We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus $${\mathbb {T}}^3$$ T 3 . In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on $${\mathbb {T}}^3$$ T 3 by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing $$\Phi ^4_3$$ Φ 3 4 -model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.

Optimal transport distances are powerful tools to compare probability distributions and have found many applications in machine learning. Yet their algorithmic complexity prevents their direct use on large scale datasets. To overcome this challenge, practitioners compute these distances on minibatches i.e. they average the outcome of several smaller optimal transport problems. We propose in this paper an analysis of this practice, which effects are not well understood so far. We notably argue that it is equivalent to an implicit regularization of the original problem, with appealing properties such as unbiased estimators, gradients and a concentration bound around the expectation, but also with defects such as loss of distance property. Along with this theoretical analysis, we also conduct empirical experiments on gradient flows, GANs or color transfer that highlight the practical interest of this strategy.

We study a family of nonlinear damped wave equations indexed by a parameter ε > 0 and forced by a space-time white noise on the two dimensional torus, with polynomial and sine nonlinearities. We show that as ε → 0, the solutions to these equations converge to the solution of the corresponding two dimensional stochastic quantization equation. In the sine nonlinearity case, the convergence is proven over arbitrary large times, while in the polynomial case, we prove that this approximation result holds over arbitrary large times when the parameter ε goes to zero even with a lack of suitable global well-posedness theory for the corresponding wave equations.

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