Moment estimates for parabolic equations in the divergence form.

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Abstract

The author proves certain generalizations of the Nash estimate [ J. Nash , Amer. J. Math. 80 (1958), 931–954; MR0100158]. Let p(s,x,t,y) be a measurable fundamental solution of ∇ t −A≡∇ t −∑ n i,j=1 ∇ i a ij ∇ j . It is assumed that ∑ n i,j=1 a ij (t,x)ξ i ξ j ≥ν|ξ| 2 and that there exist a symmetric matrix b ij (t) and a matrix c ij (t,x) , such that a ij (t,x)=b ij (t)+c ij (t,x) , and, moreover, sup 1≤j≤n ∑ n i=1 |b ij (t)|≤λ and sup 1≤i,j≤n |c ij (t,x)|≤μ/n for every (t,x)∈[0,∞)×R n . It is also assumed that the constants λ , μ , ν are independent of n . Under these assumptions the author proves the following: If m,q are nonnegative integers, then there exist positive constants C 1 , C 2 , dependent only on λ , μ , ν , m and q , such that C 1 n q+1 (t−s) (m+q)/2 ≤∫ R n (∑ i=1 n |x i −y i | m )(∑ i=1 n |x i −y i |) q p(s,x,t,y)dy≤C 2 n q+1 (t−s) (m+q)/2 for all x∈R n , 0≤s<t<∞ . Sharper estimates are proved for the particular cases m=1 , q=0 or q=1 , m=0 . The estimates of this work can be used in the solution of the problem of interacting diffusion processes in probability theory, as formulated by H. P. McKean, Jr. [Stochastic differential equations (Washington, D.C., 1967), 41–57, Air Force Office Sci. Res., Arlington, Va., 1967; MR0233437].
Original languageEnglish
Pages (from-to)473-488
Number of pages16
JournalJournal of Mathematics of Kyoto University
Volume25
Issue number3
Publication statusPublished - 1985
Externally publishedYes

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