A generalization of functional law of the iterated logarithm for \((r,p)\)-capacities on the Wiener space.
Let us consider a classical Wiener space \((W,H,{\mathcal W})\), where \(W\) is the space of all continuous functions \(w:\mathbb R_ {+}\to \mathbb R^ {d}\) satisfying \(w(0)=0\) and \(t^ {-1}w(t)\to 0\) as \(t\to \infty \), equipped with the norm \(\| w\| _ {W} = \sup _ {t\geq 0}(1+t)^ {-1}\| w(t)\| \), \(H\) consists of all \(w\in W\) of the form \(w(t) = \int ^ {t}_ {0} g(s)\,ds\), \(t\geq 0\), \(g\in L^ 2(\mathbb R_ {+}; \mathbb R^ {d})\), and \({\mathcal W}\) is the Wiener measure on \(W\). If we set \(b(t): w\mapsto w(t)\), then \(b\) is the Wiener process under the measure \({\mathcal W}\). It is known that many path properties of the process \(b\), like the functional law of the iterated logarithm or the properties of increments, hold not only \({\mathcal W}\)-almost surely, but even quasi-surely (that is, outside a set of zero capacity). Several new results of this type are established. For example, set \(K = \{g\in H\); \(\int ^ \infty _ 0 | \dot g(s)| ^ 2 \,ds \leq 1\}\), and for \(r\in \mathopen [0,\infty \mathclose [\), \(p\in \mathopen ]1,\infty \mathclose [\) denote by \(C_ {r,p}\) the \((r,p)\)-capacity. If \(M_ {t,h}(\cdot ) = (2h\log h^ {-1})^ {-1/2} \{b(t+h\cdot )-b(t)\}\), \(0\leq t\leq 1-h\), then \(\lim _ {h\to 0} \inf _ {0\leq t\leq 1-h} \| M_ {t,h} -\psi \| _ {W} = 0\) \(C_ {r,p}\)-quasi-everywhere for every \(\psi \in K\), and \(\lim _ {h\to 0} \sup _ {0\leq t\leq 1-h} \sup _ {\psi \in K} \| M_ {t,h} -\psi \| _ {W} = 0\). This implies, in particular, a quasi-sure version of Lévy's theorem on the modulus of continuity of the Brownian motion.
scientific article; zbMATH DE number 2138902