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The stationary AKPZ equation : logarithmic superdiffusivity
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Cannizzaro, Giuseppe, Erhard, Dirk and Toninelli, Fabio (2023) The stationary AKPZ equation : logarithmic superdiffusivity. Communications on Pure and Applied Mathematics . doi:10.1002/cpa.22108 ISSN 00103640. (In Press)

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Official URL: https://doi.org/10.1002/cpa.22108
Abstract
We study the two‐dimensional Anisotropic KPZ equation (AKPZ) formally given by ∂ t H = 1 2 Δ H + λ ( ( ∂ 1 H ) 2 − ( ∂ 2 H ) 2 ) + ξ , $$\begin{equation*} \hspace*{3.4pc}\partial _t H=\frac{1}{2}\Delta H+\lambda ((\partial _1 H)^2(\partial _2 H)^2)+\xi , \end{equation*}$$ where ξ is a space‐time white noise and λ is a strictly positive constant. While the classical two‐dimensional KPZ equation, whose nonlinearity is  ∇ H  2 = ( ∂ 1 H ) 2 + ( ∂ 2 H ) 2 $\nabla H^2=(\partial _1 H)^2+(\partial _2 H)^2$ , can be linearised via the Cole‐Hopf transformation, this is not the case for AKPZ. We prove that the stationary solution to AKPZ (whose invariant measure is the Gaussian Free Field (GFF)) is superdiffusive: its diffusion coefficient diverges for large times as log t $\sqrt {\mathop {\mathrm{log}}\nolimits t}$ up to log log t $\mathop {\mathrm{log}}\nolimits \mathop {\mathrm{log}}\nolimits t$ corrections, in a Tauberian sense. Morally, this says that the correlation length grows with time like t 1 / 2 × ( log t ) 1 / 4 $t^{1/2}\times (\mathop {\mathrm{log}}\nolimits t)^{1/4}$ . Moreover, we show that if the process is rescaled diffusively ( t → t / ε 2 , x → x / ε , ε → 0 $t\rightarrow t/\varepsilon ^2, x\rightarrow x/\varepsilon , \varepsilon \rightarrow 0$ ), then it evolves non‐trivially already on time‐scales of order approximately 1 /  log ε  ≪ 1 $1/\sqrt {\mathop {\mathrm{log}}\nolimits \varepsilon }\ll 1$ . Both claims hold as soon as the coefficient λ of the nonlinearity is non‐zero. These results are in contrast with the belief, common in the mathematics community, that the AKPZ equation is diffusive at large scales and, under simple diffusive scaling, converges to the two‐dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e., the case λ = 0 $\lambda =0$ ).
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics Q Science > QC Physics 

Divisions:  Faculty of Science, Engineering and Medicine > Science > Mathematics  
SWORD Depositor:  Library Publications Router  
Library of Congress Subject Headings (LCSH):  Stochastic partial differential equations, Gaussian free field , Surfaces (Physics), Interfaces (Physical sciences), Interfaces (Physical sciences)  Mathematics  
Journal or Publication Title:  Communications on Pure and Applied Mathematics  
Publisher:  John Wiley & Sons  
ISSN:  00103640  
Official Date:  2023  
Dates: 


DOI:  10.1002/cpa.22108  
Status:  Peer Reviewed  
Publication Status:  In Press  
Access rights to Published version:  Open Access (Creative Commons)  
Copyright Holders:  © 2023 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.  
Date of first compliant deposit:  4 August 2023  
Date of first compliant Open Access:  4 August 2023  
RIOXX Funder/Project Grant: 


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