Hyperbolic $P(Φ)_2$-model on the plane

Abstract

We study the hyperbolic Φ k+1 2 -model on the plane. By establishing coming down from infinity for the associated stochastic nonlinear heat equation (SNLH) on the plane, we first construct a Φ k+1 2 -measure on the plane as a limit of the Φ k+1 2 -measures on large tori. We then study the canonical stochastic quantization of the Φ k+1 2 -measure on the plane thus constructed, namely, we study the defocusing stochastic damped nonlinear wave equation forced by an additive space-time white noise (= the hyperbolic Φ k+1 2 -model) on the plane. In particular, by taking a limit of the invariant Gibbs dynamics on large tori constructed by the first two authors with Gubinelli and Koch (2021), we construct invariant Gibbs dynamics for the hyperbolic Φ k+1 2 -model on the plane. Our main strategy is to develop further the ideas from a recent work on the hyperbolic Φ 3 3 -model on the three-dimensional torus by the first two authors and Okamoto ( 2021), and to study convergence of the socalled enhanced Gibbs measures, for which coming down from infinity for the associated SNLH with positive regularity plays a crucial role. By combining wave and heat analysis together with ideas from optimal transport theory, we then conclude global well-posedness of the hyperbolic Φ k+1 2 -model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting Φ k+1 2 -measure on the plane under the dynamics of the parabolic Φ k+1 2 -model.

Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

“…While important milestones for the local well posedness theory for canonical stochastic quantisation equations are progressively been achieved, global well posedness and ergodicity results are both very rare. Indeed, local well posedness for equation (SDNLW) has been proven in [20] on T 2 , in [40,35] on R 2 , and in [34] on a general 2-dimensional compact manifold. A series of 3-dimensional results have been proven in [21] and [33] for the equation with quadratic nonlinearity, in [32,3,36] for cubic nonlinearities under the addition of some smoothing in the equation, and finally in [4] for the canonical stochastic quantisation equation for the Φ 4 3 measure.…”

Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space

“…To conquer the issue of the unboundedness of the noise, they used weighted Sobolev and Besov spaces and obtained a weighted H 2 a-priori bound for v ε with a logarithmic loss in ε. This approach of using the weighted Besov spaces in the study of stochastic PDEs was also used in [20,21,30,34]. In the case of the NLS (1.1), such an approach requires more assumptions on the regularity of the initial datum than those on the T 2 setting.…”

Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space

“…To conquer the issue of the unboundedness of the noise, they used weighted Sobolev and Besov spaces and obtained a weighted H 2 a priori bound for v ε with a logarithmic loss in ε. This approach of using the weighted Besov spaces in the study of stochastic PDEs was also used in [16,17,20,21]. In the case of the NLS (1.1), such an approach requires more assumptions on the regularity of the initial data than those on the T 2 setting.…”