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Profile cut-off phenomenon for the ergodic Feller root process

Author

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  • Barrera, Gerardo
  • Esquivel, Liliana

Abstract

The present manuscript is devoted to the study of the convergence to equilibrium as the noise intensity ɛ>0 tends to zero for ergodic random systems out of equilibrium driven by multiplicative non-linear noise of the type dXtɛ(x)=(b−aXtɛ(x))dt+ɛXtɛ(x)dBt,X0ɛ(x)=x,t⩾0,where x⩾0, a>0 and b>0 are constants, and (Bt)t⩾0 is a one dimensional standard Brownian motion. More precisely, we show the strongest notion of asymptotic profile cut-off phenomenon in the total variation distance and in the renormalized Wasserstein distance when ɛ tends to zero with explicit cut-off time, explicit time window, and explicit profile function. In addition, asymptotics of the so-called mixing times are given explicitly.

Suggested Citation

  • Barrera, Gerardo & Esquivel, Liliana, 2025. "Profile cut-off phenomenon for the ergodic Feller root process," Stochastic Processes and their Applications, Elsevier, vol. 183(C).
    • Handle: RePEc:eee:spapps:v:183:y:2025:i:c:s0304414925000286
    DOI: 10.1016/j.spa.2025.104587
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    References listed on IDEAS

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    1. Barrera, Gerardo & da-Costa, Conrado & Jara, Milton, 2025. "Gradual convergence for Langevin dynamics on a degenerate potential," Stochastic Processes and their Applications, Elsevier, vol. 184(C).
    2. Fu, Zongfei & Li, Zenghu, 2010. "Stochastic equations of non-negative processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 306-330, March.
    3. Barrera, Javiera & Lachaud, Béatrice & Ycart, Bernard, 2006. "Cut-off for n-tuples of exponentially converging processes," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1433-1446, October.
    4. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    5. Chae, Minwoo & Walker, Stephen G., 2020. "Wasserstein upper bounds of the total variation for smooth densities," Statistics & Probability Letters, Elsevier, vol. 163(C).
    6. Barrera, Gerardo, 2021. "Cutoff phenomenon for the maximum of a sampling of Ornstein–Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 168(C).
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