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Zeitschrift für Analysis und ihre Anwendungen


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Volume 18, Issue 4, 1999, pp. 1039–1064
DOI: 10.4171/ZAA/927

Published online: 1999-12-31

Representation Formulas for Non-Symmetric Dirichlet Forms

S. Mataloni[1]

(1) Università di Roma 'Tor Vergata', Italy

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 $$\bar{\mathcal E} (u, v) = \bar{\mathcal E}^c (u,v) + \int_X uvk(dx) + \iint_{XxX–d} (u(x) - u(y))((v(x) - v(y))j(dx,dy)$$ 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.

Keywords: Non-symmetric Dirichiet forms, Beurling-Deny formula, diffusion forms, energy measures, differentiation formulas, differential operators

Mataloni S.: Representation Formulas for Non-Symmetric Dirichlet Forms. Z. Anal. Anwend. 18 (1999), 1039-1064. doi: 10.4171/ZAA/927