Probability Theory and Related Fields

, Volume 157, Issue 3–4, pp 817–845 | Cite as

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



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.


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

Mathematics Subject Classification (2000)

60G15 35J08 


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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