Abstract
In the context of non-Gaussian analysis, Schneider [29] introduced grey noise measures, built upon Mittag-Leffler functions; analogously, grey Brownian motion and its generalizations were constructed (see, for example, [6, 7, 9, 27]). In this paper, we construct and study a new non-Gaussian measure, by means of the incomplete-gamma function (exploiting its complete monotonicity). We label this measure Gamma-grey noise and we prove, for it, the existence of Appell system. The related generalized processes, in the infinite dimensional setting, are also defined and, through the use of the Riemann-Liouville fractional operators, the (possibly tempered) Gamma-grey Brownian motion is consequently introduced. A number of different characterizations of these processes are also provided, together with the integro-differential equation satisfied by their transition densities. They allow to model anomalous diffusions, mimicking the procedures of classical stochastic calculus.
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Acknowledgements
The research by L.B. and L.C. was partially carried over during a visiting period at the Isaac Newton Institute in Cambridge, whose support is gratefully acknowledged. L.B. was supported, in particular, by the Kirk Distinguished Fellowship, awarded by the same institute.
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Beghin, L., Cristofaro, L. & Gajda, J. Non-Gaussian Measures in Infinite Dimensional Spaces: the Gamma-Grey Noise. Potential Anal 60, 1571–1593 (2024). https://doi.org/10.1007/s11118-023-10099-0
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DOI: https://doi.org/10.1007/s11118-023-10099-0
Keywords
- Incomplete gamma function
- Completely monotone functions
- Grey noise
- Hitting times
- Fractional Brownian motion
- Elliptically contoured measures