ADDITIVE FUNCTIONALS AS ROUGH PATHS

• 成果类型: Article
• 署名作者: Deuschel, Jean-Dominique; Orenshtein, Tal; Perkowski, Nicolas
• 署名单位: Technical University of Berlin; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Free University of Berlin
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1488
• 发表日期: 2021
• 页码: 1450-1479
• 关键词: STOCHASTIC INTEGRALS; limit; RENORMALIZATION; homogenization

摘要

We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Levy area that can be described in terms of the asymmetry (nonreversibility) of the underlying Markov process. We apply this abstract result to three model problems: First, we study random walks with random conductances under the annealed law. If we consider the Ito rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. If we consider the Stratonovich rough path, then there is no correction. The second example is a nonreversible Ornstein-Uhlenbeck process, while the last example is a diffusion in a periodic environment. As a technical step, we prove an estimate for the p-variation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path Burkholder-Davis-Gundy inequality for local martingale rough paths of (In Seminaire de Probabilites XLI (2008) 421-438 Springer; In Probability and Analysis in Interacting Physical Systems (2019) 17-48 Springer; J. Differential Equations 264 (2018) 6226-6301) to the case where only the integrator is a local martingale.