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Uniform oscillations of the local time of iterated Brownian motion. (English) Zbl 0930.60056
Consider the so-called “iterated Brownian motion”, i.e. a real-valued process $$Z(t)$$, $$t\geq 0$$, defined as $$Z(t)= X_+(Y(t))$$ if $$Y(t)\geq 0$$ and $$Z(t)= X_-(Y(t))$$ if $$Y(t)< 0$$, where $$X_+(t)$$, $$X_-(t)$$, $$Y(t)$$ are three independent standard Brownian motions. Denote by $$L^x_t(Z)$$, $$t\geq 0$$, $$x\in R$$, the local time process of $$Z$$; put $$\omega(h)$$ for the unifom modulus of continuity and $$\eta(h)$$ for the modulus of nondifferentiability of $$t\to L^x_t(Z)$$. The authors use the links between $$L^x_t(Z)$$ and Bessel processes, particularly, Ray-Knight theorems for the investigation of an asymptotics of $$\omega(h)$$ and $$\eta(h)$$ as $$h\to 0$$. They prove that $$h^{3/4}|\log h|^{3/4}$$ is a correct rate for $$\omega(h)$$ and $$h^{-3/4}|\log h|^{3/4}$$ determines the exact order of magnitude of $$\eta(h)$$ for small $$h$$.

##### MSC:
 60J65 Brownian motion
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