Fluctuations for \(\nabla \varphi \) interface model on a wall.
The following model of fluctuations of an interface on a wall is studied: Let \(\Gamma =\{1,\dots ,N-1\}\) be a one-dimensional lattice. The dynamics of random variables \(\varphi _ {t}(x)\), \(x\in \Gamma \), describing the localization of the interface at time \(t\), is governed by a stochastic differential equation of Skorokhod type \[ \begin{aligned} d\varphi _ {t}(x) = &-\bigl \{V'(\varphi _ {t}(x) -\varphi _ {t}(x-1)) + V'(\varphi _ {t}(x)-\varphi _ {t}(x+1))\bigr \}\,dt \\ &+ \sqrt {2}\,dw_ {t}(x) + d\ell _ {t}(x), \quad x\in \Gamma , \end{aligned} \tag{1} \] \[ \varphi _ {t}(x)\geq 0, \;\ell _ \bullet (x)\;\text{nondecreasing}, \;\;\int ^ \infty _ {0}\varphi _ {t}(x)\,d\ell _ {t}(x) = 0, \;\;\varphi _ {t}(0) = \varphi _ {t}(N) = 0, \] where \(V\in C^ 2(\mathbb R)\) is a symmetric strictly convex potential. There exists a unique (tempered) invariant measure for (1). Let us consider the corresponding stationary solution \(\{\varphi _ {t}(x), \;x\in \Gamma \}\) to (1) and the random field \(h^ {N}\) defined by rescaling in space and time, \[ h^ {N}(t,\theta ) = \frac 1{N}\sum _ {x\in \Gamma } \varphi _ {N^ 2 t} (x)\mathbf 1_ {[x/N- 1/2N, x/N+1/2N[}(\theta ), \qquad 0\leq \theta \leq 1. \] It may be shown that \(h^ {N}(t,\theta )\) tends to \(0\) as \(N\to \infty \); the aim of the paper is to describe the asymptotic behaviour of fluctuations of \(h^ {N}\) around its limit. It is proven that \(\sqrt {N}h^ {N}(t,\theta )\) converges weakly on a suitable function space to the unique stationary weak solution \(\Phi \) to a stochastic partial differential equation with reflection \[ \frac {\partial \Phi }{\partial t}(t,\theta ) = q\frac {\partial ^ 2 \Phi }{\partial \theta ^ 2}(t,\theta ) + \sqrt {2}\dot B(t,\theta ) + \xi (t,\theta ), \quad \theta \in [0,1], \] \[ \Phi (t,\theta )\geq 0, \quad \int _ {\left [0,\infty \right [\times [0,1]} \Phi (t,\theta )\xi (dt\,d\theta ) = 0, \;\;\Phi (t,0) = \Phi (t,1) = 0, \] where \(\xi \) is a random measure on \([0,\infty [\times [0,1]\), \(\dot B\) is a space-time white noise and the constant \(q\) depends on the potential \(V\).
stochastic partial differential equations
scientific article; zbMATH DE number 2119505
scientific article