Averaging of random parabolic operators and  
                  diffusion approximation

                    Andrey Piatnitski 
   (P.N.Lebedev Physical Inst., Russian Academy of Sciences)


             INSCC 110, 3:30pm  Monday, April 27, 1998


                          Abstract



We study the averaging problem for a random parabolic operator
of the form
$$
A^\varepsilon={\partial\over\partial t}-{\partial\over\partial x_i}
a_{ij}\Big({x\over\varepsilon},\xi_{t/\varepsilon^\alpha}\Big)
{\partial\over\partial x_j}-{1\over\varepsilon}
c\Big({x\over\varepsilon},\xi_{t/\varepsilon^\alpha}\Big),\ x\in R^n,
$$
where $\varepsilon$ is a small parameter, $\alpha>0$ and $\xi_s$
is a diffusion process in $R^d$ possessing an invariant probability
measure  with density $p(y)$. The coefficients $a_{ij}(z,y)$ and
$c(z,y)$ are periodic in $z$.
\par\noindent
The limiting behaviour of $A^\varepsilon$ depends  crucially on whether
$c(z,y)$ is a constant or not. In the former case the "classical" 
homogenization result holds and the limit operator has nonrandom
constant coefficients. Moreover, the discrepancy can be estimated.
\par\noindent
If $c(x,y)$ is not a constant then, in general, the homogenization
does not take place while weaker averaging result holds:
\par\noindent
a family of measures generated by the solutions of corresponding
Cauchy problems, converges weakly to the unique solution of the
limit martingale problem.
\par\noindent
In both cases above the coefficients of the averaged operators can be found
in terms of solutions of auxiliary problems posed in the product of  
$n$-dimensional torus and $R^d$.

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