On the Support of Diffusion Processes with Applications to the Strong Maximum Principle

A strong maximal principle for the operator (a/st) + L, is a statement of the form: "for each open 1 c [0, oo) x Rd and each (to, x0) e 9 there is a set Y(to, x0) 91 with the property that (af/lt) + LJf _ 0 on !(to, x0) and f(to, x0) = supg(toXO)f(t, x) imply f _ f(to, xO) on 9(to, x0)." Of course, in order for a strong maximum principle to be very interesting it must describe the set 9(to, x0). Further, it should be possible to show that 9(to, x0) is maximal. That is, one wants to know that if (t1, x1) e 9 W(to, x0), then there is an f satisfying (af/lt) + Ltf _ 0 on C (perhaps in a generalized sense) such that f(to, x0) = supf(t, x), andf(t1, x1) < f(to, x0)In the case when a(t, x) is positive definite for all (t, x), L. Nirenberg [6] has shown that T(to, x0) can be taken as the closure in C of the set of (t1, x1) e 1 n ([to, oo) x Rd) such that there exists a continuous map 4: [to, tl] ~ Rd with the properties that 4(to) = x0, 4(t1) = xl, and (t, 0(t)) e W for all t e (to, ti). We will give a probabilistic proof of the Nirenberg maximum principle in Section 3. Moreover, we will also prove there that Nirenberg's x(toxO) is maximal in the desired sense. If a is only nonnegative definite, the problem of finding a suitable maximum principle is more difficult. Results in this direction have been proved by J.-M. Bony [1] and C. D. Hill [3]. Both of these authors employ a modification of the technique originally introduced by E. Hopf for elliptic operators and later adapted by Nirenberg for parabolic ones. The major drawback to Bony's