Table of Contents / Contents

• Announcements
• Course Information
• How to Register
• Course Outline
• Course Materials and References
• Prerequisites — to review before class
• Related Links

Announcements

• Registration is open: send an e-mail by Friday, August 14th, 2026 (see How to Register below).
  
  
• Aug. 17th : School begins (online, via Zoom).
  
  
• Schedule: mornings 9:00–10:30 and 10:45–12:00 (lectures), afternoons 14:00–15:00 (lecture) and 15:10–16:00 (hands-on session) — all times are Kyiv time (one hour earlier in Paris).
  
  
• Aug. 21st : School ends.
  

Course Information

• Program: Online summer school, free of charge, primarily intended for Ukrainian students and open to the wider mathematical community. One intensive week: Monday 17 to Friday 21 August 2026.
  
  
• Audience: Ukrainian undergraduate and Master's students in mathematics, applied mathematics, physics, engineering or related fields; PhD students, researchers, and colleagues are most welcome.
  
  
• Format: Online (Zoom); the link will be sent to registered participants shortly before the school begins.
  
  
• Tuition: Free of charge.
  
  
• Language: Lectures and written materials in English; questions may also be asked in French or Ukrainian (written).
  
  
• Teaching Team: Victor Rabiet (Spartacus-IDH, Télécom Paris) and Prof. Olivier Rioul (Télécom Paris)
  
  
• Materials: Lecture notes and exercise sheets (with solutions, including Python simulations) are distributed to participants.
  
  
• Flyer: https://perso.telecom-paristech.fr/rioul/summer-school2026.pdf

How to Register

• Send an e-mail to victor.rabiet@telecom-paris.fr with subject “Summer school registration”, indicating your name, university, and year of study, by Friday 14 August 2026. You will receive the Zoom link and the course materials by e-mail.
  

Course Outline

• Objective and Summary : How do errors and uncertainties propagate through a computation? Gauss answered this question for finitely many small Gaussian errors; the modern answer, due to Nicolas Bouleau following his work with Francis Hirsch on Dirichlet forms, views Dirichlet forms as tools for differential calculus compatible with probability theory: to each random quantity one attaches an operator (the “carré du champ” operator Γ) that propagates its variance. The theory works directly in infinite dimensions (Wiener, Monte-Carlo and Poisson spaces) and can characterize the errors of stochastic differential equations — something Gauss-type error calculus cannot do. The field is fully alive: a conjecture stated by Bouleau and Hirsch in 1986 (energy image density, EID) was settled only in October 2025.
  
  
• Who should attend?  The course follows an inter-semester course taught for three years at Télécom Paris. It is intended for a broad audience of students in mathematics, applied mathematics, physics, engineering or related fields, as well as PhD students, researchers, and colleagues.
  
  
• Prerequisites : The course is self-contained: stochastic calculus is one of the subjects taught, not a prerequisite — a basic undergraduate background in probability suffices (see details below).
  
  
• Day-by-day programme:
  

1. Monday 17 — Probabilistic foundations and Gauss-type error calculus. A refresher on measure theory: σ-algebras, simple functions, π- and λ-systems, the Radon–Nikodym theorem; conditional expectation and conditional variance; Gauss's formulas for small errors and their connection with differential operators; classification of error calculi, examples in finite dimension.
   
2. Tuesday 18 — Construction of Brownian motion; from error germ to error structures. Random processes; the construction of Brownian motion via Schauder's tent functions; an intuitive introduction to error structures: orders of magnitude of errors, from the error germ γ to the carré du champ operator Γ; simulation of random variables and of Brownian motion.
   
3. Wednesday 19 — Stochastic calculus: Itô integral. Quadratic variation of Brownian motion; filtrations, adaptation and martingales; construction of the Itô integral: progressively measurable processes, integration of step processes, the Itô isometry and its extension by density; Itô processes and Itô's formula, with first examples of stochastic differential calculus.
4. Thursday 20 — Stochastic calculus: diffusions; semigroups and Dirichlet forms. The Markov property, diffusions and their simulation by the Euler scheme (the Langevin equation, the Black–Scholes model); strongly continuous semigroups on a Banach space; the Ornstein–Uhlenbeck semigroup; Dirichlet forms; the Dirichlet form associated with the Langevin equation as a worked example of error propagation through an SDE.
5. Friday 21 — Error structures in the general case; applications. General definition of an error structure (Ω, A, P, D, Γ): dense domain D, functional calculus of class C¹∩Lip, closedness of the form (Dirichlet norm), Markovian case; images and products of error structures; applications and outlook: sensitivity of stochastic models, the EID conjecture (settled in 2025), current research on thermal fluctuations in statistical physics.
   
   

• Exercises and simulations: Each afternoon session ends with one hour of applications: exercise sheets (with solutions) and Python notebooks (simulation of Brownian motion, Euler scheme for diffusions), so that participants can experiment on their own.
  

Course Materials and References

• Handout : lecture notes and exercise sheets (with solutions, including Python simulations) are sent to registered participants by e-mail.
• Bibliography:

1. N. Bouleau, F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, De Gruyter, 1991 (where the EID conjecture is stated).
   
   
2. N. Bouleau, Error Calculus for Finance and Physics: The Language of Dirichlet Forms, De Gruyter, 2003.
   
   
3. N. Bouleau, The Mathematics of Errors, Springer, 2022 (French original: Théorie des erreurs, Spartacus-IDH & Cassini, 2019).
   
   
4. S. Eriksson-Bique, M. Murugan, On the energy image density conjecture of Bouleau and Hirsch, arXiv:2510.13659 (preprint), 2025.
   
   
5. For probability and stochastic calculus, in Ukrainian: М. В. Карташов, Імовірність, процеси, статистика, ВПЦ «Київський університет», Kyiv, 2008 — official PDF: https://probability.knu.ua/userfiles/kmv/VPS_Pv.pdf
   

Prerequisites — to review before class

• The school is accessible to anyone with a basic knowledge of probability, as taught in the first two years of undergraduate studies (Bachelor's level). Stochastic calculus itself is taught from scratch — no prior knowledge of it is assumed. Participants may want to review or expand their knowledge in the following areas:
  
  
  
  • Basic probability theory (Bachelor's level)
    
    
  • Measure-theoretic notions — conditional expectation, π- and λ-systems, Radon–Nikodym — recalled on the first day
    
    
  • Hilbert bases and L² convergence — helpful for the construction of Brownian motion and the Itô integral, though not indispensable
    
    
• The last day, devoted to error structures in the general case, is the most advanced one; its emphasis can be adjusted to the audience.
  

Useful Links

• Nicolas Bouleau's website
  
  
• The EID conjecture preprint (Eriksson-Bique & Murugan, 2025)
  
  
• Kartashov's textbook (in Ukrainian), official PDF
  

Contacts

• Victor Rabiet (Spartacus-IDH, Télécom Paris) — victor.rabiet@telecom-paris.fr
  
  
• Olivier Rioul (Télécom Paris) — olivier.rioul@telecom-paris.fr