Brownian local time

Abstract

In dense media positrons and positronium atoms are bound in self-trapped states. These states are density fluctuations stabilised by the light quantum particles. The positronium atoms are trapped in a bubble, while positrons are trapped in a cluster. Transitions to these states at changes of the density or temperature of the medium produce an essential effect on the annihilation rate. It can be considered as a local phase transition in the vicinity of the positron (or positronium). The present article is a review of the present state of the physics of positron and positronium self-trapped states in dense gases and liquids. Basic properties of the positronium bubbles and positron clusters are considered, as well as the conditions under which they exist and their effect on the annihilation rate. Experiments in which the self-trapping was detected are discussed. The positron and positronium self-trapping is considered to be closely related to other similar electronic phenomena, known for dense gases, liquids and other dense media. The main results from the study of positron and positronium interactions with isolated atoms are also presented. The density effects in moderately dense gases anticipating self-trapping are considered.

On distributions of functionals of the brownian motion stopped at the moment inverse to local time

“…(5.6) of Theorem 5.2, Ch. I, from [5] and since the integral is a continuous function of the limit of integration, it follows that for any l, l = 1, 2,..., k, the function u(zt, y, ~) is continuous with respect to yl E R ~ . Taking into account the structure of the function f~(x, y), we have…”

“…The proofs of the results stated below about the calculation of distributions of functionals of the Brownian motion stopped'at the moment e are essentially based on Theorem 5.2, Ch. I, from [5]. "…”

Central limit theorem for the third moment in space of the Brownian local time increments

“…In the next section we recall some preliminaries on Malliavin calculus. In Section 3 we establish a stochastic integral representation for the derivative of the self-intersection local time, which has its own interest, and for the random variable F h t defined in (2). Section 4 is devoted to the proof of Theorem 1, and the Appendix contains two technical lemmas.…”

Moments estimates for local times of a class of Gaussian processes

“…It is known that joint probability density of random variables l w and w(1) has the following representation [3] …”