118344939
Convergence of isotropic scattering transport process to Brownian motion
Introduction Let us consider transporting particle in the n-dimensional Euclidian space R. It is assumed that a particle originating at a point x^R moves in a straight line with constant speed c and continues to move until it suffers a collision. The probability that the particle has a collision between t and t + Δ is kΔ + o(Δ), where k is constant. When a particle has a collision, say at y in R, it moves afresh from y with an isotropic choice of direction independent of past history. It has been proved that, when c and k grows up indefinitely under the relation k\c = 2/n + 0(1), the distribution of a particle converges weakly to that of Brownian motion for the one-dimensional case by N. Ikeda and H. Nomoto [2] (cf. M.A. Pinsky [4]), and for the two-dimensional case by To. Watanabe [6] (cf. A.S. Monin [3]). The purpose of this paper is to show that the same result is also valid for the multi-dimensional case. In section 1, we shall define the n-dimensional transport process with speed c. In section 2, we investigate the resolvent and its Fourier transform. In section 3, using the result of section 2, we shall show that the distribution of transport process with speed c converges to that of the Brownian motion as c ̂ oo under the assumption: k\c = 2/n + 0(1) (Theorem 1). In section 4, we shall show that the transport process with speed c converges weakly to the Brownian motion, considering them as the measures on the space ^ of continuous functions.
53419531
Transformation of Markov processes by multiplicative functionals
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116204420
Malliavin's calculus in terms of generalized Wiener functionals
120055632
On discontinuous additive functionals and Lévy measures of a Markov process
123713264
On two dimensional Markov processes with branching property
Introduction. A continuous state branching Markov process (C.B.P.) was introduced by Jirina [8] and recently Lamperti [10] determined all such processes on the half line. (A quite similar result was obtained independently by the author.) This class of Markov processes contains as a special case the diffusion processes (which we shall call Feller's diffusions) studied by Feller [2]. The main objective of the present paper is to extend Lamperti's result to multi-dimensional case. For simplicity we shall consider the case of 2-dimensions though many arguments can be carried over to the case of higher dimensions(2). In Theorem 2 below we shall characterize all C.B.P.'s in the first quadrant of a plane and construct them. Our construction is in an analytic way, by a similar construction given in Ikeda, Nagasawa and Watanabe [5], through backward equations (or in the terminology of [5] through S-equations) for a simpler case and then in the general case by a limiting procedure. A special attention will be paid to the case of diffusions. We shall show that these diffusions can be obtained as a unique solution of a stochastic equation of Ito (Theorem 3). This fact may be of some interest since the solutions of a stochastic equation with coefficients H65lder continuous of exponent 1/2 (which is our case) are not known to be unique in general. Next we shall examine the behavior of sample functions near the boundaries (x1-axis or x2-axis). We shall explain, for instance, the case of x1-axis. There are two completely different types of behaviors. In the first case x1-axis acts as a pure exit boundary: when a sample function reaches the x1-axis then it remains on it moving as a one-dimensional Feller diffusion up to the time when it hits the origin and then it is stopped. In the second case, there is a point x0 on x1-axis such thaty'2 = (0, x0) acts as a reflecting boundary and L1= (xo, oo) acts as a pure entrance boundary (Theorem 4 and Corollaries).
118710372
On some relations between the harmonic measure and the Levy measure for a certain class of Markov processes.
46977756
A model of mind-body relation in terms of modular logic
122989844
On hypersurfaces of spaces belonging to certain transformation group