Coordinate transformations in the FEYNMAN-KAC formula areconsidered. The contractive semigroup with transformed coordinates is expressed by nn integral with n correspondingly transformed diffusion measure. This representation is used for N-body systems transformed to clustered JACOBI coordinates implying a decomposition of the FEYNBIAN-KAC formula. ' 1 I in the sense of distribution . Denotation. H I is the generator of the usual WIENER process Sn, ~( s ) , PJ.)). Here Q l l g consists of all continuous functions to(.) mapping [O, -) into Rn with OJ(O) = x . %* is the smallest a-field generated by i f . ( . ) : f,J(Y) β¬ 8 I OJ(O) = X I , where 8 is any Bore1 set in R". Pz(.) is the WIENER measure on !ljn constructed by the transition density function P ( t , 2, y) =(znt) -nEe-Ix-u"Wt (6) with x, ycR", t z O , and 1z12= C x f . e i = I Assumption 2. Let A be a linear transformation from R" onto R" with det A =k 0. and(A-i)ij the matrix elements of A and A-1, respectively. Set u = A x . Assumption 3. Let p be the transformed transition density function defined by (7) u, vER", t > O . Denote by (L?n,u, Bn,w, L3(s), p,,(.)) the corresponding transformed process, 15 =Am. Lemma 1. T h e transformed process (52,,t,, gn, 8(s), Pu(.)) is a diffuSion process. I t s generator, denoted by R1, is generated by p(t, 21, v ) =det A-1 p(t, A -k , A -k ) , and it holds dom R1 =dom HI.
Denotation. Denote by A ,