Strassen's law of the iterated logarithm for diffusion processes for small time
The paper studies a functional form of the law of the iterated logarithm [see \textit{V. Strassen}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 3, 211-226 (1964; Zbl 0132.12903)], which has been extended to diffusion processes by \textit{P. Baldi} [Probab. Theory Relat. Fields 71, 435-453 (1986; Zbl 0587.60074)] for large values of parameter. The author extends this result for small values of parameter in the following sense. Let us consider a diffusion process \(Y\) starting from \(x\) and let \(\{\Gamma_\alpha\}_{\alpha <0}\) be a suitable family of contractions with fixed point \(x\) for every \(\alpha >0\). For \(t\in [0,1]\) and \(u\leq e^{-1}\) denote \(Z_u(t)=\Gamma_{\sqrt {u\;log\;log\;u^{-1}}}(Y_{ut})\). The author proves that the family \(\{Z_u\}\) is relatively compact for \(u\to 0^+\) and determines its limit set \(C\). For the Brownian motion a similar result can be obtained by time reversal, a technique which is not easy to reproduce for diffusion processes. A number of applications and examples are discussed.
scientific article; zbMATH DE number 1364453