Non‐local Dirichlet forms generated by pseudodifferential operators on compact abelian groups

Let G i , 1 ≤ i ≤ n , be compact abelian groups and let Γ i , 1 ≤ i ≤ n , be countable dual groups. We consider G = G 1 ⊕ G 2 ⊕ … ⊕ G n and Γ = Γ 1 ⊕ Γ 2 ⊕ … ⊕ Γ n . For 1 ≤ j ≤ n , let a j be a negative definite function on Γ j and a ( γ ) = . For φ ∈ S ( G ), the set of all generalized trigonometrical polynomials on G , we define , where ( γ ) = a j ( γ j ) ( γ ), 1 ≤ j ≤ n . Then is a Dirichlet form with the domain on L 2 ( G ). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)