The scaling limit for a stochastic PDE and the separation of phases
The stochastic partial differential equation \[ {\partial u \over \partial t} = \Delta u + \varepsilon^{-1} f(u) + \varepsilon^\gamma a(x) \dot w_t (x), \quad t > 0,\;x \in \mathbb{R}, \] is investigated, where \(a \in C^2_0 (\mathbb{R})\), the function \(f\) is smooth and has suitably bounded growth. Essential assumptions on \(f\) are that \(f\) is odd, it has exactly three roots \(1, -1,0,\) \(f'(1) < 0\), \(f'(-1) < 0\), \(f'(0) > 0\) (it is the gradient of a ``double-well'' potential). The solutions \(m\) of the steady state problem of the equation with the diffusion part omitted, and under the boundary conditions \(m(+ \infty) = 1\), \(m(- \infty) = - 1\), are functions with the only jump at the ``separation point between two phases \(-1\) and \(+1\)''. The main result describes the behaviour in time of this separation point by a stochastic differential equation which is obtained by the limit for \(\varepsilon\) going to zero if the time scale is suitably chosen and \(\gamma\) is sufficiently large. The solution uses special choice of the initial conditions, the steady states for the unperturbed equation.
stochastic partial differential equation
scientific article; zbMATH DE number 773784
scientific article