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Green Measures for a Class of non-Markov Processes
Published 2 Apr 2024 in math.PR and math.FA | (2404.02076v1)
Abstract: In this paper, we investigate the Green measure for a class of non-Gaussian processes in $\mathbb{R}{d}$. These measures are associated with the family of generalized grey Brownian motions $B_{\beta,\alpha}$, $0<\beta\le1$, $0<\alpha\le2$. This family includes both fractional Brownian motion, Brownian motion, and other non-Gaussian processes. We show that the perpetual integral exists with probability $1$ for $d\alpha>2$ and $1<\alpha\le2$. The Green measure then generalizes those measures of all these classes.
- Classical Potential Theory. Springer Monographs in Mathematics. Springer, 2001.
- Markov Processes and Potential Theory. Academic Press, 1968.
- J. L. da Silva and M. Erraoui. Generalized grey Brownian motion local time: Existence and weak approximation. Stochastics, 87(2):347–361, October 2015.
- Cameron-Martin type theorem for a class of non-Gaussian measures. Submitted, 2024.
- M. Grothaus and F. Jahnert. Mittag-Leffler Analysis II: Application to the fractional heat equation. J. Funct. Anal., 270(7):2732–2768, April 2016.
- Perpetual integral functionals of multidimensional stochastic processes. Stochastics, 93(8):1249–1260, 2021.
- A. Mura and F. Mainardi. A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics. Integral Transforms Spec. Funct., 20(3-4):185–198, 2009.
- A. Mura and G. Pagnini. Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. J. Phys. A: Math. Theor., 41(28):285003, 22, 2008.
- A. Mentrelli and G. Pagnini. Front propagation in anomalous diffusive media governed by time-fractional diffusion. J. Comput. Phys., 293:427–441, 2015.
- D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 3rd edition, 1999.
- W. R. Schneider. Fractional diffusion. In R. Lima, L. Streit, and R. Vilela Mendes, editors, Dynamics and stochastic processes (Lisbon, 1988), volume 355 of Lecture Notes in Phys., pages 276–286. Springer, New York, 1990.
- W. R. Schneider. Grey noise. In S. Albeverio, G. Casati, U. Cattaneo, D. Merlini, and R. Moresi, editors, Stochastic Processes, Physics and Geometry, pages 676–681. World Scientific Publishing, Teaneck, NJ, 1990.
- Fractional integrals and derivatives. Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications, Edited and with a foreword by S. M. Nikol’skiĭ, Translated from the 1987 Russian original, Revised by the authors.
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