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On the roughness of the paths of RBM in a wedge. (English. French summary) Zbl 1466.60168

Summary: Reflected Brownian motion (RBM) in a wedge is a 2-dimensional stochastic process \(Z\) whose state space in \(\mathbb{R}^2\) is given in polar coordinates by \(S=\{(r,\theta):r\geq0,0\leq\theta\leq\xi\}\) for some \(0<\xi<2\pi \). Let \(\alpha=(\theta_1+\theta_2)/\xi \), where \(-\pi/2<\theta_1,\theta_2<\pi/2\) are the directions of reflection of \(Z\) off each of the two edges of the wedge as measured from the corresponding inward facing normal. We prove that in the case of \(1<\alpha<2\), RBM in a wedge is a Dirichlet process. Specifically, its unique Doob-Meyer type decomposition is given by \(Z=X+Y\), where \(X\) is a two-dimensional Brownian motion and \(Y\) is a continuous process of zero energy. Furthermore, we show that for \(p>\alpha \), the strong \(p\)-variation of the sample paths of \(Y\) is finite on compact intervals, and, for \(0<p\leq\alpha \), the strong \(p\)-variation of \(Y\) is infinite on \([0,T]\) whenever \(Z\) has been started from the origin. We also show that on excursion intervals of \(Z\) away from the origin, \((Z,Y)\) satisfies the standard Skorokhod problem for \(X\). However, on the entire time horizon \((Z,Y)\) does not satisfy the standard Skorokhod problem for \(X\), but nevertheless we show that it satisfies the extended Skorkohod problem.

MSC:

60J65 Brownian motion
60J27 Continuous-time Markov processes on discrete state spaces
60G17 Sample path properties
60G52 Stable stochastic processes
60J55 Local time and additive functionals
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