Functional inequality on path space over a non-compact Riemannian manifold

Abstract

We prove the existence of the O-U Dirichlet form and the damped O-U Dirichlet form on path space over a general non-compact Riemannian manifold which is complete and stochastically complete. We show a weighted log-Sobolev inequality for the O-U Dirichlet form and the (standard) log-Sobolev inequality for the damped O-U Dirichlet form. In particular, the Poincaré inequality (and the super Poincaré inequality) can be established for the O-U Dirichlet form on path space over a class of Riemannian manifolds with unbounded Ricci curvatures. Moreover, we construct a large class of quasi-regular local Dirichlet forms with unbounded random diffusion coefficients on path space over a general non-compact manifold.Suppose M is a n-dimensional non-compact complete connected Riemannian manifold, the Riemannian path space C o,T (M) over M is defined byC o,T (M) := {γ ∈ C([0, T ]; M) : γ(0) = o}, where T is a positive constant and o ∈ M. Let d M be the Riemannian distance on M, then C o,T (M) is a Polish space under the uniform distance d(γ, σ) := sup t∈[0,T ]d M (γ(t), σ(t)), γ, σ ∈ C o,T (M).

Characterizations of the Upper Bound of Bakry–Emery Curvature

“…To overcome the difficulty, we may obtain the local the formula of integration by parts by using the cutoff method. The idea is to make a conformal change of metric such that the new Riemannian manifold is with bounded curvature(see [26,4,27]) and two metrics are the same in a compact set. In fact, for any R > 0, taking ϕ ∈ C ∞ 0 (M) such that ϕ| B R+1 (x) = 1.…”

“…According to [19, section 2](See also [19,13,27,4]) and references in, (M R , ·, · R ) is a complete Riemannian manifold under the metric ·, · R := ϕ −2 ·, · ,…”

Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

“…However, the corresponding upper bound characterizations are still open. It is known that for stochastic analysis on the path space, one needs conditions on the norm of Ric Z , see [3,4,5,7,10,15,17] and references within. Recently, A. Naber [12,9] proved that the uniform bounded condition on Ric Z for Z = −∇f is equivalent to some gradient/functional inequalities on the path space, and thus clarified the necessity of bounded conditions used in the above mentioned references.…”

Stochastic heat equations with values in a Riemannian manifold

“…In the second part of this paper, we use functional inequalities to study the properties of the solutions to the stochastic heat equations on path space over a Riemannian manifold M. Functional inequalities for Ornstein-Unlenbeck process on Riemannian path space have been well-studied (see [24,3,4,22,3,38,39,53,16] and references therein). Since the L 2 -Dirichlet form associated with the stochastic heat equation is larger than the O-U Dirichlet form E OU constructed in [19] (i.e., E OU (u, u) ≤ E (u, u) for u ∈ D(E )), all the functional inequalities with respect to E OU still hold in the stochastic heat equation case, which implies that the former requires stronger Ricci curvature conditions than the latter.…”