On the Generating Parametrix of the Stochastic Processes

Abstract

Replacing 0 by x, we now consider the function V/' as a periodic function on the real line; define a function k by: k(x) = 1 if xl < wr/2, k(x) = 0 if xl > 37r/4, k(x) linear in the remaining intervals; let h = keg4&; and define g by: g2(x) = h(x), g(x) 2 0. Using the well-known relation between absolutely convergent Fourier series and absolutely convergent Fourier integrals, together with Wiener's localization theorem,8 we see that h is the Fourier transform of a function H in L1, but g is not the Fourier transform of any such function. Also, H is an entire function (even of exponential type), since h vanishes outside a bounded interval; and H is positivedefinite,9 since h is nonnegative, so that we may state the following theorem: THEOREM 4. There exists an entire function H, of exponential type, which belongs to L1 on the real line and is positive-definite, such that the convolution equation H(x) = G(xt)G(t) dt (x real) has no positive definite solution G in L1. We conclude with the remark that g is not even the Fourier-Stieltjes transform of any bounded (complex-valued) Borel measure on the line. For, if it were, let kt'(x) = k(x/2); then ki has its Fourier transform in L1. Since the convolution of a member of Li and a bounded measure is in L1, it follows that g9 ki has its transform in L1. But g* k = g, and we arrive at a contradiction.

Some ergodic theorems for one-parameter semigroups of operators

“…We have now attained the first objective mentioned in the Introduction: to establish the equivalence of the weak and strong topology, and of several averaging processes, as far as ergodic properties of & are concerned. It will also be noticed that theorem 5 shows ^ and the cyclic semigroup {(ATa)"} to have equivalent ergodic properties when & is ultimately bounded; this fact has recently been exploited, in a special case, by Yosida (1955).…”

Markov operators and their associated semi-groups

“…We write Γ t for Γ t [F, G]. As is well known (see [11], p. 242) it is sufficient to prove that T t φ -> φ weakly as t -> 0 for all φ 6 L 2 (G). Suppose φ is continuous with compact support then using Theorem 2.3 of AF we see that (2.16…”