Representation Formulas for Non-Symmetric Dirichlet Forms

S. Mataloni

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 $(\tilde\E,D(\tilde \E))$ on $\LX$ can be expressed as $$ \tilde\E(u,v) = \tilde\E^c(u,v) + \iX uv k(dx) + \id(u(x)-u(y))((v(x) - v(y))\tilde j(dx,dy) $$ for all $u,v \in D(\tilde\E) \cap C_0(X)$ where the symmetric Dirichlet form $\tilde\E^c$, the symmetric measure $\tilde j(dx,dy)$ and the measure $k(dx)$ are uniquely determined by $\tilde\E$. 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 Dirichlet forms, Beurling-Deny formula, diffusion forms, energy measures, differentiation formulas, differential operators

Classification (MSC2000): 28A99, 31C25, 35J70

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