Traces of random variables on Wiener space and the Onsager-Machlup functional
Let \(F\) be a random variable defined over the classical Wiener space. The knowledge about \(F\) is restricted to sets of full measure; in particular, it does not give sense to speak about the value of \(F\) at a fixed point \(\omega_ 0\) of the Cameron-Martin space. The authors study sufficient conditions for the existence of the approximate limit of \(F\) at \(\omega_ 0\) (in the sense of Denjoy) and consider some weaker notion for defining \(F(\omega_ 0)\). The results are first proved for a multiple integral \(I_ n(f_ n)\), \(n\) even. Using a refined version of Young's inequality due to Brascamp and Lieb for the main estimate, the authors show that the approximate limit exists whenver \(f_ n\in L^ p([0,1]^ n)\), \(p>1+n\), and the trace of \(f_ n\) exists. Then, by controlling the summation of the limits of the multiple integrals in the Wiener chaos decomposition, the results are extended to sums \[ F=\sum_{n=0}^ \infty I_ n n(f_ n), \quad \sum_{n=0}^ \infty \sqrt{n!} K^ n\| f_ n\|_ \infty<\infty,\quad \text{for all real}\quad K>0. \] The last section of the paper applies these results to the computation of the Onsager-Machlup functional of a special nonanticipative stochastic differential equation.
scientific article; zbMATH DE number 65349