A method of averaging the processes in random frames is described. An averaged operator splitting principle is stated, whereby explicit expressions can be obtained for calculating the effective characteristics.
Mathematicallyrigorous averaging of equations in random media has been dealt with earlier in /1-4/. 1. A geometric model. Let ~...... ~>-be domains in R",-l with piecewise smooth boundaries; B b j=1. 2โข... , N. are cylinders B;= {xER'"IX1ER. (X. I ellโข...โข x.,J ell) E~;}; Bh;a is the cylinder obtained from Bj by orthogonal transformation of the space with transformation I:latrix a=lIa yll and shift by he=(h,eโข...โข hme). As the geometric model of the periodic m-dimensional frame we take the domain N B=U U U Bht, j-J a.eA.lllellj
where A)is a set of orthogonal m-th order matrices; Hl=Rm; e. It are small parameters; A).ll) are independent of e. J-1. and do not contain two sets (a l '), hI"), (a l ". hI%) with the same first columns in the matrices all) and a l " with hll)=h l ". We assume that B is connected and periodic, period e , with respect to each variable Xi' the intersection of B with the cube of periodicity Q,={xER"'IO~XI~e. l=1, 2โข...โข m} being connected and expressible as the finite union I a UBhp~pnQ, . (1.1) P-l In Fig.l we show two examples of two-dimensional frames. The frame B= where N B. =U U U Bhja (CJl), j -I c,oA J hells take where Bht(CJl) is Bhja with probability 1-p~,j and is ยข with probability p~.j; Bht(liJ) are