2018/10/15 by Charles-Édouard Bréhier, Bréhier, Charles-Edouard
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1810.06448
openalex publication_date 2018/10/15 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28
This article is devoted to the analysis of semilinear, parabolic, Stochastic\nPartial Differential Equations, with slow and fast time scales. Asymptotically,\nan averaging principle holds: the slow component converges to the solution of\nanother semilinear, parabolic, SPDE, where the nonlinearity is averaged with\nrespect to the invariant distribution of the fast process.\n We exhibit orders of convergence, in both strong and weak senses, in two\nrelevant situations, depending on the spatial regularity of the fast process\nand on the covariance of the Wiener noise in the slow equation. In a very\nregular case, strong and weak orders are equal to frac12 and 1. In a less\nregular case, the weak order is also twice the strong order.\n This study extends previous results concerning weak rates of convergence,\nwhere either no stochastic forcing term was included in the slow equation, or\nthe covariance of the noise was extremely regular.\n An efficient numerical scheme, based on Heterogeneous Multiscale Methods, is\nbriefly discussed.\n