Moment estimates for parabolic equations in the divergence form (Q1075510)

The purpose of the paper is to derive results similar to an estimate obtained by Nash. Denote by P(s,x,t,y) the fundamental solution of \(\partial /\partial t-\sum^{n}_{i,j=1}(\partial /\partial x_ i)a_{ij}(x,t)\partial /\partial x_ j=0\). It is shown that for nonnegative integers p,q there exist positive constants \(C_ 1,C_ 2\) depending only on \(a_{ij}\), p,q such that \[ C_ 1 n^{q+1} (t- s)^{(p+q)/2}\leq \] \[ \int_{R^ n}(\sum^{n}_{i=1}| x_ i- y_ i|^ p)(\sum^{n}_{i=1}| x_ i-y_ i|)\quad^ q P(s,x,t,y)dy\leq C_ 2 n^{q+1} (t-s)^{(p+q)/2} \] for all \(x\in R^ n\), \(0\leq s<t<\infty\). In the case \(p=0\), \(q=1\) a more precise estimate is given. This study of general moments of P(s,x,t,y) is motivated by a problem of interacting diffusion processes in probability theory.