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Bell, "A quasi-invariance theorem for measures on Banach spaces," Trans. Amer. Math. , 290, No. 2, 841-845 (1985). 5. D. Bell, "The Malliavin calculus," In: Pitman Monogr. Surv. Pure Appl. , Vol. 34, Wiley, New York (1987). 6. D. Bell, "Transformations of measure on an infinite-dimensional vector space," In: Seminar on Stochastic "Processes, 1990 (Vancouver, BC, 1990), Progr. , Vol. 24, Birkhs Boston, Boston, MA (1991). 7. V. I. Bogachev, "Locally convex, spaces with the CLT property and supports of measures," Moscow Univ.

Instead, following [11], we shall introduce assumptions much weaker than those in [39, 40], which are intrinsic and apply to general locally convex spaces. ) the equation t V,(x) = Q,(x) + j Q,_sA(U,(x)) ds. , then this equation corresponds to the ordinary differential equation x'(t) = A(x(t)) + Lx(t), x(O) = x. Our approach enables us to obtain the following more general statement, where Qt are not assumed to be continuous on X; only the one-fold differentiability of A is required and the exponential integrability of A is dropped.

Y) on L + y , y E Y. As we have already explained, there is a flow {Ut} acting in L + y , y E Y, such that the images of these conditional measures under {Ut} have densities rty with respect to the measures themselves. ly)), 4exp{ f exp(clSuA(z)l)#(dx]y)}. 17) need not be u-integrable. For this reason, in order to obtain an estimate of Ilrtllp, we shall use the following trick. For every N, let us define the measure #N on X by ~N(E) = f . ( E l y ) u(dy), I'N where For y E YN, the measure /~N has the same conditional measures on L + y as #; for other y it has zero conditional measures.

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