Large-time and small-time behaviors of the spectral heat content for time-changed stable processes. (English) Zbl 1507.60064

Summary: We study the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.


60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
60K50 Anomalous diffusion models (subdiffusion, superdiffusion, continuous-time random walks, etc.)
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[1] L. Acuña Valverde. On the one dimensional spectral heat content for stable processes. J. Math. Anal. Appl., 441 (2016), 11-24. · Zbl 1337.60092
[2] L. Acuña Valverde. Heat content for stable processes in domains of \[{\mathbb{R}^d} \]. J. Geom. Anal., 27 (2017), 492-524. · Zbl 1361.60033
[3] N.H. Bingham, C.M. Goldie, and J.L. Teugels. Regular Variation, volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1987. · Zbl 0617.26001
[4] T. Grzywny, H. Park, and R. Song. Spectral heat content for Lévy processes. Math. Nachr. 292 (2019), 805-825. · Zbl 1472.35209
[5] K. Kobayashi. Small ball probabilities for a class of time-changed self-similar processes. Statist. Probab. Lett. 110 (2016), 155-161. · Zbl 1337.60058
[6] K. Kobayashi and H. Park. Spectral heat content for time-changed killed Brownian motions. To appear in J. Theor. Probab.
[7] M.M. Meerschaert and A. Sikorskii. Stochastic Models for Fractional Calculus. Volume 43 of De Gruyter Studies in Mathematics, 2012. · Zbl 1247.60003
[8] H. Park. Higher order terms of spectral heat content for killed subordinate and subordinate killed Brownian motions related to symmetric \(α\)-stable processes in \[\mathbb{R} \]. Potential Anal., 57 (2022), 283-303. · Zbl 1495.60038
[9] H. Park and R. Song. Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic \(α\)-stable processes. Bull. London Math. Soc. 51 (2019), 371-384. · Zbl 1481.60165
[10] H. Park and R. Song. Spectral heat content for \(α\)-stable processes in \[{C^{1,1}}\] open sets. Elect. J. Probab., Vol. 27 No. 22, (2022) 1-19. · Zbl 1486.60066
[11] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, 1999. · Zbl 0973.60001
[12] S. Umarov, M. Hahn, and K. Kobayashi. Beyond the Triangle: Brownian Motion, Itô Calculus, and Fokker-Planck Equation — Fractional Generalizations. World Scientific, 2018. · Zbl 1403.60001
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