Gibbs measures 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 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.
In the second part of the talk, 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). In particular, I will describe the paracontrolled approach to study stochastic nonlinear wave equations, introduced in my previous work with Gubinelli and Koch.
If time permits, I will briefly describe the globalization part, 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 (Edinburgh), while the second part is based on a joint work with Mamoru Okamoto (Hiroshima) and Leonardo Tolomeo (Edinburgh).
15:15-16:15
佐野 めぐみ 氏 (奈良女子大学)
A simple proof of attainability for the Sobolev inequality