ABSTRACT
In the case dim X 2: 2 there is no direct connection with Brownian motion and the energy representation is gi\Tn "only algebraically" in the L 2(f-L)-spacc with respect to the natural Gaussian measure associated \Vith the scalar product given by the energy form fld'P 'P-1l(x)2 dx by translation by the !vlaurer-Cartan 1-form. There is a well-known unitary equivalence of U(f-L) with the Fock space '3' over the Sobolcv space Hj(IR"); see, e.g., [Gui].
