Journal of Statistical Physics

, Volume 173, Issue 6, pp 1587–1597 | Cite as

Persistence Exponents for Gaussian Random Fields of Fractional Brownian Motion Type

  • G. MolchanEmail author


The fractional Brownian motion of index 0 < H < 1, H-FBM, with d-dimensional time is considered on an expanding set \( T\Delta \), where \( \Delta \) is a bounded convex domain that contains 0 at its boundary. The main result: if 0 is a point of smoothness of the boundary, then the log-asymptotics of the probability that H-FBM does not exceed a fixed positive level in \( T\Delta \) is \( (H - d + o(1))\log T,T \to \infty \). Some generalizations of this result to isotropic but not self-similar Gaussian fields with stationary increments are also considered.


Fractional Brownian motion Persistence probability One-sided exit problem 



I am very grateful to two anonymous reviewers for a careful reading of the manuscript and for their suggestions for improving the presentation. This research was supported by the Russian Science Foundation through the Research Project 17-11-01052.


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Authors and Affiliations

  1. 1.Institute of Earthquake Prediction Theory and Mathematical GeophysicsRussian Academy of ScienceMoscowRussian Federation
  2. 2.National Research University Higher School of EconomicsMoscowRussian Federation

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