RATE OF CONVERGENCE TO EQUILIBRIUM OF FRACTIONAL DRIVEN STOCHASTIC DIFFERENTIAL EQUATIONS WITH ROUGH MULTIPLICATIVE NOISE

• 成果类型: Article
• 署名作者: Deya, Aurelien; Panloup, Fabien; Tindel, Samy
• 署名单位: Universite de Lorraine; Universite d'Angers; Purdue University System; Purdue University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1265
• 发表日期: 2019
• 页码: 464-518
• 关键词: stationary solutions; sdes driven; ergodicity; approximation

摘要

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 is an element of (1/3, 1) and multiplicative noise component sigma. When sigma is constant and for every H is an element of (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 is an element of (0, 1) (depending on H). In [Ann. Inst. Henri Poincare 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 is an element of (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 Poincare 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.