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The spatial sojourn time for the solution to the wave equation with moving time: Central and non-central limit theorems

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  • Tudor, Ciprian A.
  • Zurcher, Jérémy

Abstract

We consider the sojourn time of the solution to the stochastic wave equation with space–time white noise on the spatial domain [−T,T]. We analyze its asymptotic behavior in distribution when T→∞ and the time variable also tends to infinity with T, i.e.t=Tα with α>0. For α<1, we prove that the properly renormalized sojourn time satisfies a Central Limit Theorem and we derive its rate of convergence under the Wasserstein distance by using the techniques of the Stein–Malliavin calculus. When the time exceeds the critical value t=T, we show that the renormalized sojourn time converges in law to a non-Gaussian limit.

Suggested Citation

  • Tudor, Ciprian A. & Zurcher, Jérémy, 2024. "The spatial sojourn time for the solution to the wave equation with moving time: Central and non-central limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    • Handle: RePEc:eee:spapps:v:172:y:2024:i:c:s0304414924000395
    DOI: 10.1016/j.spa.2024.104333
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    References listed on IDEAS

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    1. Marie F. Kratz & José R. León, 2001. "Central Limit Theorems for Level Functionals of Stationary Gaussian Processes and Fields," Journal of Theoretical Probability, Springer, vol. 14(3), pages 639-672, July.
    2. Krzysztof Dȩbicki & Peng Liu & Zbigniew Michna, 2020. "Sojourn Times of Gaussian Processes with Trend," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2119-2166, December.
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