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Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. (English) Zbl 1229.60044

Large deviations of local times and intersection local times have been extensively studied in the context of Markov processes. The present paper contains large deviation results for (intersection) local times of two natural families of Gaussian processes without the Markov property, the fractional Brownian motions and the Riemann-Liouville processes.
Let \(\{B(t): t\geq 0\}\) be a standard \(\mathbb R^d\)-valued fractional Brownian motion with Hurst index \(H\in (0,1)\). The local time \(\{L_t^x: t\geq 0, x\in \mathbb R^d\}\) of \(B\) is defined heuristically as
\[ L_t^x=\int_{0}^t \delta_x(B(s))\,ds \]
and is known to exist if \(Hd<1\). The authors prove that the limit
\[ \lim_{a\to\infty} a^{-1/(Hd)}\log\operatorname{P}[L_1^0\geq a] \]
exists and provide bounds for this limit. Given \(p\) independent \(\mathbb R^d\)-valued standard fractional Brownian motions, their intersection local time is a random measure \(\alpha\) on \(\mathbb R_+^p\) given heuristically by
\[ \alpha(A)=\int_A\prod_{j=1}^{p-1} \delta_0(B_j(s_j)-B_{j+1}(s_{j+1}))\,ds_1\dots ds_{p},\quad A\subset \mathbb R_+^p. \]
The authors prove the existence of \(\alpha\) provided that \(Hd<p/(p-1)\). They show that the limit
\[ \lim_{a\to\infty} a^{-1/((p-1)Hd)}\log\operatorname{P}[\alpha([0,1]^p)\geq a] \]
exists and provide bounds for it. Using the above large deviation results, the authors obtain the laws of the iterated logarithm for \(L_t^0\) and \(\alpha([0,t]^d)\) as \(t\to\infty\). Similar results are obtained if \(B\) is an \(\mathbb R^d\)-valued Riemann-Liouville process.

MSC:

60G22 Fractional processes, including fractional Brownian motion
60J55 Local time and additive functionals
60F10 Large deviations
60G15 Gaussian processes
60G18 Self-similar stochastic processes
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