We consider Galton-Watson trees associated with a critical offspring distribution and conditioned to have exactly n vertices. These trees are embedded in the real line by affecting spatial positions to the vertices, in such a way that the increments of the spatial positions along edges of the tree are independent variables distributed according to a symmetric probability distribution on the rea...

Tags for today's lecture: Donsker's invariance priciple, Stochastic integration, Itô's formula In this lecture we show an application of Donsker's invariance principle and then proceed to the construction of Itô's stochastic integral. We recall the definitions and give a simple example of an application of the invariance principle. Consider a random walk S n = Σ n i=1 x i with E(x) = 0, E(x 2) ...

Noether’s invariance principle is one of the most helpful and fundamental results of physics. It describes the universal fact that “invariance with respect to some family of parameter transformations gives rise to the existence of certain conserved quantities.” Such relation is used to explain everything from the fusion of hydrogen to the motion of planets orbiting the sun [6]. For a modern acc...

Background: Dust is one of the most important forms of air pollution. Pursuant to Article 21 of the declaration on the principle of non-harmful use of land, states are responsible, in accordance with the charter of the United Nations and the principles of international law, for arranging for activities within their competence or oversight to cause harm to the environment of other countries or r...

Abstract. Invariance principle in L(0, 1) is studied using signed random measures. This approach to the problem uses an explicit isometry between L(0, 1) and a reproducing kernel Hilbert space giving a very convenient setting for the study of compactness and convergence of the sequence of Donsker functions. As an application, we prove a L(0, 1) version of the invariance principle in the case of...

We consider a ballistic random walk in an i.i.d. random environment that does not allow retreating in a certain fixed direction. We prove an invariance principle (functional central limit theorem) under almost every fixed environment. The assumptions are nonnestling, at least two spatial dimensions, and a 2 + ε moment for the step of the walk uniformly in the environment. The main point behind ...