Stability of the instanton under small random perturbations
The following stochastic partial differential equation is called the Allen-Cahn equation: \[ \begin{cases} \partial_ t u(x,t) = {1\over 2} \partial_{xx} u(x,t) - V(u(x,t)),\quad x \in \mathbb{R}, \;t\geq 0,\\ u(x,0) = \varphi_ 0(x),\end{cases}\tag{1} \] where \(V(u) = u^ 3 - u\). When \(\varphi_ 0\) is a stationary solution of (1) which interpolates between the two constants \(\pm 1\), it is called an instanton. This paper considers the Allen-Cahn equation perturbed by a white noise with initial condition an instanton, \[ \begin{cases} \partial_ t u^ \varepsilon(x,t) = {1\over 2} \partial_{xx} u^ \varepsilon(x,t) - V(u^ \varepsilon(x,t)) + \sqrt{\varepsilon} \mu(x) \alpha(x,t),\\ u^ \varepsilon(x,0) = \varphi_ 0(x),\end{cases}\tag{2} \] where \(\alpha(x,t)\) is a space-time white noise and \(\mu(x)\) is a bounded function in \(L^ 2(\mathbb{R},x^ 2 dx)\). When \(\varphi_ 0\) is bounded, equation (2) has a unique solution. The author proves that, as \(\varepsilon\) goes to zero, the process solution of (2) remains close to the initial instanton in the following form: \[ u^ \varepsilon (x,t \varepsilon^{-\gamma}) = \varphi_ 0(x + \varepsilon^ {(1-\gamma)/2} W(t)) + r^ \varepsilon (x,t) \] where \(\gamma < 1/3\), \(W(t)\) is a Brownian motion and \(r^ \varepsilon\) is of smaller order than \(\varepsilon^{(1-\gamma)/2}\).
stochastic partial differential equation
scientific article; zbMATH DE number 724750