Space–time fluctuation of the Kardar–Parisi–Zhang equation in d≥3 and the Gaussian free field

Abstract

We study the solution ${\mathit{h}}_{\mathit{\epsilon }}$ of the Kardar–Parisi–Zhang (KPZ) equation in ${\mathbb{R}}^{\mathit{d}}\times [0,\infty )$ for $\mathit{d}\ge 3$:

$$\frac{\partial }{\partial \mathit{t}}{\mathit{h}}_{\mathit{\epsilon }}=\frac{1}{2}\mathrm{\Delta }{\mathit{h}}_{\mathit{\epsilon }}\mathbf{+}[\frac{1}{2}|\nabla {\mathit{h}}_{\mathit{\epsilon }}{|}^2-{\mathit{C}}_{\mathit{\epsilon }}]\mathbf{+}\mathit{\beta }{\mathit{\epsilon }}^{\frac{\mathit{d}-2}{2}}{\mathit{\xi }}_{\mathit{\epsilon }},{\mathit{h}}_{\mathit{\epsilon }}(0,\mathit{x})=0.$$

Here $\mathit{\beta }>0$ is a parameter called the disorder strength, ${\mathit{\xi }}_{\mathit{\epsilon }}=\mathit{\xi }\star {\mathit{\varphi }}_{\mathit{\epsilon }}$ is a spatially smoothened (at scale ε) Gaussian space–time white noise and ${\mathit{C}}_{\mathit{\epsilon }}$ is a divergent constant as $\mathit{\epsilon }\to 0$. When β is sufficiently small and $\mathit{\epsilon }\to 0$, ${\mathit{h}}_{\mathit{\epsilon }}(\mathit{t},\mathit{x})-{\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})\to 0$ in probability where ${\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})$ is the stationary solution of the KPZ equation – more precisely, ${\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})$ solves the above equation with a random initial condition (that is independent of the driving noise ξ) and its marginal law is constant in $(\mathit{\epsilon },\mathit{t},\mathit{x})$. In the present article we quantify the rate of the above convergence in this regime and show that the fluctuations ${({\mathit{\epsilon }}^{1-\frac{\mathit{d}}{2}}[{\mathit{h}}_{\mathit{\epsilon }}(\mathit{t},\mathit{x})-{\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})])}_{\mathit{x}\in {\mathbb{R}}^{\mathit{d}},\mathit{t}>0}$ about the stationary solution converge pointwise (with finite dimensional distributions in space and time) to a Gaussian free field convoluted with the deterministic heat equation. We also identify the fluctuations of the stationary solution itself and show that the rescaled averages ${\int }_{{\mathbb{R}}^{\mathit{d}}}\mathrm{d}\mathit{x}\mathit{\phi }(\mathit{x}){\mathit{\epsilon }}^{1-\frac{\mathit{d}}{2}}[{\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})-\mathbb{E}{\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})]$ converge to that of the stationary solution of the stochastic heat equation with additive noise, but with (random) Gaussian free field marginals (instead of flat initial condition).

Nous étudions la solution ${\mathit{h}}_{\mathit{\epsilon }}$ de l’équation de Kardar–Parisi–Zhang (KPZ) sur ${\mathbb{R}}^{\mathit{d}}\times [0,\infty )$ avec $\mathit{d}\ge 3$ :

$$\frac{\partial }{\partial \mathit{t}}{\mathit{h}}_{\mathit{\epsilon }}=\frac{1}{2}\mathrm{\Delta }{\mathit{h}}_{\mathit{\epsilon }}\mathbf{+}[\frac{1}{2}|\nabla {\mathit{h}}_{\mathit{\epsilon }}{|}^2-{\mathit{C}}_{\mathit{\epsilon }}]\mathbf{+}\mathit{\beta }{\mathit{\epsilon }}^{\frac{\mathit{d}-2}{2}}{\mathit{\xi }}_{\mathit{\epsilon }},{\mathit{h}}_{\mathit{\epsilon }}(0,\mathit{x})=0.$$

Ici $\mathit{\beta }>0$ est un paramètre appelé la force du désordre, ${\mathit{\xi }}_{\mathit{\epsilon }}=\mathit{\xi }\star {\mathit{\varphi }}_{\mathit{\epsilon }}$ est un bruit blanc gaussien espace-temps régularisé en espace (à l’échelle ε) et ${\mathit{C}}_{\mathit{\epsilon }}$ est une constante qui diverge lorsque $\mathit{\epsilon }\to 0$. Lorsque β est suffisamment petit et $\mathit{\epsilon }\to 0$, ${\mathit{h}}_{\mathit{\epsilon }}(\mathit{t},\mathit{x})-{\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})\to 0$ en probabilité où ${\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})$ est la solution stationnaire de l’équation KPZ – plus précisément, ${\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})$ est solution de l’équation ci-dessus avc une condition initiale aléatoire (laquelle est indépendante du bruit ξ) et dont la loi marginale est constante en $(\mathit{\epsilon },\mathit{t},\mathit{x})$. Dans cet article nous quantifions la vitesse de la convergence ci-dessus et nous montrons que les fluctuations ${({\mathit{\epsilon }}^{1-\frac{\mathit{d}}{2}}[{\mathit{h}}_{\mathit{\epsilon }}(\mathit{t},\mathit{x})-{\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})])}_{\mathit{x}\in {\mathbb{R}}^{\mathit{d}},\mathit{t}>0}$ autour de la solution stationnaire converge ponctuellement (de manière jointe pour un nombre fini de points de l’espace et du temps) vers un champ libre gaussien convolé à l’équation de la chaleur déterministe. Nous identifions également les fluctuations de l’équation stationnaire autour de sa moyenne et montrons que ${\int }_{{\mathbb{R}}^{\mathit{d}}}\mathrm{d}\mathit{x}\mathit{\phi }(\mathit{x}){\mathit{\epsilon }}^{1-\frac{\mathit{d}}{2}}[{\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})-\mathbb{E}{\mathfrak{h}}_{\mathit{\epsilon }}^{\mathrm{st}}(\mathit{t},\mathit{x})]$ converge vers la solution stationnaire de l’équation de la chaleur avec bruit additif, dont la loi marginale est donnée par le champ libre gaussien (au lieu de la condition initiale plate).