Onsager Machlup Functional for Stochastic Evolution Equations in a Class of Norms

Abstract

In this note, we show that the Onsager Machlup functional for stochastic evolution equations on a given separable Hilbert space H, computed in Bardina, Rovira, and Tindel (Onsager Machlup functional for stochastic evolution equations. In Annales Institut Henri Poincaré, 2003, 39, 69–93.), does not depend on the measurable norm considered on the functions from [0, 1] to H, dominating the norm on L 2([0, 1]; H), and satisfying some lower bound conditions for the stochastic convolution associated to our evolution system. The main ingredient of the proof is the extension of some Gaussian correlation inequalities of Hargé (Une inégalité de décorrélation pour la mesure gaussienne, C. R. Acad. Sci. Paris Sér. I Math. 326, no. 11, (1998), 1325–1328) and S˘idak (On multivariate normal probabilities on rectangles (their dependence on correlation), Ann. Math. Statist. 39, (1968), 1425–1434) to the infinite dimensional case.