Representation Formulas for Non-Symmetric Dirichlet Forms

  • S. Mataloni

    Università di Roma 'Tor Vergata', Italy


As well-known, for XX a given locally compact separable Hausdorff space, mm a positive Radon measure on XX with supp[mm] = XX and C0(X)C_0(X) the space of all continuous functions with compact support on XX the Beurling and Deny formula states that any regular Dirichlet form (Eˉ,D(Eˉ))(\bar{\mathcal E}, D(\bar{\mathcal E})) on L2(X,m)L^2(X,m) can be expressed as

Eˉ(u,v)=Eˉc(u,v)+Xuvk(dx)+XxXd(u(x)u(y))((v(x)v(y))j(dx,dy)\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,vD(EˉC0(X)u,v \in D(\bar{\mathcal E} \cap C_0 (X) where the symmetric Dirichiet form Eˉc\bar{\mathcal E}^c, the symmetric measure j(dx,dy)j(dx,dy) and the measure k(dx)k(dx) are uniquely determined by Eˉ\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.

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