Sample Path and Fractal Properties of Lévy Processes

The sample paths of Lévy processes generate various random fractals and many of their properties have been studied since 1960’s [see the survey papers of Fristedt (1974), Taylor (1986) and Xiao (2004)]. The lectures will cover the following topics: Hausdorff dimension and exact Hausdorff measure functions for random fractals determined by Lévy processes; Hausdorff dimension computation using potentialtheoretic methods; recent results on potential theory of multiparameter Lévy processes; packing dimension of the images of Lévy processes; local times and intersection local times of Lévy processes. Partial List of References [1] J. Bertoin (1999), Subordinators: examples and applications. In: Lectures on Probability Theory andStatistics (Saint-Flour, 1997), pp. 1–91, Lecture Notes in Math., 1717, Springer-Verlag, Berlin.[2] R. M. Blumenthal and R. Getoor (1961), Sample functions of stochastic processes with stationaryindependent increments. J. Math. Mech. 10, 493–516.[3] S. N. Evans (1987a), Multiple points in the sample paths of a Lévy process. Probab. Th. Rel. Fields76, 359–367.[4] K. J. Falconer and J. D. Howroyd (1997), Packing dimensions for projections and dimension profiles.Math. Proc. Cambridge Philo. Soc. 121, 269–286.[5] P. J. Fitzsimmons and T. S. Salisbury (1989), Capacity and energy for multiparameter processes. Ann.Inst. Henri Poincaré Probab. Statist. 25, 325–350.[6] B. E. Fristedt (1974), Sample functions of stochastic processes with stationary, independent increments.Adv. in Probab. III, pp. 241–396, Dekker.[7] B. Fristedt and S. J. Taylor (1992), The packing measure of a general subordinator. Probab. Th. Rel.Fields 92, 493–510.[8] J. Hawkes (1971), On the Hausdorff dimension of the intersection of the range of a stable process witha Borel set. Z. Wahrsch. Verw. Gebiete 19, 90–102.[9] J. Hawkes (1998), Exact capacity results for stable processes. Probab. Th. Rel. Fields 112, 1–11.[10] J. Hawkes and W. E. Pruitt (1974), Uniform dimension results for processes with independent incre-ments. Z. Wahrsch. Verw. Gebiete 28, 277–288.[11] D. Khoshnevisan, R. L. Schilling and Y. Xiao (2010), Packing dimension profiles and Lévy processes.Preprint.[12] D. Khoshnevisan, N.-R. Shieh and Y. Xiao (2008), Hausdorff dimension of the contours of symmetricadditive Lévy processes. Probab. Th. Rel. Fields 140, 169–193.[13] D. Khoshnevisan and Y. Xiao (2002), Level sets of additive Lévy processes. Ann. Probab. 30, 62–100.[14] D. Khoshnevisan and Y. Xiao (2005), Lévy processes: capacity and Hausdorff dimension. Ann. Probab.33, 841–878.