Organized by: Dominique Bakry (Toulouse), Szymon Peszat (Cracow), and Bogusław Zegarliński (London).

The meeting will review recent results in the area of discrete and continuous probability/stochastics intertwined with related problems and methods of mathematical analysis. Besides a number of presentation by international participants, the meeting will include one 6 hrs mini-course (suitable for PhD students and young researchers).

The venue is the Institute of Mathematics, Polish Academy of Sciences, Św. Tomasza 30/7.

**Previous and related
activities:
**

- Workshop on Stochastic Analysis and Related Fields, Cracow 20 - 22 July, 2009
- Workshop on Analysis and Stochastic Analysis, Cracow 11 - 15 July, 2011
- CNRS-PAN Mathematics Summer Institute, Cracow 9 - 13 July, 2012
- Noncommutative Workshop, Cracow 9-12 September 2013
- CNRS-PAN Mathematics Summer Institute, Cracow 1 - 7 July, 2013

**Minicourse:**

- Thierry Lévy
(Paris) Connections in physics: an introduction to the
stochastic quantisation
of gauge theories

**Abstract: The best available physical models of the world at the subatomic scale are gauge theories : theories in which part of the physical substance (the electromagnetic potential, for instance) is modelled by a connection on a principal bundle. On the mathematical side, gauge theories provide deep insights into low-dimensional geometry and topology, and are a rich source of intuition (for a handful of privileged people, at least) and of fascinating hard problems (for the rest of us). The goal of these lectures is to outline a path which connects classical nineteenth century physics to something of the formidable edifice of quantum field theory, and more specifically of the formulation in terms of functional integrals of gauge theories. We will try to understand how physicists, who once were solving the Maxwell equations, ended up integrating functionals of geometric nature over the infinite dimensional space of connections over space-time. We will then discuss some of the mathematical issues and perspectives which arise from the consideration of these functional integrals.**

**
List of participants:**

- D. Bakry (Toulouse)
- D. Chafaï (Paris)

- C. Delan (Toulouse)

- T. Lévy (Paris)

- S. Li (Toulouse)

- E. Marciniak (Cracow)

- M. Ottobre (London)

- S. Peszat (Cracow)

- M. Pitera (Cracow)

- P. Przybyłowicz (Cracow)

- M. Riedle (London)

- P. Rola (Cracow)

- D. Stroock (Cambridge, USA)

- A. Sulima (Cracow)

- D. Tarłowski (Cracow)

- J. Trybuła (Cracow)

- D. Zawisza (Cracow)

- B. Zegarliński (London)

For more information contact S. Peszat, e-mail napeszat[AT]cyf-kr.edu.pl

**Tentative Program: **

**Wednesday 25 June Arrival day
**

**Thursday 26 June**

- 10.00 - 11.30 Registration
- 11.30 - 11.50 T&C
- 11.50 - 12.30 D. Zawisza: Monotone mean-variance preferences in continuous time stochastic factor model
- 14.30 - 16.30 T. Lévy: (Mini-course)
- 16.30 - 16.50 T&C
- 16.50 - 18.00 B. Zegarliński: Analysis of reaction diffusion systems via log-Sobolev inequalities

- 10.00 - 11.30 S. Peszat: Smoothening effects of transition semigroups
- 11.30 - 11.50 T&C
- 11.50 - 12.50 D. Stroock: Gauss applied to Banach

**Saturday 28 June**

- Excursion

**Sunday 29 June**

- Excursion

**Monaday 30 June**

- 10.00 - 11.30 D. Bakry: About hypergroup property
- 11.30 - 11.50 T&C
- 11.50 - 12.20 P. Rolla: Arbitrage in markets with bid-ask spreads
- 14.30 - 16.30 T. Lévy: (Mini-course)
- 16.30 - 16.50 T&C
- 16.50 - 17.20 D. Tarłowski: Nonautonomous dynamical systems in stochastic global optimization

**Tuesday 1 July
**

- 10.00 - 11.30 D. Chafaï: Random walks on regular graphs and the semicircle law
- 11.30 - 11.50 T&C
- 11.50 - 12.30 M. Pitera: On time consistency for dynamic utility measures
- 15.00 - 16.30 M. Ottobre: Diffusion limits for Random Walk Metropolis algorithm
- 16.30 - 16.50 T&C
- 16.50 - 17.20 E. Marciniak: On optimal dividend problem for an insurance risk models with surplus-dependent premiums
- 18.30 - 20.30 History of Science Evening, Aula Jagiellońska, Collegium Maius, D. Stroock: "Mark Kac, a Simple Pole in Ithaca New York".

**Wednesday 2 July**

- 10.00 - 12.00 T. Lévy: (Mini-course)
- 14.30 - 16.00 M. Riedle: Cylindrical Lévy processes
- 19.00 - Conference Dinner

**Thursday 3 July**

- 10.00 - 11.00: Discussion groups
- 11.00 - 11.30 T&C
- 11.30 - 12.30: Discussion groups
- 15.00 - 17.30 Discussion Groups
- 17.30 - 18.00 T&C

**Friday 4 July**

- 10.00 - 11.00 Discussion groups
- 11.00 - 11.50 T&C
- 11.50 - 12.30 Discussion groups
- 14.30 - 16.30 Discussion Groups
- 16.30 - 16.50 T&C