ABSTRACT
Define the probability space i i = C ( 0 , l ; i i ) x U endowed with the natural filtration Elements of i i will be denoted by u = (X^q). Notice that J : i i —> R+ is l.s.c. Denote by $ the space of all functions <f> : H —* R of the form
<l>(X(s)) = <t>o(< X ( s ) , ei > , . . . , < X (s ) , e„ > ) ,
where <f>o £ C o°(R n), e* £ D(A*) for any ¿ = 1 ,2 , . . . and n = 1 ,2 ,3 , — For any 0 < t < 1 define the functional C* : $ x i i —> R by
C t (<f>,X,q) - 0 (X (i) ) - ¿ 1 ^ , 0 - L 2<f>(X,q,t)
where the operators Li and L 2 are of the form
= j f ( < A * V ^ (X (5) ) ,X ( 5) > + iT r[f ir* (X (5))V 20 (X (5))^ (X (S) ) ] ) ^ ,
and
L 2<t>{X,q,t) = f ( < V 4 > ( X ( s ) ) , f ( X ( s ) , u ) > q ( d u , d s ) , Jo J v
where V</> and V 2</i> denote the first and second Frechet derivatives of the functional </>.
