On transient Bessel processes and planar Brownian motion reflected at their future infima
Let \(R_t\), \(t \geq 0\), be a Bessel process of dimension \(d > 2\) and \(J_t = \inf \{R_s; s \geq t\}\) be its future infimum, it is known that these two processes as well as \(R_t - J_t\) satisfy the same law of iterated logarithm, namely \(\limsup_{t \to 0} {X_t \over (2t \log \log t)^{1/2}} = 1\) a.s. for any of these processes. It is shown that the asymptotic behaviour of these processes can be distinguished by means of their upper functions, namely, it is known that the following integral test (Dvoretsky, Erdös, Orey, Pruitt test) holds \[ P \bigl( R_t > t^{1/2} f(t) \text{ i.o. as } t \to \infty \bigr) = 1 \quad \text{(resp 0)} \] if and only if \[ \int^\infty f(t) \exp \left( - {f^2 (t) \over 2} f^d (t) {dt \over t} \right) = \infty \quad \text{(resp} < \infty), \] and the main result of the paper is that \[ P \bigl( R_t - J_t > t^{1/2} f(t) \text{ i.o. as } t \to \infty) = 1 \quad \text{(resp 0)} \] if and only if \[ \int^\infty f(t) \exp \left( - {f^2 (t) \over 2} f^{4 - d} (t) {dt \over t} \right) = \infty \quad \text{(resp} < \infty), \] thus distinguishing the processes \(R\) and \(R - J\). An integral test for the process \(J\) had been obtained earlier by \textit{Khoshnevisan}, \textit{Lewis} and \textit{Shi} (to appear in Ann. Probab.). Some similar results are also obtained in the case of 2-dimensional Bessel process. Since this process is not transient, this time the infimum is taken on the interval \([t,1]\) for \(t \leq 1\) and the asymptotic behaviour for \(t \to 0\) are explored.
scientific article; zbMATH DE number 859949
scientific article