Optimal Gaussian solutions of nonlinear stochastic partial differential equations
We present a linearization procedure of a stochastic partial differential equation for a vector field \((X_ i(t,x))\) (t\(\in [0,\infty)\), \(x\in R^ d\), \(i=1,...,n):\) \(\partial_ tX_ i(t,x)=b_ i(X(t,x))+D_ i\Delta X_ i(t,x)+\sigma_ if_ i(t,x).\) Here \(\Delta\) is the Laplace-Beltrami operator in \(R^ d\), and \((f_ i(t,x))\) is a Gaussian random field with \(<f_ i(t,x)f_ j(t',x')>=\delta_{ij}\delta (t- t')\delta (x-x').\) The procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations. The linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the Kullback-Leibler entropy. The procedure is applied to the scalar-valued Ginzburg-Landau model in \(R^ 1\) with \(b_ 1(z)=\mu z-vz^ 3\). Stationary values of mean, variance, and correlation length are calculated. They almost agree with exact ones if \(\mu\lesssim 1.24\) \((v^ 2\sigma^ 4_ 1/D_ 1)^{1/3}:=\mu_ c\). When \(\mu \geq \mu_ c\), there appear quasistationary states fluctuating around one of the bottoms of the potential \(U(z)=\int b_ 1(z)dz\). The second moment at the quasistationary states almost agrees with the exact one. Transient phenomena are also discussed. Half-width at half-maximum of a structure function decays like \(t^{-1/2}\) for small t. The diffusion term \(\partial^ 2_ xX\) accelerates the relaxation from the neighborhood of an unstable initial state X(0,x)\(\simeq 0\).
scientific article; zbMATH DE number 3942633