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# Journal of Operator Theory

Volume 68, Issue 1, Summer 2012  pp. 85-100.

Gaussian upper bounds on heat kernels of uniformly elliptic operators on bounded domains

Authors Narinder S. Claire
Author institution: Global Equities and Commodity Derivatives\break Quantitative Research, BNP Paribas London, London, NW1 6AA, U.K.

Summary:  We obtain Gaussian upper bounds for heat kernels of higher order differential operators with Dirichlet boundary conditions on bounded domains in $\R^N$. The bounds exhibit explicitly the nature of the spatial decay of the heat kernel close to the boundary as well as the long-time exponential decay implied by the spectral gap. We make no smoothness assumptions on our operator coefficients which we assume only to be bounded and measurable.

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