Exact L2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

• 成果类型: Article
• 署名作者: Nazarov, AL; Nikitin, YY
• 署名单位: Saint Petersburg State University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-004-0337-z
• 发表日期: 2004
• 页码: 469-494
• 关键词: probabilities; LAW

摘要

We find the exact small deviation asymptotics for the L-2-norm of various m- times integrated Gaussian processes closely connected with the Wiener process and the Ornstein - Uhlenbeck process. Using a general approach from the spectral theory of linear differential operators we obtain the two-term spectral asymptotics of eigenvalues in corresponding boundary value problems. This enables us to improve the recent results from [15] on the small ball asymptotics for a class of m-times integrated Wiener processes. Moreover, the exact small ball asymptotics for the m-times integrated Brownian bridge, the m-times integrated Ornstein - Uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory.