@article{21894,
  abstract     = {The Dean–Kawasaki equation—one of the most fundamental SPDEs of
fluctuating hydrodynamics—has been proposed as a model for density fluctuations in weakly interacting particle systems. In its original form, it is highly
singular and fails to be renormalizable, even by approaches such as regularity structures and paracontrolled distributions, hindering mathematical approaches to its rigorous justification. It has been understood recently that it is
natural to introduce a suitable regularization, for example, by applying a formal spatial discretization or by truncating high-frequency noise: This yields
well-posed equations that should still precisely approximate the law of the
particle density fluctuations.
In the present work, we prove that a regularization in the form of a formal
discretization of the Dean–Kawasaki equation indeed accurately describes
density fluctuations in systems of weakly interacting diffusing particles: We
show that, in suitable weak metrics, the law of fluctuations as predicted by
the discretized Dean–Kawasaki SPDE approximates the law of fluctuations
of the original particle system, up to an error that is of arbitrarily high order in
the inverse particle number and a discretization error. In particular, the Dean–
Kawasaki equation provides a means for efficient and accurate simulations of
density fluctuations in weakly interacting particle systems.},
  author       = {Cornalba, Federico and Fischer, Julian L and Ingmanns, Jonas and Raithel, Claudia},
  issn         = {2168-894X},
  journal      = {The Annals of Probability},
  keywords     = {Weakly interacting particle systems, fluctuating hydrodynamics, Dean-Kawasaki equation, stochastic PDEs, numerical approximation},
  number       = {1},
  pages        = {155--215},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation}},
  doi          = {10.1214/25-aop1763},
  volume       = {54},
  year         = {2026},
}

@article{21271,
  abstract     = {For general non-Hermitian large random matrices X and deterministic deformation matrices A, we prove that the local eigenvalue statistics of A+X close to the typical edge points of its spectrum are universal. Furthermore, we show that, under natural assumptions, on A the spectrum of A+X does not have outliers at a distance larger than the natural fluctuation scale of the eigenvalues. As a consequence, the number of eigenvalues in each component of Spec(A+X) is deterministic.},
  author       = {Campbell, Andrew J and Cipolloni, Giorgio and Erdös, László and Ji, Hong Chang},
  issn         = {2168-894X},
  journal      = {The Annals of Probability},
  number       = {6},
  pages        = {2256--2308},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{On the spectral edge of non-Hermitian random matrices}},
  doi          = {10.1214/25-aop1761},
  volume       = {53},
  year         = {2025},
}

@article{11354,
  abstract     = {We construct a recurrent diffusion process with values in the space of probability measures over an arbitrary closed Riemannian manifold of dimension d≥2. The process is associated with the Dirichlet form defined by integration of the Wasserstein gradient w.r.t. the Dirichlet–Ferguson measure, and is the counterpart on multidimensional base spaces to the modified massive Arratia flow over the unit interval described in V. Konarovskyi and M.-K. von Renesse (Comm. Pure Appl. Math. 72 (2019) 764–800). Together with two different constructions of the process, we discuss its ergodicity, invariant sets, finite-dimensional approximations, and Varadhan short-time asymptotics.},
  author       = {Dello Schiavo, Lorenzo},
  issn         = {2168-894X},
  journal      = {Annals of Probability},
  number       = {2},
  pages        = {591--648},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold}},
  doi          = {10.1214/21-AOP1541},
  volume       = {50},
  year         = {2022},
}

@article{11418,
  abstract     = {We consider the quadratic form of a general high-rank deterministic matrix on the eigenvectors of an N×N
Wigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large N limit. The proof is a combination of the energy method for the Dyson Brownian motion inspired by Marcinek and Yau (2021) and our recent multiresolvent local laws (Comm. Math. Phys. 388 (2021) 1005–1048).},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {2168-894X},
  journal      = {Annals of Probability},
  number       = {3},
  pages        = {984--1012},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Normal fluctuation in quantum ergodicity for Wigner matrices}},
  doi          = {10.1214/21-AOP1552},
  volume       = {50},
  year         = {2022},
}