Optimal Gaussian solutions of nonlinear stochastic partial differential equations

Abstract

We present a linearization procedure of a stochastic partial differential equation for a vector field (X _i (t,x)) (t∈[0, ∞),x∈R ^d,i=l,...,n): ∂ _t X _i (t,x)=b _i (X(t, x)) +D, ΔX _i (t, x) + σ _i f _i (t, x). HereΔ is the Laplace-Beltrami operator inR ^d, and (f _i (t,x)) is a Gaussian random field with 〈f _i (t,x)f _j (t′,x′)〉 = δ _ij δ(t − t′)δ(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 R¹ withb ₁(z) =μz - vz ³. Stationary values of mean, variance, and correlation length are calculated. They almost agree with exact ones ifμ ≲ 1.24 (ν ²θ ⁴₁ /D ₁ ^1/3:=μ _c . Whenμ⩾μ _c , there appear quasistationary states fluctuating around one of the bottoms of the potentialU(z) = ∫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 liket ^−1/2 for small t. The diffusion term∂ ² _x X accelerates the relaxation from the neighborhood of an unstable initial stateX(0,x) ∼ 0.