The APDE seminar on Monday, 4/18, will be given by Tadahiro Oh (University of Edinburgh) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (email@example.com).
Title: Gibbs measures, canonical stochastic quantization,
and singular stochastic wave equations
In this talk, I will discuss the (non-)construction of the focusing
Gibbs measures and the associated dynamical problems. This study was
initiated by Lebowitz, Rose, and Speer (1988) and continued by Bourgain
(1994), Brydges-Slade (1996), and Carlen-Fröhlich-Lebowitz (2016). In
the one-dimensional setting, we consider the mass-critical case, where a
critical mass threshold is given by the mass of the ground state on the
real line. In this case, I will show that the Gibbs measure is indeed
normalizable at the optimal mass threshold, thus answering an open
question posed by Lebowitz, Rose, and Speer (1988).
In the three dimensional-setting, I will first discuss the construction
of the $\Phi^3_3$-measure with a cubic interaction potential. This
problem turns out to be critical, exhibiting a phase transition:
normalizability in the weakly nonlinear regime and non-normalizability
in the strongly nonlinear regime. Then, I will discuss the dynamical
problem for the canonical stochastic quantization of the
$\Phi^3_3$-measure, namely, the three-dimensional stochastic damped
nonlinear wave equation with a quadratic nonlinearity forced by an
additive space-time white noise (= the hyperbolic $\Phi^3_3$-model). As
for the local theory, I will describe the paracontrolled approach to
study stochastic nonlinear wave equations, introduced in my work with
Gubinelli and Koch (2018). In the globalization part, I introduce a new,
conceptually simple and straightforward approach, where we directly work
with the (truncated) Gibbs measure, using the variational formula and
ideas from theory of optimal transport.
The first part of the talk is based on a joint work with Philippe Sosoe
(Cornell) and Leonardo Tolomeo (Bonn), while the second part is based on
a joint work with Mamoru Okamoto (Osaka) and Leonardo Tolomeo (Bonn).