On the behaviour of metrics $H_s$ on loop groups

The heat measures with respect to metric $H_s$ on loop groups were introduced by P. Malliavin; its behaviour as $s\downarrow 1/2$ of finite dimensional distributions was studied by Y. Inahama [J. Funct. Anal., 198 (2003), p. 311-340]. In this note, we shall transfer this problem to the long time behaviour of diffusion processes. We conclude the result by using a metric equivalent form for the lower bound of Ricci tensors.