Time Averaging for Random Nonlinear Abstract Parabolic Equations

It is well known that the averaging principle is a powerful tool of investigation of ordinary differential equations, containing high frequency time oscillations, and a vast work was done in this direction (cf. [1]). This principle was extended to many other problems, like ordinary differential equations in Banach spaces, delayed differential equations, and so forth (for the simplest result of such kind we refer to [2]). It seems to be very natural to apply such an approach to the case of parabolic equations, either partial differential, or abstract ones. However, only a few papers deal with such equations. Most of them deal with linear and quasilinear equations in the case when high oscillations in coefficients and/or forcing term are of periodic or almost periodic nature [4, 6, 9, 8, 13, 14, 18]. Moreover, many applications give rise naturally to parabolic equations with highly oscillating random coefficients. For linear equations of such kind the averaging principle was studied in [15, 16, 17]. Note that, in [17] the so-called spatial and space-time averaging (homogenization) is investigated, while the time averaging is also considered. In the present paper, we study the averaging problem for an abstract monotone parabolic equation, the operator coefficient of which is a stationary (operator valued) stochastic process. We prove that in this case the averaging takes place almost surely, that is, with probability 1. As a consequence, we get an averaging result for the case of almost periodic coefficients (almost periodic functions may be regarded as a particular case of a stationary process). This result is, so to speak, individual, in contrast to the main theorem which is statistical in its nature. Our approach differs from those used in the references we pointed out above, except [17], and is based on the theory of G-convergence of abstract parabolic operators. The last theory was developed in [7] in