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Lower bounds for densities of uniformly elliptic random variables on Wiener space

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Abstract.

 In this article, we generalize the lower bound estimates for uniformly elliptic diffusion processes obtained by Kusuoka and Stroock. We define the concept of uniform elliptic random variable on Wiener space and show that with this definition one can prove a lower bound estimate of Gaussian type for its density. We apply our results to the case of the stochastic heat equation under the hypothesis of unifom ellipticity of the diffusion coefficient.

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Received: 6 November 2001 / Revised version: 27 February 2003 / Published online: 12 May 2003

Key words or phrases: Malliavin Calculus – Density estimates – Aronson estimates

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Kohatsu-Higa, A. Lower bounds for densities of uniformly elliptic random variables on Wiener space. Probab. Theory Relat. Fields 126, 421–457 (2003). https://doi.org/10.1007/s00440-003-0272-4

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Keywords

  • Diffusion Coefficient
  • Lower Bound
  • Diffusion Process
  • Heat Equation
  • Gaussian Type