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