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2022 Weak universality of dynamical Φ34: polynomial potential and general smoothing mechanism
Dirk Erhard, Weijun Xu
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Electron. J. Probab. 27: 1-43 (2022). DOI: 10.1214/22-EJP833


We consider a class of stochastic reaction-diffusion equations on the three dimensional torus. The non-linearities are odd polynomials in the weakly non-linear regime, and the smoothing mechanisms are very general higher order perturbations of the Laplacian. The randomness is the space-time white noise without regularisation. We show that these processes converge to the dynamical Φ34(λ) model, where the coupling constant λ has an explicit expression involving non-trivial interactions between all details of the smoothing mechanism and the non-linearity, even though they all formally vanish in the limit.

Funding Statement

D. E. gratefully acknowledges financial support from the National Council for Scientific and Technological Development – CNPq via a Universal grant 409259/2018-7 and a Bolsa de Produtividade 303520/2019-1. W.X. gratefully acknowledges financial support from the Engineering and Physical Sciences Research Council through the fellowship EP/N021568/1.


Part of the work was completed when both authors visited The Hausdorff Research Institute for Mathematics during the junior trimester programme “Randomness, PDEs and Nonlinear Fluctuations”. We thank the hospitality and the financial support of HIM.


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Dirk Erhard. Weijun Xu. "Weak universality of dynamical Φ34: polynomial potential and general smoothing mechanism." Electron. J. Probab. 27 1 - 43, 2022.


Received: 19 January 2022; Accepted: 26 July 2022; Published: 2022
First available in Project Euclid: 29 August 2022

Digital Object Identifier: 10.1214/22-EJP833

Primary: 60H15

Keywords: dynamical Φ34 , general smoothing mechanism , Weak universality


Vol.27 • 2022
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