2018/09/04 by Chatterjee, Sourav, Dunlap, Alexander · 4 citations
#35R60 #60H15 #81T15 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.1809.00803
The (d+1)-dimensional KPZ equation is the canonical model for the growth of rough d-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for d=1 has been achieved in recent years, and the case d≥ 3 has also seen some progress. The most physically relevant case of d=2, however, is not very well-understood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the d=2 case is neither ultraviolet superrenormalizable like the d=1 case nor infrared superrenormalizable like the d≥ 3 case. Moreover, unlike in d=1, the Cole-Hopf transform is not directly usable in d=2 because solutions to the multiplicative stochastic heat equation are distributions rather than functions. In this article we show the existence of subsequential scaling limits as ε → 0 of Cole-Hopf solutions of the (2+1)-dimensional KPZ equation with white noise mollified to spatial scale ε and nonlinearity multiplied by the vanishing factor |logε|-1/2. We also show that the scaling limits obtained in this way do not coincide with solutions to the linearized equation, meaning that the nonlinearity has a non-vanishing effect. We thus propose our scaling limit as a notion of KPZ evolution in 2+1 dimensions.