## S4F2 - Graduate Seminar on Stochastic Analysis:

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

**SS21: Tuedays 10–12. **

Due to the CoVid-19 pandemic, we must arrange the seminar via video (zoom). The link has been sent to the participans via email.

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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:

**20.4: Lukas Gehring:**potential theory, Feynman-Kac formula [S, Ch.1.1-1.4] (Prerequisites for this: basics in continuous-time Markov chains)**27.4: Mike Schäfer:**Gaussian free field, its properties, some potential theory [S, Ch.1.3, Propositions 1.10 and 1.11] and [S, Ch.2.1] (Prerequisites for this: basic probability, Gaussian process should suffice)**4.5: Janis Papewalis:**The classical First and Second Ray-Knight Theorems [MR, Ch.2.6-2.7] (Prerequisites for this: Brownian motion and Brownian local time)**11.5: Hannah Westhoff:**Measures of paths from x to y, Dynkin isomorphism theorem [S, Ch.1.5 and Ch.2.2-2.3] (Prerequisites for this: continuous-time Markov processes, definition of Gaussian free field)**18.5: Julian Gödtel:**(rooted, pointed) loops, restriction property [S, Ch.3.1-3.3], see also [LJ] (Prerequisites for this: continuous-time Markov processes)**1.6: Strahinja Kajganic:**Local time for the loops [S, Ch.3.4] (Prerequisites for this: continuous-time Markov processes, definition of Markovian loops)**8.6: Panos Zografos:**Poisson point process of unrooted loops [S, Ch.3.5 and 4.1] (Prerequisites for this: continuous-time Markov processes, stochastic processes (PPP))**29.6: Johanna Thierkopf:**Occupation field of Poisson gas of loops, Le Jan’s Isomorphism theorem [S, Ch.4.2] (Prerequisites for this: Poisson point process of loops and local time)**6.7: Chunqiu Song:**Symanzik’s representation formula for moments, other identities [S, Ch.4.3-4.4] (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 [MR, Ch.3.6] (Prerequisites for this: continuous-time Markov processes and some stochastic calculus)
- Eisenbaum isomorphism theorem and generalizations [S, Ch.2.3-2.4] (Prerequisites for this: continuous-time Markov processes, definition of Gaussian free field)
- Inverting Ray-Knight identity <https://arxiv.org/abs/1311.6622> (Prerequisites for this: Gaussian free field, continuous-time Markov processes, local time, martingales, uniform integrability)
- Discrete-time loop soup and occupation times <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 Perlman: Loop measures and the Gaussian free field <http://arxiv.org/pdf/1403.4285.pdf>
- Lawler and Limic: Random Walk: A Modern Introduction (book), Chapter 9

*Further reading:*

- Lupu: From loop clusters and random interlacement to the free field <http://arxiv.org/abs/1402.0298>
- Lupu: Convergence of the two-dimensional random walk loop soup clusters to CLE <http://arxiv.org/abs/1502.06827>
- Lawler & Werner: The Brownian loop soup <http://arxiv.org/abs/math/0304419>
- Lawler: Proc. ICM2018 <http://eta.impa.br/dl/PL009.pdf>