Stratonovitch Calculus with Spatial Parameters and Anticipative Problems in Multiplicative Ergodic Theory

Let u(t; x); t 2 R; be an adapted process parametrized by a variable x in some metric space X, (!; dx) a probability kernel on the product of the probability space and the Borel sets of X. We deal with the question whether the Stratonovich integral of u(:; x) with respect to a Wiener process on and the integral of u(t; :) with respect to the random measure (:; dx) can be interchanged. This question arises for example in the context of stochastic diierential equations. Here (:; dx) may be a random Dirac measure (dx), where appears as an anticipative initial condition. We give this random Fubini type theorem a treatment which is mainly based on ample applications of the real variable continuity lemma of Garsia, Rodemich and Rumsey. As an application of the resulting "uniform Stratonovich calculus" we give a rigorous veriication of the diagonalization algorithm of a linear system of stochastic diierential equations. stochastic diierential equations; random dynamical systems; multiplicative ergodic theory .