# Representation Formulas for Non-Symmetric Dirichlet Forms

### S. Mataloni

Università di Roma 'Tor Vergata', Italy

## Abstract

As well-known, for $X$ a given locally compact separable Hausdorff space, $m$ a positive Radon measure on $X$ with supp[$m$] = $X$ and $C_{0}(X)$ the space of all continuous functions with compact support on $X$ the Beurling and Deny formula states that any regular Dirichlet form $(Eˉ,D(Eˉ))$ on $L_{2}(X,m)$ can be expressed as

for all $u,v∈D(Eˉ∩C_{0}(X)$ where the symmetric Dirichiet form $Eˉ_{c}$, the symmetric measure $j(dx,dy)$ and the measure $k(dx)$ are uniquely determined by $Eˉ$. It is our aim to prove this formula in the non-symmetric case. For this we consider certain families of non-symmetric Dirichlet forms of diffusion type and show that these forms admit an integral representation involving a measure that enjoys some important functional properties as well as in the symmetric case.

## Cite this article

S. Mataloni, Representation Formulas for Non-Symmetric Dirichlet Forms. Z. Anal. Anwend. 18 (1999), no. 4, pp. 1039–1064

DOI 10.4171/ZAA/927