On some limit theorems for occupation times of one dimensional Brownian motion and its continuous additive functionals locally of zero energy (Q1822419)

Let \(B_ t\) be a standard Brownian motion and f be a continuous function. The author investigates the convergence in law on the space of continuous functions for the family of stochastic processes \(\xi_{\lambda}(t)=a_{\lambda}\int^{\lambda t}_{0}f(B_ s)ds\) as \(\lambda\to \infty\). The limit process depends on the nonrandom norming function \(a_{\lambda}\) and assumptions on f(\(\cdot)\). The known results are connected with the case when f has a compact support and \(I=\int f(x)dx\neq 0\) or \(I=0\) but \(<f,f>=2\int^{\infty}_{- \infty}(\int^{x}_{-\infty}f(u)du)^ 2dx<\infty.\) In this article the case when \(f\not\in L^ 1({\mathbb{R}})\) or the case when \(I=0\) but \(<f,f>=\infty\) are discussed. For this purpose some classes of continuous additive functionals locally of zero energy are introduced: class \(C^ a_ t\) of functionals corresponding to Cauchy's principle value for 1/(x-a), class \(H^ a(-1-\alpha,t)\) corresponding to Hadamard's finite part (f.p.) \((x-a)_+^{-1-\alpha}\) and class \(H^ a(1+\beta,t)\) corresponding to f.p. \((x-a)_+^{\beta -1}.\) Under some additional assumptions on f(\(\cdot)\) the limit process for the \(\xi_{\lambda}\) can be expressed with the help of this classes of functionals.