Representation Formulas for Non-Symmetric Dirichlet Forms

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 $(\bar{\mathcal{E}},D(\bar{\mathcal{E}}))$ on $L^2(X,m)$ can be expressed as

for all $u,v\in D(\bar{\mathcal{E}}\cap C_0(X)$ where the symmetric Dirichiet form ${\bar{\mathcal{E}}}^c$, the symmetric measure $j(dx,dy)$ and the measure $k(dx)$ are uniquely determined by $\bar{\mathcal{E}}$. 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.