×

Survival exponents for fractional Brownian motion with multivariate time. (English) Zbl 1355.60053

Summary: Fractional Brownian motion of index \(0 < H < 1\), \(H\)-FBM, with \(d\)-dimensional time is considered in a spherical domain that contains 0 at its boundary. The main result: the log-asymptotics of the probability that H-FBM does not exceed a fixed positive level is \((H-d + o(1))\log T\), where \(T \gg 1\) is the radius of the domain.

MSC:

60G22 Fractional processes, including fractional Brownian motion
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] F. Aurzada and S. Dereich. Universality of the asymptotics of the one-sided exitproblem for integrated processes. Ann. Inst. Henri Poincar´e Probab. Stat. 49 (1),236-251 (2013) · Zbl 1285.60042
[2] F. Aurzada, N. Guillotin-Plantard and F. Pene. Persistence probabilities for station-ary increment processes. ArXiv Mathematics e-prints (2016) · Zbl 1396.60037
[3] F. Aurzada and T. Simon. Persistence probabilities and exponents. In L´evy matters.V, volume 2149 of Lecture Notes in Math., pages 183-224. Springer, Cham (2015) · Zbl 1338.60077
[4] A. J. Bray, S. N. Majumdar and G. Schehr. Persistence and first-passage propertiesin nonequilibrium systems. Advances in Physics 62 (3), 225-361 (2013).DOI:10.1080/00018732.2013.803819
[5] X. Fernique. Regularit´e des trajectoires des fonctions al´eatoires gaussiennes. In´Ecole d’ ´Et´e de Probabilit´es de Saint-Flour, IV-1974, pages 1-96. Lecture Notesin Math., Vol. 480. Springer, Berlin (1975) · Zbl 0331.60025
[6] M. Lifshits.Lectures on Gaussian processes.Springer Briefs in Mathemat-ics. Springer, Heidelberg (2012). ISBN 978-3-642-24938-9; 978-3-642-24939-6 · Zbl 1248.60002
[7] G. Molchan. Maximum of a fractional Brownian motion: probabilities of smallvalues. Comm. Math. Phys. 205 (1), 97-111 (1999) · Zbl 0942.60065
[8] G. Molchan. Survival exponents for some Gaussian processes. Int. J. Stoch. Anal.pages Art. ID 137271, 20 (2012) · Zbl 1260.60073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.