The I-function for diffusion processes with boundaries
Suppose \(X_ t\) is a diffusion process with generator (L,\({\mathcal D})\) on the compact set \(A\subset R^ n\). Let \(\nu_ t(B)\) be the proportion of (0,t) that \(X_ t\) spends in \(B\subset A\), so that \(\nu_ t\) is a (random) probability measure on A. Then it is known that, under suitable conditions, a large deviation principle holds for \(\nu_ t\) with action/entropy/I-function I(\(\mu)\) given by minus the infimum of \(\int (Lu/u)d\mu\) over those \(u\in {\mathcal D}\) bounded below by a strictly positive constant, where \(\mu\) is a probability measure on A. Here a more explicit representation of I is provided, subject to certain conditions, for those \(\mu\) with a density in \(C^ 1(A)\). A further result relaxes the condition that the density be in \(C^ 1(A)\) and the author conjectures that \(I(\mu)=\infty\) when this weaker condition does not hold.
scientific article; zbMATH DE number 3921666