Numerical Solutions of Stochastic PDEs Driven by Ornstein-Uhlenbeck Noise

Ning Guan, Zhongqiang Zhang, Xinghui Zhong

Abstract: This paper investigates polynomial chaos methods for the linear parabolic stochastic partial differential equations (SPDEs) driven by a special type of colored noises, i.e., the Ornstein–Uhlenbeck (OU) noise. Unlike the study for the SPDEs driven by white noises, the stochastic analysis for colored noises suffers from great difficulties due to the lack of orthogonality among the “basis” functions. We take special care for such scenario and prove the existence and uniqueness of the chaos solution and establish the convergence analysis of the truncated numerical solutions. We show that the truncation error exhibits exponential convergence in the expansion order and cubic convergence in the number of random variables. This improved convergence rate, surpassing the linear rate described in Lototsky and Stemmann (Q Appl Math 66(3):499–520, 2008), underscores the significance of our findings. Moreover, we consider first-order differential operators as the coefficients of noises, which are more challenging in analysis. Our study also extends the numerical study in Chen et al. (Stoch Partial Differ Equ Anal Comput 7(1):1–39, 2019) for SPDEs driven by white noises of various distributions to a more general type of noise. The results can provide useful guidance for distribution-free SPDEs driven by colored noise of various distributions. Numerical results are provided to validate our theoretical findings and to illustrate different convergence rates for commutative noises and noncommutative noises.