosonic systems with linear noise operators and quadratic Hamilto- nians. They play a prominent role in quantum statistical mechanics, open quantum systems and noncommutative probability, since they preserve the family of Gaussian states and admit an explicit descrip- tion through their action on Weyl operators.
In this talk we discuss the problem of characterizing Gelfand– Naimark–Segal (GNS) symmetry for Gaussian Quantum Markov Semi- groups with respect to a faithful invariant Gaussian state. Exploiting the Gaussian structure, we derive necessary and sufficient algebraic conditions on the coefficients of the generator that are equivalent to GNS symmetry. A key feature of the analysis is that symmetry can be tested directly at the level of Weyl operators, leading to a significant simplification with respect to the general theory of quantum Markov semigroups.
We then show that, under natural assumptions on the invariant Gaussian state, every symmetric GQMS is unitarily equivalent to a direct sum of quantum Ornstein–Uhlenbeck semigroups. This provides a complete structural description of the symmetric case and highlights the distinguished role played by Ornstein–Uhlenbeck dynamics within the Gaussian framework. We also discuss the relationship between symmetry and modular theory, showing how GNS symmetry implies commutation with the modular automorphism group associated with the invariant state.
Finally, we illustrate how Bogoliubov and metaplectic transforma- tions can be used to reduce the problem to a canonical form, clarifying the geometry underlying symmetric Gaussian quantum dynamics.
Hyperbolic partial differential equations (PDEs) play a fundamental role in mathematical physics and serve as models for many physical phenomena, particularly those involving wave propagation, causality, and finite propagation speeds. Traditionally studied with local interactions, it has become increasingly clear in many different areas of mathematics and physics that it would be highly desirable to extend this type of equations by incorporating also nonlocal interactions, thereby providing a more flexible framework for modelling complex phenomena where long-range correlations or memory effects play a significant role. In this talk, I will present a brief overview of some recent results obtained in the analysis of such equations, with an emphasis on their Cauchy problem and the propagation speed of their solutions.
McKean--Vlasov-type stochastic differential equations (SDEs) are characterised by coefficients depending on both the state and the law of the solution. In our work, we focus on a class of such equations where the coefficients depend on a linear combination of the expected signature of the geometric p-rough path lift of its solution, with p ∈ (2,3). After establishing the strong existence and uniqueness of a solution, we prove how such an equation can approximate a general class of path-dependent McKean–Vlasov SDEs. Finally, we consider the associated particle system and establish propagation of chaos. This is a joint work with F.E. Benth (BI Norwegian Business School and University of Oslo) and S. Ortiz-Latorre (University of Oslo).
The Stochastic Burgers Equation (SBE) was introduced in the eighties by van Beijeren, Kutner and Spohn as a mesoscopic model for driven diffusive systems with one conserved quantity. In the subcritical dimension d=1, it coincides with the derivative of the KPZ equation whose large-scale behaviour is polynomially superdiffusive and given by the KPZ Fixed Point, and in the super-critical dimensions d>2, it was recently shown to be diffusive and rescale to an anisotropic Stochastic Heat equation. At the critical dimension d=2, the SBE was conjectured to be logarithmically superdiffusive with a precise exponent but this has only been shown up to lower order corrections. This talk is based on the work https://arxiv.org/abs/2501.00344 joint with Giuseppe Cannizzaro and Quentin Moulard where we pin down the logarithmic superdiffusivity by identifying exactly the large-time asymptotic behaviour of the so-called diffusion matrix and show that, once the logarithmic corrections to the scaling are taken into account, the solution of the SBE satisfies a central limit theorem. This is the first superdiffisive scaling limit result for a critical SPDE, beyond the weak coupling regime.
We study the high contrast convolution-type operators with rapidly oscillating coefficients in the diffusive scaling and discuss the spectral convergence when the small parameter tends to zero. We discuss deterministic and stochastic (ergodic) case.
Rough paths and regularity structures study pathwise S(P)DEs using local expansions. The associated remainders are usually estimated via analytic methodsbut one can also write them explicitly, and in fact there are many possible ways of expressing them. With Lorenzo Agabiti and Alberto Bonicelli we have found a particularly nice way of rearranging the terms in this sum, which allows to give seemingly optimal a priori bounds on the solution. The formula is however surprisingly difficult and involves several non-trivial combinatorial tools.
We establish strong Feller property and irreducibility for the transition semigroup associated to a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove uniqueness of the invariant measure under very mild assumptions. The drift of the equation diverges exactly where the noise coefficient vanishes, resulting in a competition between the dissipative effects and the degeneracy of the noise. The main idea is to introduce a mathematical method to measure the accumulation of the solution towards the potential barriers, allowing to give rigorous meaning to the inverse of the noise operator even in the degenerate case. If the singularity of the drift and the degeneracy of the noise are suitably balanced, the dynamics are shown to stabilise for large times. From the mathematical point of view, the results provide a first generalisation of the classical work by Peszat & Zabczyk [1] to the case of degenerate multiplicative diffusions. From the application perspective, the models cover interesting scenarios in physics, in the context of evolution of relative concentrations of mixtures, under the influence of thermodynamically-relevant potentials of Flory-Huggins type. The talk is based on a joint work with L.Scarpa.
[1] Peszat, S., Zabczyk, J.: Strong Feller property and irreducibility for diffusions on Hilbert spaces. The Annals of Probability, 157–172, (1995).