REPRESENTATION OF PATHWISE STATIONARY SOLUTIONS OF STOCHASTIC BURGERS' EQUATIONS

In this paper, we show that the stationary solution u(t, omega) of the differentiable random dynamical system U: R(+) x L(2)[0, 1] x Omega -> L(2)[0, 1] generated by the stochastic Burgers' equation with large viscosity, denoted by nu, driven by a Brownian motion in L(2)[0, 1], is given by: u(t, omega) = U(t, Y(omega), omega) = Y (theta(t, omega)), where Y (omega) can be represented by the following integral equation: Y(omega) = 1/2 integral(0)(-infinity) T(nu)(-s)partial derivative(Y(theta(s, omega))(2)/partial derivative xds + integral(0)(-infinity) T(nu)(-s) dW(s)(omega). Here theta is the group of P-preserving ergodic transformations on the canonical probability space (Omega, F, P) such that theta(t, omega)(s) = W(t + s) W(t), where W is the L(2)[0, 1]-valued Brownian motion on the probability space (Omega, F, P), T(nu) is the linear operator semigroup on L(2)[0, 1] generated by nu Delta.Statistics & ProbabilitySCI(E)8ARTICLE4613-634