Conformally invariant cluster value theory
J. L. Doob · Illinois Journal of Mathematics · 1961
DOOBeral theorem when applied to the classical case yields a result which is at least as strong as the classical one.In other applications to classical situations, how- ever, the situation is less clear.The appropriate invariant concept corre- sponding to a cluster value of a function on a disc, along a nontangential se- quence to a perimeter point seems to be a fine cluster value of a function on a hyperbolic Riemann surface R at a minimal point of R M (see Section 4).The relation between the two concepts if R is a disc has not yet been worked out.Thus, as might be expected, a conformally invariant theory, an intrinsic theory, is more than a generalization.It is to some extent a recasting of the theory from a different point of view.We remark that the inappropriateness of sectorial approach to disc boundary points in potential theory and allied subjects can be seen from the well-known fact that the classical boundary limit theorems for positive harmonic and superharmonic functions on a disc differ (angular approach is admissible for the first class, only radial or similar approach is admissible for the second), whereas approach in the fine topology serves in both cases.The methods used will be probabilistic, corresponding to the fact that in the present state of mathematics certain potential-theoretic results are easier to prove probabilistically than by purely potential-theoretic methods.A non- probabilistic statement of each theorem will be given, however.To clarify the historical background, references will be made to original papers, but most of the cluster value theorems referred to can be found, with proofs, in the books of Noshiro [2] and Tsuii [2].2. Functions and paths on Riemann surfaces A Riemann surface is, roughly, a connected Hausdorff space in which each point is in an open set, called a parametrized neighborhood below, which is the one-to-one conformal image of a plane disc.The nomenclature of the book of Ahlfors-Sario [1] will be used."The Riemann surface R has a posi- tive boundary, is hyperbolic, has a Green function, has a nonconstant pos- itive superharmonic function" are equivalent assertions.In the contrary case, R "has a null boundary, is parabolic".An open connected set R0 on a Riemann surface R is itself a Riemann surface, under the obvious conven- tions.If R is hyperbolic, so is R0.If R is parabolic, R0 is parabolic if and only if R R0 has capacity zero.If f is a continuous function from one Riemann surface R1 into a second, R2, and is regular in terms of the local parameters, we shall call f an analytic function from R1 to R2, or an R-valued analytic function on R,.It is trivial that R must then be parabolic if R is.If a path on a noncompact Riemann surface leaves every compact set, we shall say it "goes to oo ".If a Riemann surface R is hyperbolic, a more inter- esting compactification than that implied by the above is obtained by the adiunction of the Martin boundary RM, whose properties are fundamental in the work of this paper.CONFORMALLY INVARIANT CLUSTER VALUE THEORY 523 Brownian motion paths on abstract Euclidean spaces have already been discussed by Kakutani [1] and by the author [3].These discussions yielded rather awkward and unnatural definitions of Brownian motion on a Riemann surface (considered as a covering surface of the plane).We shall need a more appropriate definition, a conformally invariant one, which we now give.Let R be a Riemann surface, and let p be a function of the triple (t, , A), where is a strictly positive number, e R, and A is a Borel subset of R. If p satisfies the following conditions (a)-(e), p will be called a Brownian motion transition function on R.(a) p(t, i," is a measure of Borel subsets of R, with p(t, i, R) <= 1.(b) p(t,., A) is a Baire function, for each pair (t, A).(c) If0 < s,t, p(s z7 t,,A) fR p(t,v,A)p(s,,dv).(d) If e R, there is a strong Markov process with state space R, initial point , transition function p, and continuous sample functions.(e) Let be a point of R.There is then a parametrized neighborhood A of with the following properties.(el) Almost every path of the process described in (d) meets R A. (e2) Let R0 be an open subset of A whose closure is a compact subset of A. Let u be a function defined and superhar- monic on a neighborhood of this closure.Let z(t), 0 =< < } be a sto- chastic process as in (d), with initial point in R0, and let T be the time that a path from first meets the boundary of R0.Define zl(t) z[min (t, T)], and let ff (t) be the least Borel field of sets with respect to which zl (s) is meas- Theorems 8.1, 8.3, and 8.5.The methods used are purely function-theoretic, with no use of or application to probability theory.