Conditional measure on the Brownian path and other random sets

Abstract

Let $B$ denote the range of the Brownian motion in $\mathbb{R}^{d}$ ($d\geq3$). For a deterministic Borel measure $\nu$ on $\mathbb{R}^{d}$ we wish to find a random measure $\mu$ such that the support of $\mu$ is contained in $B$ and it is a solution to the equation $E(\mu(A))=\nu(A)$ for every Borel set $A$. We discuss when it is possible to find a solution $\mu$ and in that case we construct the solution. We study several properties of $\mu$ such as the probability of $\mu\neq0$ and we establish a formula for the expectation of the double integral with respect to $\mu\times\mu$. We calculate $\mu$ in terms of the occupation measure when $\nu$ is the Lebesgue measure, i.e. we provide an explicit deterministic density function of $\mu$ with respect to the occupation measure. As a conclusion we calculate an explicit formula for the expectation of the double integral with respect to the occupation measure. We generalise the theory for more general random sets in separable, metric, Radon spaces. As an additional example, we also apply our results to percolation limit sets on boundaries of trees.