The Edwards–Wilkinson Limit of the Random Heat Equation in Dimensions Three and Higher

Abstract

We consider the heat equation with a multiplicative Gaussian potential in dimensions d ≥ 3. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic heat equation (SHE) with additive white noise, with an effective variance.

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

“…Our result in d ≥ 3 can be viewed as a continuation of the previous works on the stochastic heat equation (SHE) [16,18] and as a counterpart of the recent work of Magnen and Unterberger [27], where the driving force is mollified in both temporal and spatial variables. While the proof in [27] is based on a multiscale expansion and a calculation of multi-point correlation functions, we present a probabilistic proof using the tools of Malliavin calculus.…”

“…In [18,Theorem 1.2], it was proved that the fluctuations of u ε are given by the same Edwards-Wilkinson model: 12) which can also be viewed as a special case of Theorem 1.2 with f(y) = y. In other words, when viewed as random fields, u ε (t, ⋅) and log u ε (t, ⋅) have the same limiting distribution!…”

“…The effective diffusion matrix a eff comes from the temporal correlation of the randomness, which was previously discussed in [5,19]. In [18,31], as a crucial ingredient of the proof, a Markov chain was constructed to model the evolution of the path increments and the fast mixing of the Markov chain drives a central limit theorem which gives rise to the effective diffusivity. We believe that the approach developed in this paper, combined with the Markov chain techniques used in [18], will give another proof of the result in [27] for random potentials that are not white in time.…”

“…In [18,31], as a crucial ingredient of the proof, a Markov chain was constructed to model the evolution of the path increments and the fast mixing of the Markov chain drives a central limit theorem which gives rise to the effective diffusivity. We believe that the approach developed in this paper, combined with the Markov chain techniques used in [18], will give another proof of the result in [27] for random potentials that are not white in time. We choose to work in the white in time setting since it is technically simpler to explain, but is also illustrative enough to reveal the main idea of the proof for the general case.…”

“…1 , B 2 . By[18, Corollary 4.4] and the fact that d ≥ 3, the proof is complete.◻ , A λ (t, x)) = inf{ W ε −W ε 2 ∶W ε ∈ A λ (t,x)}. Now we can apply [24, Lemma 4.5] to obtain > 0, where λ, c > 0 are chosen as in Lemma B.2.…”