The maximum Markovian self-adjoint extensions of Dirichlet oper- ators for interacting particle systems

Let μ be a Ruelle measure on the configuration space ΓRd with a pair potential φ. Then the generator of the corresponding intrinsic Dirichlet form (Eμ, H 0 (ΓRd ;μ)) is an extension of the Dirichlet operator ∆φ on L (ΓRd ;μ) with domain FC∞ b defined by ∆φ = div Γ φ∇. Here FC∞ b is the set of smooth cylinder functions, ∇Γ the gradient of the Riemannian structure on ΓRd and div Γ φ the corresponding divergence. For a large class of nonnegative (singular) potentials φ, we give a convergence characterization for the weak Sobolev spaces W 1,2 ∞ (ΓRd ;μ) and prove that the generator of (Eμ, W 1,2 ∞ (ΓRd ;μ)) is the maximum Markovian self-adjoint extension of (∆φ,FC b ). Furthermore, we construct stationary diffusion processes associated with (Eμ, W 1,2 ∞ (ΓRd ;μ)) by approximation. 1991 Mathematical Subject Classification: 60G60, 82C22; 31C25, 47D07. Running Title: Markovian self-adjoint extensions of Dirichlet operators.