Ergodicity of homogeneous Brownian flows

Let M be a finite-dimensional smooth-oriented paracompact manifold and ζk, 0⩽k⩽d, be a family of complete smooth vector fields on M so that the Brownian flow associated with D=∑k12ζkζk+ζ0 exists globally. We prove that any volume form μ on M is irreducible for the Brownian flows if and only if there exists only constant functions ψ∈L∞(M,μ) satisfying the following equation:ψ=ψ∘α(ζk,t) ∀t∈R, 0⩽k⩽d,where (α(ζ,t) ∀t∈R) is the one-parameter group of diffeomorphism on M associated with the complete vector field ζ. In such a case, an invariant finite volume form μ is ergodic for the flow.