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We study the noise, in the sense of Tsirelson, generated by Harris flows. A criterion is given for the noise to be non-white, and in this case we study the associated spectral sets., Comment: 15 pages

Ito’s theory of excursion point processes is reviewed and the following topics are discussed: Application of the theory to one-dimensional diffusion processes on half-intervals satisfying Feller’s boundary conditions, and its multi-dimensional extension, i.e., the application of the theory to multi-dimensional diffusion processes satisfying Wentzell’s boundary conditions.

By a noise in continuous time t∈R, we mean a family {Fs,t, s⩽t} of sub σ-fields of events on a separable probability space with properties given in Definition 1.2, a property of independence, in particular. A noise is black if there does not exist any nontrivial Lévy process ξt such that ξt−ξs is Fs,t-measurable. We give an example of black noise for which the proof that it is black may be simpler than any other known examples of black noises.

An analogy of Levi sums is considered for a class of stochastic partial differential equations to construct their stochastic fundamental solutions. These notions are shown to coincide with Donsker's delta functions, typical generalized Wiener functionals, which have been studied in the frame of the Malliavin calculus. © 1994 John Wiley & Sons, Inc.

Fractional order Sobolev spaces are introduced on an abstract Wiener space and Donsker's delta functions are defined as generalized Wiener functionals belonging to Sobolev spaces with negative differentiability indices. By using these notions, the regularity in the sense of Holder continuity of a class of conditional expectations is obtained.

We discuss the spectral theory of second-order differential operators that describe the vibration of strings, diffusion processes, and others. M. G. Krein established a one-to-one correspondence between the spectral measure and the string in the case of regular left boundaries, and this correspondence was extended by S. Kotani to a certain class of strings with singular left boundary. In this article we study the relationship between the asymptotic behavior of the spectral measure and that of the corresponding string. Although the results are basically for Kotani’s strings, some are also applicable to Krein’s.

In the study of spectral functions of Strum-Liouville operators, S. Kotani found a one-to-one correspondence between the operators and Herglotz functions determining the spectral measures. In the present paper we give another correspondence to a class of Levy processes obtained as a compensated integrals of Brownian local times. As an application the continuity theorem of Kotani's correspondence is extended.

It is well known that a class of subordinators can be represented using the local time of Brownian motions. An extension of such a representation is given for a class of Lévy processes which are not necessarily of bounded variation. This class can be characterized by the complete monotonicity of the Lévy measures. The asymptotic behavior of such processes is also discussed and the results are applied to the generalized arc-sine law, an occupation time problem on the positive side for one-dimensional diffusion processes.

We show the existence and continuity of the density and obtain its formula for the law of the fraction of time spent on the positive side of a very general class of one-dimensional (generalized or gap) diffusion processes. We also study the asymptotic properties of the density at the extreme values $x = 0$ and $x = 1$, which extend, in particular, some recent results by Kasahara and Yano [8].

Publisher Summary It o's stochastic calculus is one of the most important methods in modern probability theory to analyze various probability models. In up-to-date applications, it provides with a basic tool in the theory of mathematical finance such as theory of derivative pricing. A general framework of this calculus is usually given as a sample functions analysis for a class of stochastic processes called “semimartingales” or “generalized Ito processes.” A characteristic feature of a semimartingale is that it represents a random motion in time perturbed by a noise; the noise is assumed to be a martingale that represents a fluctuation around the main motion and the main motion is assumed to be adapted and smooth in the sense that its sample function is of bounded variation in time. Here, the martingale property and adaptedness are referred to a given filtration on the probability space.

Let (W, P) be the d-dimensional Wiener space: W be the space of continuous paths W = {w ∈ C([0, ∞) → R d )|w(0) = 0} and P be the standard d-dimensional Wiener measure on W. Then w = (w k(t)) k=1 d in W is a canonical realization of a d-dimensional Wiener process. Levy’s stochatic area is defined on the two-dimensional Wiener space by Ito’s stochastic integral as follows: $$S\left( t.\omega \right)=\frac{1}{2}\int{_{0}^{t}}{{w}^{1}}\left( s \right)d{{w}^{2}}\left( s \right)-{{w}^{2}}\left( s \right)d{{w}^{1}}\left( s \right)$$ .

The regularity in the sense of Holder continuity of a class of conditional expectation is obtained on a Wiener space by using the notion of Donsker’s delta functions, typical generalized Wiener functionals belonging to fractional order Sobolev spaces with negative differentiability indices.

Publisher Summary This chapter presents Donsker's δ -functions in the Malliavin calculus. Dirac δ -function is a well-defined mathematical object in the Schwartz theory of distributions. The chapter discusses a satisfactory disintegration theory on the foliation of submanifolds imbedded in the Wiener space and describes the exact Sobolev spaces, to which Donsker's δ -functions belong. It reviews notions and results in the Malliavin calculus relevant to Donsker's δ -functions.

O. Introduction By a Bessel diffusion process with index ~ (~ > 0), we mean a conservative onedimensional diffusion process on [0, o0) determined by the local generator

A statistical theory of the neutron transport process has been discussed by Harris [[I] and Mullikin Q7J. They considered the neutron population by generation rather than in real time and formulate them as a discretetime branching process. In this paper, we will formulate the neutron transport process as a continuous time branching process and study asymptotic properties of the neutron population. As a fundamental equation, we have a non-linear equation like (1.3) and our results may be regarded, as a problem in analysis, to be concerned with the asymptotic properties of solutions of such a non-linear equation. In such study, a linearized Boltzman equation, which is the dual of the evolution equation for expectation semigroup, plays important role. In this paper, we are mainly concerned with a monoenergetic and isotropic transport process on bounded domain in R, though a more general case can be treated by the same method. Also, our method can be applied to a general class of branching processes including branching diffusion processes but we will not go further into such generalizations.

application/pdf, The long time asymptotics of the time spent on the positive side are discussed for one-dimensional diffusion processes in random environments. The limiting distributions under the log–log scale are obtained for the diffusion processes in the stable medium as well as for the Brox model. Similar problems are discussed for random walks in random environments and it is proved that the limiting laws are the same as in the case of diffusions.