  We study rigidity phenomenons for infinite dimension diffusion operators of
positive curvature using semigroup interpolations. In particular, for such
diffusions, an analogous statement of Obata's theorem is established. Moreover,
the same rigidity holds for the Bakry--Ledoux isoperimetric comparison Theorem
- a result due to Franck Morgan. Recently, Mossel and Neeman have exploited the
semigroup proof of Bakry and Ledoux to derive dimension free bounds for the
Gaussian isoperimetry. We extend theirs arguments to obtain in particular new
quantitative bounds on the spherical isoperimetric inequality in large
dimension.
