## S4F2 - Graduate Seminar on Stochastic Analysis:

## Topics on Markov paths, Occupation times, and Fields

NB: Time has changed to TUESDAY 10-12

NB: You should register for the seminar via BASIS from 01 to 30 April 2021

**SS21: Tuedays 10–12. **

Due to the CoVid-19 pandemic, the seminar might be arranged via video (zoom).

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The goal is to explore links between occupation times, Gaussian free fields, Poisson gases of Markovian loops, etc. For example, Ray-Knight type theorems and Dynkin isomorphism theorems relate local time of a Markov process to an associated Gaussian process. One can also consider Markovian loops, whose occupation times are related to the Gaussian free field via Le Jan's isomorphism theorem. Such identities have been applied in mathematical physics (quantum field theory) and were further developed, becoming an important part of modern probability theory, in the last 20 years or so.

**Prerequisites:** Basics of Stochastic processes and Markov chains suffices. Most topics concern continuous-time processes, but there is some literature on discrete-time analogs. For some topics, Stochastic Analysis (Brownian motion, martingales, stochastic calculus) is needed. The topics will be distributed according to people's background.

If you have any questions or wishes, please contact me:

*firstname.lastname @ hcm.uni-bonn.de*

## Schedule (if you want to participate and didn't get a slot, or you are missing from the list, please contact me):

**Participants: please check that your slot is OK. I will send more detailed instructions during February.**

**13.4: Lukas:**Some potential theory and Feynman-Kac formula, from Chapters 1.1-1.4 of the lecture notes [S] (Prerequisites for this: basics in continuous-time Markov chains)**20.4: TBA:**Local time for Brownian motion, from Chapter 2.4 of the book [MR] (Prerequisites for this: Brownian motion and some stochastic calculus)**27.4: Mike:**Gaussian free field, its properties, and some potential theory from Chapter 1.3 and Chapter 2.1 of the lecture notes [S] (Prerequisites for this: basic probability, Gaussian process should suffice)**4.5: Janis:**The classical First and Second Ray-Knight Theorems, from Chapters 2.6-2.7 of the book [MR] (Prerequisites for this: Brownian motion and Brownian local time)**11.5: Hannah:**Measures of paths from x to y and Dynkin isomorphism theorem, from Chapter 2.2-2.3 of the lecture notes [S] (Prerequisites for this: continuous-time Markov processes, definition of Gaussian free field)**18.5: Julian:**Markovian loops and restriction property, from Chapter 3.1-3.3 of the lecture notes [S], see also [LJ] (Prerequisites for this: continuous-time Markov processes)**1.6: Strahinja:**Local time for Markovian loops, from Chapter 3.4 of the lecture notes [S] (Prerequisites for this: continuous-time Markov processes, definition of Markovian loops)**8.6: Panos:**Poisson point process of unrooted Markovian loops, from Chapter 3.5 and 4.1 of the lecture notes [S] (Prerequisites for this: continuous-time Markov processes, stochastic processes (PPP))**29.6: Johanna:**Occupation field of the Poisson gas of loops and Le Jan’s Isomorphism theorem, from Chapter 4.2 of the lecture notes [S] (Prerequisites for this: Poisson point process of loops and local time)**6.7: Chunqiu:**Symanzik’s representation formula for moments and other identities, from Chapter 4.3-4.4 of the lecture notes [S] (Prerequisites for this: Gaussian processes, definition of occupation field of the Poisson gas of loops)

## Additional Topics:

Some of these can be skipped, some can be added, and the order can be changed.

Local time for Markov processes, from Chapter 3.6 of the book [MR] (Prerequisites for this: continuous-time Markov processes and some stochastic calculus)

Inverting Ray-Knight identity, from the paper <https://arxiv.org/abs/1311.6622> (Prerequisites for this: Gaussian free field, continuous-time Markov processes, local time, martingales, uniform integrability)

Eisenbaum isomorphism theorem and generalizations, from Chapter 2.3-2.4 of the lecture notes [S] (Prerequisites for this: continuous-time Markov processes, definition of Gaussian free field)

Discrete-time loop soup and occupation times. from Lawler http://www.math.uchicago.edu/~lawler/lejan.pdf

## Literature:

*Background literature continuous-time:*

- [S]: Sznitman, Topics in Occupation Times and Gaussian Free Fields:
*<*https://ethz.ch/content/dam/ethz/special-interest/math/department/Research/Research_Groups/Sznitman/Publications/2011_special_topics.pdf*>* - [LJ]: Le Jan, Markov Paths, Loops, and Fields:
*<*https://arxiv.org/pdf/0808.2303.pdf*>* - [MR]: Marcus and Rosen, Markov Processes, Gaussian Processes and Local Times (book)

*Background literature discrete-time:*

- Lawler: lecture notes <http://www.math.uchicago.edu/~lawler/lejan.pdf>
- Lawler and Limic: Random Walk: A Modern Introduction (book), Chapter 9