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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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