2D ANISOTROPIC KPZ AT STATIONARITY: SCALING, TIGHTNESS AND NONTRIVIALITY

• 成果类型: Article
• 署名作者: Cannizzaro, Giuseppe; Erhard, Dirk; Schonbauer, Philipp
• 署名单位: University of Warwick; Universidade Federal da Bahia; Imperial College London
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1446
• 发表日期: 2021
• 页码: 122-156
• 关键词: stochastic burgers; growth-process; heat-equation; (2+1)d growth; limit

摘要

In this work we focus on the two-dimensional anisotropic KPZ (aKPZ) equation, which is formally given by partial derivative(t)h = v/2 Delta h + lambda(partial derivative(1)h)(2) - (partial derivative(2)h)(2)) + v(1/2)xi, where xi denotes a noise which is white in both space and time, and lambda and nu are positive constants. Due to the wild oscillations of the noise and the quadratic nonlinearity, the previous equation is classically ill posed. It is not possible to linearise it via the Cole-Hopf transformation and the pathwise techniques for singular SPDEs (the theory of regularity structures by M. Hairer or the paracontrolled distributions approach of M. Gubinelli, P. Imkeller, N. Perkowski) are not applicable. In the present work we consider a regularised version of aKPZ which preserves its invariant measure. We prove the existence of subsequential limits once the regularisation is removed, provided lambda and nu are suitably renormalised. Moreover, we show that, in the regime in which v is constant and the coupling constant lambda converges to 0 as the inverse of the square root logarithm, any limit differs from the solution to the linear equation obtained by simply dropping the nonlinearity in aKPZ.