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My main area of research is stochastic analysis, especially in connection with questions in constructive quantum field theory. I develop tools and concepts to describe and analyse the pathwise behaviour of random and quantum fields, including formulations based on partial differential equations and renormalisation-group ideas. I introduced controlled rough paths and developed branched rough paths, and later co-developed paracontrolled calculus for singular stochastic PDEs. These ideas have found applications to KPZ-type equations, regularisation by noise, stochastic quantisation and Euclidean quantum field theory. Much of my recent work concerns stochastic quantisation and variational methods for the construction and analysis of Euclidean fields. This research is currently supported by the UKRI Frontier Research grant StochFields (“Stochastic Analysis of Quantum Fields”) and by the Simons Collaboration “Probabilistic Paths to QFT”.
More broadly, I am interested in the statistical mechanics of multiscale systems, PDEs involving randomness and homogenisation theory, mathematical quantum mechanics, path-integral formalisms, non-commutative probability and non-commutative geometry. I also have a side interest in the formalisation of mathematics. A common theme is how randomness can be used both as an analytic tool and as a framework for constructing and understanding quantum fields.
with F. De Vecchi, L. Fresta and M. Gordina. Non-commutative \(L^p\) spaces and Grassmann stochastic analysis. Probability Theory and Related Fields, may 2025. 10.1007/s00440-025-01379-4
with G. Cannizzaro and F. Toninelli. Gaussian Fluctuations for the Stochastic Burgers Equation in Dimension \(d \geqslant 2\). Communications in Mathematical Physics, 405(4):89, apr 2024. 10.1007/s00220-024-04966-z
with H. Koch and T. Oh. Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity. Journal of the European Mathematical Society, jan 2023. 10.4171/jems/1294
with M. Hofmanová. A PDE Construction of the Euclidean \(\Phi^4_3\) Quantum Field Theory. Communications in Mathematical Physics, apr 2021. 10.1007/s00220-021-04022-0
with L. Galeati. Noiseless regularisation by noise. Revista Matemática Iberoamericana, jul 2021. 10.4171/RMI/1280
with N. Perkowski. The infinitesimal generator of the stochastic Burgers equation. Probability Theory and Related Fields, aug 2020. 10.1007/s00440-020-00996-5
with N. Barashkov. A variational method for \(\Phi^4_3\). Duke Mathematical Journal, 169(17):3339–3415, nov 2020. 10.1215/00127094-2020-0029
with P. Imkeller and N. Perkowski. Paracontrolled distributions and singular PDEs. Forum of Mathematics. Pi, 3:0, 2015. 10.1017/fmp.2015.2
with F. Flandoli and E. Priola. Well-posedness of the transport equation by stochastic perturbation. Inventiones Mathematicae, 180(1):1–53, 2010. 10.1007/s00222-009-0224-4
Ramification of rough paths. Journal of Differential Equations, 248(4):693–721, 2010. 10.1016/j.jde.2009.11.015
Controlling rough paths. Journal of Functional Analysis, 216(1):86–140, 2004. 10.1016/j.jfa.2004.01.002
Notes from the lectures of the 2nd workshop (URL) of the Program “New developments and challenges in Stochastic Partial Differential Equations” at the Bernoulli Center, EPFL, Lausanne
Notes from the seminar held during the SRQ – Scaling limits, rough paths, quantum field theory research period in 2018 at the Newton Institute, Cambridge
My PhD thesis “Finite-size scaling in non-equilibrium critical phenomena” (University of Pisa, 2002) (pdf)