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.
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 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 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.
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.
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.