Various tools can be used to calculate or estimate the dimension of measures. Using a probabilistic interpretation, we propose very simple proofs for the main inequalities related to this notion. We also discuss the case of quasi-Bernoulli measures and point out the deep link existing between the calculation of the dimension of auxiliary measures and the multifractal analysis.
This paper is devoted to computational problems related to Markov chains (MC) on a finite state space. We present formulas and bounds for characteristics of MCs using directed forest expansions given by the Matrix Tree Theorem. These results are applied to analysis of direct methods for solving systems of linear equations, aggregation algorithms for nearly completely decomposable MCs and the Markov chain Monte Carlo procedures.
The sequence of random probability measures νn that gives a path of length n, times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment. Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.
We define thin equivalence relations ∼ on shift spaces and derive Dirichlet forms on the quotient space in terms of the nearest neighbour averaging operator. We identify the associated Laplace operator. The conditions are applied to some non-self-similar extensions of the Sierpiński gasket.
We give a complete analytical characterization of the functions transforming reflected Brownian motions to local Dirichlet processes.
We study a linear second order parabolic equation in an open subset of a separable Hilbert space, with the Dirichlet boundary condition. We prove that a probabilistic formula, analogous to one obtained in the finite-dimensional case, gives a solution to this equation. We also give a uniqueness result.
The convergence of discrete approximations of generalized reflected backward stochastic differential equations with random terminal time in a general convex domain is studied. Applications to investigation obstacle elliptic problem with Neumann boundary condition for partial differential equations are given.