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Lower Deviation for the Supremum of the Support of Super-Brownian Motion

Author

Listed:
  • Yan-Xia Ren

    (Peking University)

  • Renming Song

    (University of Illinois at Urbana-Champaign)

  • Rui Zhang

    (Capital Normal University)

Abstract

We study the asymptotic behavior of the supremum $$M_t$$ M t of the support of a supercritical super-Brownian motion. In our recent paper (Ren et al. in Stoch Proc Appl 137:1–34, 2021), we showed that, under some conditions, $$M_t-m(t)$$ M t - m ( t ) converges in distribution to a randomly shifted Gumbel random variable, where $$m(t)=c_0t-c_1\log t$$ m ( t ) = c 0 t - c 1 log t . In the same paper, we also studied the upper large deviation of $$M_t$$ M t , i.e., the asymptotic behavior of $$\mathbb {P} (M_t>\delta c_0t) $$ P ( M t > δ c 0 t ) for $$\delta \ge 1$$ δ ≥ 1 . In this paper, we study the lower large deviation of $$M_t$$ M t , i.e., the asymptotic behavior of $$\mathbb {P} (M_t\le \delta c_0t|\mathcal {S}) $$ P ( M t ≤ δ c 0 t | S ) for $$\delta

Suggested Citation

  • Yan-Xia Ren & Renming Song & Rui Zhang, 2024. "Lower Deviation for the Supremum of the Support of Super-Brownian Motion," Journal of Theoretical Probability, Springer, vol. 37(2), pages 1079-1123, June.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:2:d:10.1007_s10959-023-01292-3
    DOI: 10.1007/s10959-023-01292-3
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