An estimate on the Hessian of the heat kernel

  • Daniel W. Stroock


Let M be a compact, connected Riemannian manifold, and let p t (x, y) denote the fundamental solution to Cauchy initial value problem for the heat equation \(\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u\), where Δ is the Levi-Civita Laplacian. The purpose of this note is to show that the Hessian of log p t (·, y) at x is bounded above by a constant times \(\frac{1}{t}+\frac{dist{{\left( x,y \right)}^{2}}}{{{t}^{2}}}\) for t ∈ (0, 1].


Heat Kernel Ricci Flow Heat Kernel Estimate Brownian Path Connected Riemannian Manifold 
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Copyright information

© Springer-Verlag Tokyo 1996

Authors and Affiliations

  • Daniel W. Stroock
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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