Random Walks and A Sojourn Density Process of Brownian Motion

The sojourn times for the Brownian motion process in 1 dimension have often been investigated by considering the distribution of the length of time spent in a fixed set during a fixed time interval.In this approach the sojourn time is either treated as a functional leading to a differential equation (as in [9, Kac]) or else the method of moments is used (as in [3, Darling and Kac] in a more general situation).An alternative approach, however, is suggested by a theorem of H. Trotter [14] that as a set function, the sojourn time up to a time t is a. s. absolutely continuous with respect to Lebesgue measure, and that, in fact, it has a density function f(x, t, w) which is continuous in (x, t).This theorem creates the possibility of considering the sojourn density function as a stochastic process with parameter x, coexistent with the Brownian motion in t from which it is defined.The study of sojourn times then becomes equivalent to the study of this new process.The possible advantage of such an approach depends on the fact that when this process is considered in a suitable random time interval it proves to be very amenable to investigation.If one sets, for example, T= inf, : /(0, t,w) > a, then f(x, T, w) turns out to be the diffusion process with x _ 0 as parameter, initial value a, infinitesimal generator 4y(d2/dy2), and an absorbing barrier at 0. This fact illustrates our main result, which is contained below in Theorem 2.2.Its proof, together with a new proof of the theorem of [14], is accomplished by using a strong approximation to Brownian motion by means of the classical random walks (i.e., "coin tossing") which appeared in [10].It seems of methodological interest that special properties of this construction are used essentially, so that although the result concerns only Brownian motion there is apparently no easy way to avoid using discrete random walks in the proof (2).The first part of the paper introduces the random walk versions of the limiting sojourn density processes and includes a weak limit of their joint distributions, Received by the editors September 14, 1962.(!) This work was supported by the Air Force Office of Scientific Research.(2) After the completion of this work, the author learned of the recent but prior paper [13] of D. Ray, which contains a proof of Theorem 2.2 carried out briefly (and elegantly) by analytical methods.It thus emerges that the random walks are not required.Even according to, [13] however, they provide useful motivation for the theorem.F. B. KNIGHTUsing the easily verified facts that £(A(x)) = x and E(X(x1)X(x2)) = x,(x2 + x,) it can be shown directly that the first two moments of the distributions of (4.9) and (4.10) are the same.To verify directly the identity of the two distributions, however, remains an open problem.