Ergodicity of homogeneous Brownian flows

Anilesh Mohari

Stochastic Processes and their Applications, 2003, vol. 105, issue 1, 99-116

Abstract: Let M be a finite-dimensional smooth-oriented paracompact manifold and [zeta]k, 0[less-than-or-equals, slant]k[less-than-or-equals, slant]d, be a family of complete smooth vector fields on M so that the Brownian flow associated with exists globally. We prove that any volume form [mu] on M is irreducible for the Brownian flows if and only if there exists only constant functions [psi][set membership, variant]L[infinity](M,[mu]) satisfying the following equation:[psi]=[psi]o[alpha]([zeta]k,t) [for all]t[set membership, variant]R, 0[less-than-or-equals, slant]k[less-than-or-equals, slant]d,where ([alpha]([zeta],t) [for all]t[set membership, variant]R) is the one-parameter group of diffeomorphism on M associated with the complete vector field [zeta]. In such a case, an invariant finite volume form [mu] is ergodic for the flow.

Keywords: Brownian; flows; Stochastically; complete; Horizontal; Brownian; flows; Irreducible; Ergodic; Manifold; Holonomy; group; Holonomy; algebra (search for similar items in EconPapers)
Date: 2003
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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