(e_{n})

H_{0}^{1} (D)

and write the GFF formally as

h = \sum_{n} ξ_{n} e_{n}

, where the

ξ_{n}

are independent standard normal variables. The Green-kernel expansion in the Dirichlet space gives

\sum_{n \geq 1} (\int_{D} e_{n} (x) ρ (d x))^{2} = \iint_{D \times D} G_{D} (x, y) ρ (d x) ρ (d y) < \infty.

Consequently the series

(h, ρ) := \sum_{n \geq 1} ξ_{n} \int_{D} e_{n} (x) ρ (d x)

converges in

L^{2}

. Its partial sums are centered Gaussian and their variances converge to the displayed Green energy. The

L^{2}

limit is therefore Gaussian with mean zero and variance

\iint_{D \times D} G_{D} (x, y) ρ (d x) ρ (d y) .

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.