Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
The authors consider the stochastic Allen-Cahn equation \[ \partial_t u=\Delta u+Cu-u^3+\sqrt{\varepsilon}\;\xi_\delta, \] where \(C\in\mathbb{R}\) and \(\xi_\delta\) is a random driving noise with weak intensity \(\sqrt{\varepsilon}\) and almost white, i.e., the correlation length scale is \(\delta\ll 1\). The limiting behaviour of the solutions is studied as \(\varepsilon\) and \(\delta\) approach zero. The theory of regularity structures is used to analyse the behaviour of the solutions for fixed \(\varepsilon\) and \(\delta\rightarrow 0\). In order to obtain nontrivial limits as \(\delta\) vanishes, they consider the modified equation \[ \partial_t u=\Delta u+(C+3\varepsilon C_\delta^{(1)}-9\varepsilon^2 C_\delta^{(2)})u-u^3+\sqrt{\varepsilon}\;\xi_\delta. \] The main result of the paper states a large deviation principle for these renormalised solutions. A remarkable feature is that the diverging renormalisation constants \(C_\delta^{(1)}\) and \(C_\delta^{(2)}\) disappear on the level of large deviations. This result is applied to derive a sharp condition on \(\delta,\varepsilon\) that guarantees a large deviation principle for the equation without renormalisation.
stochastic partial differential equations
scientific article