Averaging for Fundamental Solutions of Parabolic Equations

Herein, an averaging theory for the solutions to Cauchy initial value problems of arbitrary order, "-dependent parabolic partial di erential equations is developed. Indeed, by directly developing bounds between the derivatives of the fundamental solution to such an equation and derivatives of the fundamental solution of an \averaged" parabolic equation, we bring forth a novel approach to comparing x-derivatives of @tu (x; t) = X jkj 2p Ak(x; t=") @ k xu (x; t); u(x; 0) = '(x) on < [0; T ] to like derivatives of @tu(x; t) = X jkj 2p A0k(x) @ k xu (x; t); u(x) = '(x) (as "! 0) under general regularity conditions and our basic hypothesis that Z t 0 Ak(x; s=") A 0 k(x) ds "!0 ! 0 for each x; t (i.e. pointwise). The exibility a orded by studying fundamental visa-vis speci c solutions of these equations not only permits "-dependent Cauchy data and provides a uni ed method of treating all x derivatives of u up to order 2p 1 but also proves an invaluable tool when considering related problems of stochastic averaging. Our development was motivated by and retains a strong resemblance to the classical theory of parabolic partial di erential equations. However, it will turn out that the classical conditions under which fundamental solutions are known to exist are somewhat unsuitable for our purposes and a modi ed set of conditions must be used. The author gratefully acknowledges support from NSF, LORAL Defense Systems, the Canadian Laboratory for Research in Statistics and Probability, and NSERC (in the form of a postdoctoral fellowship and funding from a collaborative grant in stochastic partial di erential equations). This work would not have been possible without this support.