In this note, we show that the Onsager Machlup functional for stochastic evolution equations on a given separable Hubert 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 L2([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 Sidak (On multivariate normal probabilities on rectangles (their dependence on correlation), Ann. Math. Statist. 39, (1968), 1425-1434) to the infinite dimensional case.
- Correlation inequalities
- Onsager Machlup functionals
- Stochastic evolution equations