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Summary. A diffusion equation approach is investigated for the study of stochastic monotonicity, positive correlations and the preservation of Lipschitz functions. Necessary and sufficient conditions are given for diffusion semigroups to be stochastically monotonic and to preserve the class of positively correlated measures. Applications are given which discuss the shape of the ground state for Schr6dinger operators -A + V with FKG potentials V. O. IntroductionIn this paper we investigate the use of diffusion equations as a tool for establishing stochastic monotonicity and correlation inequalities. Our approach to these problems is through diffusion semigroups on IR n which preserve the cone ~/of bounded monotonic increasing functions on IR" and (or) which preserve the class of probability measures on 1R n which have positive correlations in the sense that [,f9 d# > 0 holds for all f g ~ ~/~ with ~fd# = ~9 dl~ = O.The problem of characterizing those diffusion generators G = 89 + ~bJ(x)c~j on IR" for which the corresponding semigroup preserves monotonic functions leads to conditions on the {alJ(x)} and {bJ(x)} of a specific intuitive nature. These are: each aiJ(x) depends only on the coordinates xl and x j, so aiJ(x) = a~J(xl, xj) and aU(x) = ai(xi) while each drift coefficient bJ(x) must be an increasing function of the other coordinates xl for i ~ j. Here, 1 < i, j < n.Since aii(x) = ai(xl), 89 ~2 generates a one-dimensional diffusion Xi(t) without drift. The independent diffusions {X~(t)} may be coupled with pair interactions so that the resulting multidimensional diffusion X(t)= (Xl(t),..., X,(t)) has generator 89 with the infinitesimal correlations a~J(x) satisfying a~J(x) = aiJ (x i, x J). Finally, if an order-preserving drift term ~ bJ(x)c~j is introduced, *

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