Constructing a solution of the $(2+1)$-dimensional KPZ equation

Abstract

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 wellunderstood 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 | log ε| − 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.where ν, λ and D are strictly positive parameters andẆ denotes a standard space-time white noise on the two-dimensional torus T 2 = R 2 /Z 2 . More precisely, we define W to be a cylindrical Wiener process on L 2 (T 2 ) whose covariance operator is the identity, as in [13] or [3], and thenẆ is its (distributional) derivative in time. Thus, formally we have

Gaussian fluctuations from the 2D KPZ equation

“…A comparison with [6] and [7]. The basic ideas behind our approach and the one in [6] are similar, and the key is to modify the partition function so that log Z( t ε 2 , x ε ) can be "linearized" in some sense.…”

“…The approach in [7] relies on the Feynman-Kac formula and the concentration inequality to control the intersection time of two polymer paths. While a naive application of the Gaussian-Poincaré inequality fails for a similar reason as our Lemma 4.1 does not allow δ → 0, the authors have designed a clever recursive scheme that is similar to perturbative renormalization, using which they obtained the desired estimates to prove the tightness.…”

“…For the KPZ equation, progresses in d ≥ 3 can be found in [15,20], where results similar to Theorem 1.1 were proved. In two dimensions, the tightness of {h ε } ε∈(0, 1) , as a sequence of random distributions, was proved in the recent work of Chatterjee-Dunlap [7]. To prove Theorem 1.1, we implement the same strategy laid out in [15].…”

“…While their result is more general and covers the entire "subcritical" regime, the proof presented here seems to be simpler and offers another perspective. We discuss the approaches of [6,7] in Section 2.4.…”

Fluctuations of the solutions to the KPZ equation in dimensions three and higher

“…We refer to the reviews [15,36] for a more complete list of references. In d = 2, some relevant results can be found in [4,7,8,9,10,17,37].…”