The Annals of Probability

Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise

Aurélien Deya, Fabien Panloup, and Samy Tindel

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in(1/3,1)$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in(0,1)$, it was proved in [Ann. Probab. 33 (2005) 703–758] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-\alpha}$ where $\alpha\in(0,1)$ (depending on $H$). In [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503–538], this result has been extended to the multiplicative case when $H>1/2$. In this paper, we obtain these types of results in the rough setting $H\in(1/3,1/2)$. Once again, we retrieve the rate orders of the additive setting. Our methods also extend the multiplicative results of [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503–538] by deleting the gradient assumption on the noise coefficient $\sigma$. The main theorems include some existence and uniqueness results for the invariant distribution.

Article information

Source
Ann. Probab., Volume 47, Number 1 (2019), 464-518.

Dates
Received: October 2016
Revised: November 2017
First available in Project Euclid: 13 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1544691626

Digital Object Identifier
doi:10.1214/18-AOP1265

Mathematical Reviews number (MathSciNet)
MR3909974

Zentralblatt MATH identifier
07036342

Subjects
Primary: 60G22: Fractional processes, including fractional Brownian motion 37A25: Ergodicity, mixing, rates of mixing

Keywords
Stochastic differential equations fractional Brownian motion multiplicative noise ergodicity rate of convergence to equilibrium Lyapunov function total variation distance

Citation

Deya, Aurélien; Panloup, Fabien; Tindel, Samy. Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise. Ann. Probab. 47 (2019), no. 1, 464--518. doi:10.1214/18-AOP1265. https://projecteuclid.org/euclid.aop/1544691626


Export citation

References

  • [1] Alòs, E., Mazet, O. and Nualart, D. (2000). Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than $\frac{1}{2}$. Stochastic Process. Appl. 86 121–139.
  • [2] Arnold, L. (1998). Random Dynamical Systems. Springer, Berlin.
  • [3] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Springer, Cham.
  • [4] Cohen, S. and Panloup, F. (2011). Approximation of stationary solutions of Gaussian driven stochastic differential equations. Stochastic Process. Appl. 121 2776–2801.
  • [5] Cohen, S., Panloup, F. and Tindel, S. (2014). Approximation of stationary solutions to SDEs driven by multiplicative fractional noise. Stochastic Process. Appl. 124 1197–1225.
  • [6] Crauel, H. (1993). Non-Markovian invariant measures are hyperbolic. Stochastic Process. Appl. 45 13–28.
  • [7] Davie, A. M. (2008). Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. AMRX 2008 Art. ID abm009.
  • [8] Deya, A., Gubinelli, M., Hofmanova, M. and Tindel, S. (2016). A priori estimates for rough PDE with application to rough conservation laws. Preprint. Available at arXiv:1604.00437.
  • [9] Deya, A., Panloup, F. and Tindel, S. (2019). Supplement to “Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise.” DOI:10.1214/18-AOP1265SUPP.
  • [10] Down, D., Meyn, S. P. and Tweedie, R. L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 1671–1691.
  • [11] Fontbona, J. and Panloup, F. (2017). Rate of convergence to equilibrium of fractional driven stochastic differential equations with some multiplicative noise. Ann. Inst. Henri Poincaré Probab. Stat. 53 503–538.
  • [12] Friz, P. K., Gess, B., Gulisashvili, A. and Riedel, S. (2016). The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory. Ann. Probab. 44 684–738.
  • [13] Friz, P. K. and Hairer, M. (2014). A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, Cham.
  • [14] Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge.
  • [15] Garrido-Atienza, M. J., Kloeden, P. E. and Neuenkirch, A. (2009). Discretization of stationary solutions of stochastic systems driven by fractional Brownian motion. Appl. Math. Optim. 60 151–172.
  • [16] Guasoni, P. (2006). No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16 569–582.
  • [17] Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal. 216 86–140.
  • [18] Hairer, M. (2005). Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab. 33 703–758.
  • [19] Hairer, M. and Mattingly, J. C. (2008). Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Ann. Probab. 36 2050–2091.
  • [20] Hairer, M., Mattingly, J. C. and Scheutzow, M. (2011). Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab. Theory Related Fields 149 223–259.
  • [21] Hairer, M. and Ohashi, A. (2007). Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35 1950–1977.
  • [22] Hairer, M. and Pillai, N. S. (2013). Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Ann. Probab. 41 2544–2598.
  • [23] Jeon, J.-H., Tejedor, V., Burov, S., Barkai, E., Selhuber-Unkel, C., Berg-Sørensen, K., Oddershede, L. and Metzler, R. (2011). In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106 048103.
  • [24] Kou, S. C. (2008). Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins. Ann. Appl. Stat. 2 501–535.
  • [25] Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.
  • [26] Mattingly, J. C. (2002). Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Comm. Math. Phys. 230 421–462.
  • [27] Odde, D. J., Tanaka, E. M., Hawkins, S. S. and Buettner, H. M. (1996). Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth. Biotechnol. Bioeng. 50 452–461.
  • [28] Röckner, M. and Wang, F.-Y. (2001). Weak Poincaré inequalities and $L^{2}$-convergence rates of Markov semigroups. J. Funct. Anal. 185 564–603.
  • [29] Viens, F. G. and Zhang, T. (2008). Almost sure exponential behavior of a directed polymer in a fractional Brownian environment. J. Funct. Anal. 255 2810–2860.

Supplemental materials

  • Supplement to “Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise”. Our supplement develops the proofs of several crucial but technical results in our paper. We first prove a Lyapunov-type property for rough differential equations with inward looking drifts. Then we handle rough differential equations involving singular drifts, a type of system which arises when one tries to condition in the highly non-Markovian fractional Brownian motion setting. Next, we show how to lift rough paths involving singularities. Finally, we evaluate some effects of our conditioning procedure on the underlying fractional Brownian motion $X$ in equation (1.3).