The Gaussian free field and Hadamard's variational formula
  We relate the Gaussian free field on a planar domain to the variational formula of Hadamard which explains the change of the Green function under a perturbation of the domain. This is accomplished by means of a natural integral operator related to Hadamard's formula. 
Haakan Hedenmalm, Pekka J. Nieminen
math.PR math.FA