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We study rigidity phenomenons for infinite dimension diffusion operators ofpositive curvature using semigroup interpolations. In particular, for suchdiffusions, 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 thesemigroup proof of Bakry and Ledoux to derive dimension free bounds for theGaussian isoperimetry. We extend theirs arguments to obtain in particular newquantitative bounds on the spherical isoperimetric inequality in largedimension.

Rigidity phenomenons for an infinite dimension diffusion operator and cases of near equality in the Bakry--Ledoux isoperimetric comparison Theorem