Weak Poincaré Inequalities on Domains Defined by Brownian Rough Paths

where 0 ≤ s ≤ t ≤ 1 and ⊗ denotes a tensor product. Lyons [17] proved that solutions of stochastic differential equations (SDEs) are continuous functions of the Brownian rough path w(s, t) = ( w(s, t)1, w(s, t)2). We give a precise definition of the Brownian rough path in the next section; see also [18] and [15]. The discontinuity of solutions of SDEs in the uniform convergence topology of w causes difficulties in analysis on Wiener spaces. However, the Lyons results provide a good topology on Wiener space and may be applied to problems which have difficulties because of the discontinuity of Wiener functionals; for example, see [16]. The present paper is an attempt to apply the Lyons continuity theorem to problems in infinite-dimensional analysis and we prove weak Poincaré inequalities (WPIs) on some “connected” domain on a Wiener space defined by a continuous function of Brownian rough paths. The WPI (actually, equivalent uniform positivity improving property of the corresponding diffusion semigroup) on a connected domain was first proved by Kusuoka [14] and led to abundant research on analysis on Wiener space and loop space. The WPI itself was introduced in [21] and the equivalence to uniform positivity improving property of the semigroup was proved therein. Aida [4] proved that WPI holds on a domain