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Let L be a second order elliptic differential operator in Rd with no zero order terms and let E be a bounded domain in Rd with smooth boundary 'E. We say that a function h is L-harmonic if Lh = 0 in E. Every positive L-harmonic function has a unique representation h(x) J(x, y)i(dy), E where k is the Poisson kernel for L and z' is a finite measure on &E. We call z the trace of h on ME. Our objective is to investigate positive solutions of a nonlinear equation Lu = u' in E for 1 2]. We associate with every solution u a pair (IF, i), where F is a closed subset of &E and zis a Radon measure on 0 -E \ F. We call (F, zi) the trace of u on ME. F is empty if and only if u is dominated by an L-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions for a pair (F, z') to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.

Let L be a second order elliptic differential operator on a Riemannian manifold E with no zero order terms. We say that a function h is L-harmonic if Lh 0. Every positive L-harmonic function has a unique representation

Let ϕ \phi be a real valued function defined on the state space of a Markov process x t {x_t} . Let τ t {\tau _t} be the first time x t {x_t} gets to a level set of ϕ \phi which is t t units higher than the one on which it started. We call the time changed process x ~ t = x τ t \tilde {x}_{t} = x_{{\tau _t}} a stochastic wave. We give conditions under which this process is Markovian and we evaluate its infinitesimal operator.

Let X = ( X t , P ) X = ({X_t},P) be a measure-valued stochastic process. Linear functionals of X X are the elements of the minimal closed subspace L L of L 2 ( P ) {L^2}(P) which contains all X t ( B ) {X_t}(B) with ∫ X t ( B ) 2 d P > ∞ \smallint {{X_t}{{(B)}^2}\;dP\; > \infty } . Various classes of L L -valued additive functionals are investigated for measure-valued Markov processes introduced by Watanabe and Dawson. We represent such functionals in terms of stochastic integrals and we derive integral and differential equations for their Laplace transforms. For an important particular case—"weighted occupation times"—such equations have been established earlier by Iscoe. We consider Markov processes with nonstationary transition functions to reveal better the principal role of the backward equations. This is especially helpful when we derive the formula for the Laplace transforms.

The class of regular Markov processes is very close to the class of right processes studied by Meyer, Getoor and others. We say that a transition function p p is regular if it is the transition function of a well-defined regular Markov process. A characterization of regular transition functions is given which implies that, if p p is regular, then the Dawson-Watanabe and the Fleming-Viot supertransition functions over p p belong to the same class.

A simple proposition (Theorem 1) on flows allows the investigation of random time shift and time reversal in Markov processes without assuming any regularity of paths. Theorem 5 is a generalization of Nagasawa's time reversal theorem and Theorem 4 generalizes a recent result of Getoor and Glover. 1. Flows. 1.1. We consider a flow in a measurable space (2, Y) that is a family of transformations at, t E R, such that OAt = 0s+t for all s, t E R and {(t, c): Oto E A} E R X -4 for all A E F(R is the Borel a-algebra in R). We put (OtY)(X) = Y(Otw), for every function Y and (POt)(A) = P(0t71A), A E Y, P=| POt dt, for every measure P. We denote by _ the collection of all sets A E Y such that t1A = A for all t E R. A function Y is measurable with respect to the a-algebra Wif and only if it is -measurable and invariant under all transformations Ot. Obviously P(A) = 0 or + oo for all A E sl. Nevertheless we prove THEOREM 1. If P is a-finite (over Y), it determines uniquely the values P(A) for all A e . 1.2. As a tool we use stationary times. A stationary time T is a measurable mapping from 2 to the extended real line [oo, + oo] such that OtT = T t for all t E R. Suppose that Y coincides with its completion with respect to the class of all probability measures. Then the first hitting time of a set A E Y, TA = inf{t: Ot, e A), and the last exit time from A, TA = sup{t: Ot co A) are stationary times2 (the measurability follows from [5, Chapter 3, ?1]). Received by the editors March 13, 1984. 1980 Mathematics Subject Classification. Primary 60G, 601. 1 Supported in part ;by National Science Foundation Grant MCS-8202286. 2We put inf= + oo, sup = -oo. (C1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page