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A simple method for finite range decomposition of quadratic forms and Gaussian fields

Abstract

We present a simple method to decompose the Green forms corresponding to a large class of interesting symmetric Dirichlet forms into integrals over symmetric positive semi-definite and finite range (properly supported) forms that are smoother than the original Green form. This result gives rise to multiscale decompositions of the associated Gaussian free fields into sums of independent smoother Gaussian fields with spatially localized correlations. Our method makes use of the finite propagation speed of the wave equation and Chebyshev polynomials. It improves several existing results and also gives simpler proofs.

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Acknowledgments

The author thanks David Brydges and Gordon Slade for many helpful discussions, advice, and careful proofreading. He also thanks Martin Barlow for helpful discussions.

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Correspondence to Roland Bauerschmidt.

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Bauerschmidt, R. A simple method for finite range decomposition of quadratic forms and Gaussian fields. Probab. Theory Relat. Fields 157, 817–845 (2013). https://doi.org/10.1007/s00440-012-0471-y

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Keywords

  • Green’s function
  • Positive definite
  • Gaussian free field
  • Dirichlet form
  • Elliptic operator
  • Renormalization group

Mathematics Subject Classification (2000)

  • 60G15
  • 35J08