The local structure of q-Gaussian processes

• Autorzy: Bryc W. , Wang Y.
• Języki publikacji: EN

Abstrakty

EN

The local structure of q-Ornstein-Uhlenbeck process and q-Brownian motion are investigated for all q ϵ (−1, 1). These are classical Markov processes that arose from the study of noncommutative probability. These processes have discontinuous sample paths, and the local small jumps are characterized by tangent processes. It is shown that, for all q ϵ (−1, 1), the tangent processes in the interior of the state space are scaled Cauchy processes possibly with drifts. The tangent processes at the boundary of the state space are also computed, but they are not well-known processes in classical probability theory. Instead, they can be associated with the free 1/2-stable law, a well-known distribution in free probability, via Biane’s construction.

Słowa kluczowe

EN

q-Brownian motion; q-Ornstein-Uhlenbeck process; inhomogeneous Markov process; tangent process; self-similar process; Cauchy process; free stable law; Biane’s construction

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2016
• Tom: Vol. 36, Fasc. 2
• Strony: 335--352
• Opis fizyczny: Bibliogr. 20 poz., wykr.

Twórcy

autor

Bryc W.

• Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati, OH, 45221-0025, USA

autor

Wang Y.

• Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati, OH, 45221-0025, USA

Bibliografia

• [1] M Anshelevich, Generators of some non-commutative stochastic processes, Probab. Theory Related Fields 157 (3-4) (2013), pp. 777-815.
• [2] H. Bercovici and V. Pata, Stable laws and domains of attraction in free probability theory. With an appendix by Philippe Biane, Ann. of Math. (2) 149 (3) (1999), pp. 1023-1060.
• [3] H. Bercovici and D. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (3) (1993), pp. 733-773.
• [4] P. Biane, Processes with free increments, Math. Z. 227 (1) (1998), pp. 143-174.
• [5] P. Billingsley, Probability and Measure, third edition, Wiley, New York 1995.
• [6] P. Billingsley, Convergence of Probability Measures, second edition, Wiley, New York 1999.
• [7] M. Bożejko and W. Bryc, On a class of free Lévy laws related to a regression problem, J. Funct. Anal. 236 (1) (2006), pp. 59-77.
• [8] M. Bożejko, B. Kümmerer, and R. Speicher, q-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1) (1997), pp. 129-154.
• [9] W. Bryc, Stationary random fields with linear regressions, Ann. Probab. 29 (1) (2001), pp. 504-519.
• [10] W. Bryc and A. Hassairi, One-sided Cauchy-Stieltjes kernel families, J. Theoret. Probab. 24 (2) (2011), pp. 577-594.
• [11] W. Bryc, W. Matysiak, and P. J. Szabłowski, Probabilistic aspects of Al-Salam-Chihara polynomials, Proc. Amer. Math. Soc. 133 (4) (2005), pp. 1127-1134 (electronic).
• [12] W. Bryc and J. Wesołowski, Conditional moments of q-Meixner processes, Probab. Theory Related Fields 131 (3) (2005), pp. 415-441.
• [13] S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York 1986.
• [14] K. J. Falconer, The local structure of random processes, J. London Math. Soc. (2) 67 (3) (2003), pp. 657-672.
• [15] M. E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia Math. Appl., Vol. 98, Cambridge University Press, Cambridge 2009.
• [16] H. Maassen, Addition of freely independent random variables, J. Funct. Anal. 106 (2) (1992), pp. 409-438.
• [17] P. Mörters and Y. Peres, Brownian Motion. With an appendix by Oded Schramm andWendelin Werner, Cambridge University Press, Cambridge 2010.
• [18] V. Pérez-Abreu and N. Sakuma, Free generalized gamma convolutions, Electron. Commun. Probab. 13 (2008), pp. 526-539.
• [19] P. J. Szabłowski, q-Wiener and (α, q)-Ornstein-Uhlenbeck processes. A generalization of known processes, Theory Probab. Appl. 56 (4) (2012), pp. 634-659.
• [20] D. Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (3) (1986), pp. 323-346.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).