Real Half-line Research Articles

An extension of watanabe's theorem of characterization of poisson processes over the positive real half line

We give an elementary proof of the martingale characterization theorem for Poisson processes over the positive real half line. This theorem is due to Watanabe [8] in the case where the mean measure associated to the Poisson process is the Lebesgue measure.

Nonoscillation of arbitrary order retarded differential equations of non-homogeneous type

The object of the present paper is to study the delay differential equation of arbitrary order namely y(n)(t) + a(t)yτ(t) = f(t), n≥2 (an integer) and prove a nonoscillation theorem under the general situation in which a(t) and f(t) are allowed to oscillate arbitrarily often on some positive half real line. This is accomplished by way of two differential inequalities of nth order.

Spectral Representation for Branching Processes with Immigration on the Real Half Line

§ Oo Introduction In the previous paper JJT], we have obtained the spectral representation for the semigroups of continuous state branching processes (CBprocesses). In this paper, we shall obtain the analogous results for continuous state branching processes with immigration (CBI-processes). CBI-processes were introduced by Kawazu and Watanabe Q3] as a continuous version of Galton-Watson processes with immigration and our results are similar to those of Karlin and McGregor []2] for GaltonWatson processes with immigration. When CBI-processes are diffusions, our representations are some concrete examples of the general represen

Asymptotic Behavior of Solutions of Parabolic Equations with Unbounded Coefficients

Let Rn be the n-dimensional Euclidean space, each point of which is denoted by its coordinate x = (x1,...,xn). The variable t is in the real half line [0, ∞).

Spectral Representation for Branching Processes on the Real Half Line

The spectral theory for the semigroups of Galton-Watson processes (G.W.P.'s) was developed by Karlin and McGregor [4] [5]. On the other hand, a continuous state branching process (C.B.P.), which may be regarded as a continuous version of a G.W.P., was introduced by Jirina [3] and recently discussed by Lamperti [7], S. Watanabe [8], etc. The object of this paper is to obtain the spectral representation theorems for C.B.P.'s similar to those of Karlin and McGregor for G.W.P.'s. These may be of some interest, since there are many C.B.P.'s with discontinuous sample functions and their semigroups are nonsymmetrizable. In § 1, we obtain a representation (see (1.10)) of a so called ^-semigroup by means of the stationary measure, under the assumption that the extinction probability is positive. In § 2, we shall prepare several lemmas. The most important one is Lemma 2.4, which asserts that, under Condition A, the representation of a M>semigroup in § 1 turns out to be a spectral representation of the transition function. In §3, the spectral representation theorem for sub- and supercritical C.B.P.'s is given. In this representation, only the discrete spectrum appears which consists of powers of the largest eigenvalue. In § 4, the critical case is considered. In this case, the spectrum is continuous. The asymptotic behavior of the semigroup is also obtained. In § 5, we give a few examples which contain all diffusion C.B.P.'s. The author wishes to express his thanks to Professor K. Ito

Asymptotic behavior of the solution of an abstract evolution equation and some applications

in a Banach space X. The functions u(t) andf(t) are defined on the real half line [0, co) and have values in X. The function A(t) has the same domain of definition and has values in a set of unbounded linear operators acting in X. Tanabe [I] investigated the behavior as t + co of the solution of Eq. (1.1). He stated that under certain natural conditions on the behavior of A(t) and f(t), one can prove that if both converge in some sense as t -+ 03, then the solution u(t)of Eq. (1.1) aJs 0 converges to some element-of X as t -+ 00. In the present note we assume a certain asymptotic behavior of A(t) and f(t) as t -+ co and obtain a corresponding asymptotic behavior of the solution u(t). This result is used in Section 4 to obtain an asymptotic expansion of the solution of a parabolic equation in a cylindrical domain. Results similar to the results of Section 4 were obtained by Friedman [4].