Average and deviation for the stochastic FitzHugh--Nagumo system

Abstract

An averaged system for the slow-fast stochastic FitzHugh--Nagumo system is derived in this paper. The rate of convergence in probability is obtained as a byproduct. Moreover the deviation between the original system and the averaged system is studied. A martingale approach proves that the deviation is described by a Gaussian process. The deviation gives a more accurate asymptotic approximation than previous work. References S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction--diffusion equations, to appear in Probab. Th. and Rel. Fields . doi:10.1007/s00440-008-0144-z R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys J. , 1 (1961). H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Commun. Math. Phys. , 65 (1979), 79--128. doi:10.1007/3-540-08853-9 R. Z. Khasminskii, On the principle of averaging the Ito stochastic differential equations (Russian), Kibernetika , 4 (1968), 260--279. P. E. Kloeden and E. Platen. Numerical solution of stochastic differential equations , volume 23 of Applications of Mathematics . Springer--Verlag, 1992. M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces , Scuola Normale Superiore, Pisa, 1988. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions , Cambridge University Press, 1992. J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. , 146 (1987), 65--96. doi:10.1007/BF01762360 V. M. Volosov, Averaging in systems of ordinary differential equations. Russ. Math. Surv. , 17 (1962), 1--126. doi:10.1070/RM1962v017n06ABEH001130