# Probability Seminar

# Spring 2020

**Thursdays in 901 Van Vleck Hall at 2:30 PM**, unless otherwise noted.
**We usually end for questions at 3:20 PM.**

If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu

## January 23, 2020, Timo Seppalainen (UW Madison)

**Non-existence of bi-infinite geodesics in the exponential corner growth model**

Whether bi-infinite geodesics exist has been a significant open problem in first- and last-passage percolation since the mid-80s. A non-existence proof in the case of directed planar last-passage percolation with exponential weights was posted by Basu, Hoffman and Sly in November 2018. Their proof utilizes estimates from integrable probability. This talk describes an independent proof completed 10 months later that relies on couplings, coarse graining, and control of geodesics through planarity and increment-stationary last-passage percolation. Joint work with Marton Balazs and Ofer Busani (Bristol).

## January 30, 2020, Scott Smith (UW Madison)

**Quasi-linear parabolic equations with singular forcing**

The classical solution theory for stochastic ODE's is centered around Ito's stochastic integral. By intertwining ideas from analysis and probability, this approach extends to many PDE's, a canonical example being multiplicative stochastic heat equations driven by space-time white noise. In both the ODE and PDE settings, the solution theory is beyond the scope of classical deterministic theory because of the ambiguity in multiplying a function with a white noise. The theory of rough paths and regularity structures provides a more quantitative understanding of this difficulty, leading to a more refined solution theory which efficiently divides the analytic and probabilistic aspects of the problem, and remarkably, even has an algebraic component.

In this talk, we will discuss a new application of these ideas to stochastic heat equations where the strength of the diffusion is not constant but random, as it depends locally on the solution. These are known as quasi-linear equations. Our main result yields the deterministic side of a solution theory for these PDE's, modulo a suitable renormalization. Along the way, we identify a formally infinite series expansion of the solution which guides our analysis, reveals a nice algebraic structure, and encodes the counter-terms in the PDE. This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.

## February 6, 2020, Cheuk-Yin Lee (Michigan State)

**Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points**

In this talk, we will discuss sample path properties of stochastic partial differential equations (SPDEs). We will present a sharp regularity result for the stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. We prove the exact modulus of continuity via the property of local nondeterminism. We will also discuss the existence problem for multiple points (or self-intersections) of the sample paths of SPDEs. Our result shows that multiple points do not exist in the critical dimension for a large class of Gaussian random fields including the solution of a linear system of stochastic heat or wave equations.

## February 13, 2020, Jelena Diakonikolas (UW Madison)

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## February 20, 2020, Philip Matchett Wood (UC Berkeley)

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## February 27, 2020, No seminar

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## March 5, 2020, Jiaoyang Huang (IAS)

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## March 12, 2020, No seminar

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## March 19, 2020, Spring break

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## March 26, 2020, Philippe Sosoe (Cornell)

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## April 2, 2020, Tianyu Liu (UW Madison)

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## April 9, 2020, Alexander Dunlap (Stanford)

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## April 16, 2020, Jian Ding (University of Pennsylvania)

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## April 22-24, 2020, FRG Integrable Probability meeting

3-day event in Van Vleck 911

## April 23, 2020, Martin Hairer (Imperial College)

Wolfgang Wasow Lecture at 4pm in Van Vleck 911

## April 30, 2020, Will Perkins (University of Illinois at Chicago)

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