REPRESENTATION OF PATHWISE STATIONARY SOLUTIONS OF STOCHASTIC BURGERS' EQUATIONS

Abstract

In this paper, we show that the stationary solution u(t, ω) of the differentiable random dynamical system U: ℝ+ × L2[0, 1] × Ω → L2[0, 1] generated by the stochastic Burgers' equation with large viscosity, denoted by ν, driven by a Brownian motion in L2[0, 1], is given by: u(t, ω) = U(t, Y(ω), ω) = Y(θ(t, ω)), where Y(ω) can be represented by the following integral equation: [Formula: see text] Here θ is the group of P-preserving ergodic transformations on the canonical probability space [Formula: see text] such that θ(t, ω)(s) = W(t + s) - W(t), where W is the L2[0, 1]-valued Brownian motion on the probability space [Formula: see text], Tν is the linear operator semigroup on L2[0, 1] generated by νΔ.