Lecturer. Massimiliano Gubinelli (University of Oxford).
These eight lectures were given at the 53rd Saint-Flour Probability Summer School (Saint-Flour, France), together with the courses of Justin Salez and Peter Bartlett.
The goal of these lectures is to introduce Euclidean quantum field theory as natural objects in probability theory: higher dimensional generalisations of Markov processes. I will explain the motivations and problems inherent in a rigorous theory of these objects and indicate how it is possible to use the fundamental ideas of stochastic analysis to tackle them, at least in dimensions two and three. We will explore various ways to construct Euclidean fields by solving certain stochastic analysis problems, a method called stochastic quantisation. If time permits I will also touch upon the open problem of giving a mathematical characterisation of Euclidean fields.
A stochastic analysis perspective on Euclidean fields. [PDF] [HTML]. Gaussian free field and Wick powers, the variational approach, the quartic and exponential interactions in two dimensions, a scale-by-scale interpolation and the quartic interaction in three dimensions.
The lectures were recorded by the school; all eight are in the YouTube playlist. Each lecture lasts about ninety minutes. The times below link to the start of each topic in the recording, and the section numbers refer to the lecture notes. All dates are in 2025.
Lecture 1 (Tuesday 1 July): Euclidean fields as Markov random fields. Euclidean fields as perturbations of the Gaussian free field; the problems of renormalisation, large fields and infinite volume; the models and the plan of the course. Notes: Sections 1 and 2.1.
00:05 Motivation: Euclidean fields as higher-dimensional Markov processes
07:25 Formal Gibbs measures and the Markov property
14:44 The Gaussian free field as a random distribution
27:14 Renormalisation, large fields and infinite volume
40:01 Regularisation and the models: \(\varphi^4_2\), \(\varphi^4_3\), \(P (\varphi)_2\), exponential and sine-Gordon
53:25 Plan of the course; questions on locality and on dimension four
1:13:16 Regularity: Hölder and Besov spaces
Lecture 2 (Wednesday 2 July): Besov spaces, the Gaussian free field and Wick powers. Besov spaces, the regularity of the Gaussian free field on the torus, and Wick powers in two dimensions. Notes: Sections 2.1–2.3.
02:31 Littlewood–Paley blocks and Besov spaces
13:08 Bernstein inequality, embeddings and Fourier multipliers
29:39 Regularity of the Gaussian free field on the torus
54:38 Local functionals and their regularisation
1:05:24 Hermite polynomials, Wick powers and renormalisation constants
1:23:36 Regularity of the Wick powers in two dimensions by hypercontractivity
Lecture 3 (Thursday 3 July): Nelson's construction and the Boué–Dupuis formula. Wick powers in three dimensions and in weighted spaces, Nelson's construction of \(\varphi^4_2\) on \(\mathbb{T}^2\), and the variational approach. Notes: Sections 2.1–2.4 and 3.
00:06 Wick powers, renormalisation and Wiener chaos
09:57 Wick powers in three dimensions
19:06 Weighted Besov spaces and Wick powers on \(\mathbb{R}^2\)
35:35 Nelson's construction of \(\varphi^4_2\) on \(\mathbb{T}^2\)
1:03:24 The Gibbs variational principle
1:10:45 Relative entropy on Wiener space, Föllmer drift and Girsanov theorem
1:21:21 The Boué–Dupuis formula; an example with Lipschitz functionals
Lecture 4 (Saturday 5 July): Variational bounds for \(\varphi^4_2\). The martingale embedding of the Gaussian free field and bounds on the variational problem for \(\varphi^4_2\) that are uniform in the cutoff; first steps in infinite volume. Notes: Sections 3.2 and 4.
00:05 The Gibbs variational principle and the Girsanov density
07:31 Weak formulation and Boué–Dupuis controls
20:37 Martingale embedding of the Gaussian free field
30:27 The Boué–Dupuis formula for \(\varphi^4_2\)
44:22 Shifted Wick powers and the coercive term
50:58 Pathwise estimates uniform in the cutoff
1:15:24 Infinite volume and the Euler–Lagrange equation
Lecture 5 (Tuesday 8 July): \(\varphi^4_2\) in infinite volume. Construction of \(\varphi^4_2\) on \(\mathbb{R}^2\) from a priori estimates on the Euler–Lagrange equation that do not depend on the volume; beginning of large deviations. Notes: Sections 4 and 4.1.
00:05 Volume cutoff and tightness
06:42 Uniqueness and phase transitions
10:01 The Euler–Lagrange equation as a forward–backward equation; open problems
17:27 Minimisers and a coupling of \(\varphi^4_2\) with the Gaussian free field
35:24 Weighted energy estimates uniform in the volume
1:03:34 Tightness and the equation in infinite volume
1:17:30 Large deviations in the semiclassical limit
Lecture 6 (Wednesday 9 July): Large deviations and exponential integrability. The large deviation principle for \(\varphi^4_2\), exponential functionals through perturbed variational problems, and the exponential interaction. Notes: Sections 3.1, 4.1 and 5.
00:05 Large deviations as a semiclassical limit
08:21 The small-noise forward–backward equation
23:09 The limiting elliptic equation and uniqueness for convex functionals
33:02 Exponential functionals through perturbed variational problems
47:21 Exponential integrability of \(\varphi^4_2\)
59:07 Completion of the large deviation principle; discussion
1:10:55 The exponential model and Gaussian multiplicative chaos
Lecture 7 (Thursday 10 July): The exponential interaction; \(\varphi^4_3\). Construction and uniqueness of the exponential model in infinite volume; set-up of the variational problem for \(\varphi^4_3\) on the torus. Notes: Sections 5, 6 and 7.
00:56 Gaussian multiplicative chaos
04:20 The exponential model and its variational formula
14:50 The Euler–Lagrange equation, the sign of the drift and infinite volume
28:12 Uniqueness by convexity
40:28 Why the argument fails for \(\varphi^4_2\); a remark on sine-Gordon
51:09 \(\varphi^4_3\): scale-by-scale decomposition of the Gaussian free field
58:36 The Boué–Dupuis functional for \(\varphi^4_3\) and its counterterms
1:12:33 Completion of the square and change of control
Lecture 8 (Saturday 12 July): Paraproducts and renormalisation of \(\varphi^4_3\). Paraproducts and commutator estimates, and the second renormalisation of \(\varphi^4_3\). Notes: Sections 7, 7.1 and 7.2.
00:05 Recap of the variational problem for \(\varphi^4_3\)
05:04 The new control and the regularity of the drift
28:55 Paraproducts and resonant products
49:27 Duality and commutator lemmas
58:53 Commutators with Fourier multipliers
1:03:46 Decomposition of the quadratic term
1:21:05 The renormalised product and the logarithmic counterterm
1:29:17 Second completion of the square; outlook