Related papers: Absolute continuity and singularity of two probabi…

Let $M$ be a von Neumann algebra equipped with a normal semi-finite faithful trace (nsf trace in short) and let $T\colon M\to M$ be a contraction. We say that $T$ is absolutely dilatable if there exist another von Neumann algebra $M'$…

The measurability by means of continuous measurements, of an observable $\A(t_0)$, at an instant, and of a time averaged observable, $\bar \A=1/T\int \A(t')dt'$, is examined for linear and in particular for non-linear quantum mechanical…

Let $\mu$ be a measure on the Euclidean space $\R^d$ of unbounded total variation that is positive or translation bounded and has a pure point Fourier transform in the sense of distributions $\hat\mu$. We prove that the measure $\nu$ with…

We present an in-depth study of the problem of multiple-shot discrimination of von Neumann measurements in finite-dimensional Hilbert spaces. Specifically, we consider two scenarios: minimum error and unambiguous discrimination. In the case…

We consider iterated function systems on the interval with random perturbation. Let $Y_\epsilon$ be uniformly distributed in $[1- \epsilon, 1 + \epsilon]$ and let $f_i \in C^{1+\alpha}$ be contractions with fixpoints $a_i$. We consider the…

Let $A_r=\{r<|z|<1\}$ be an annulus. We consider the class of operators $\mathcal{F}_r:=\{T\in\mathcal{B}(H): r^2T^{-1}(T^{-1})^*+TT^*\le r^2+1,\hspace{0.08 cm}\sigma(T)\subset A_r\}$ and show that for every bounded holomorphic function…

We show that for uniform domains $\Omega\subseteq \mathbb{R}^{d+1}$ whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to $\alpha$-dimensional Hausdorff…

A metric measure space $(X,d,\mu)$ is said to satisfy the strong annular decay condition if there is a constant $C>0$ such that $$ \mu\big(B(x,R)\setminus B(x,r)\big)\leq C\, \frac{R-r}{R}\, \mu (B(x,R)) $$ for each $x\in X$ and all $0<r…

In this paper, we consider probability measures $\mu$ and $\nu$ on a $d$--dimensional sphere in $\Rd, d \geq 1,$ and cost functions of the form $c(\x,\y)=l(\frac{|\x-\y|^2}{2})$ that generalize those arising in geometric optics where…

Let S be a nonexceptional oriented surface of finite type. We construct an uncountable family of probability measures on the space of area on holomorphic quadratic differentials over the moduli space for S containing the usual Lebesgue…

A classical result of Arne Beurling states that the Fourier transform of a nonzero complex Borel measure $\mu$ on the real line cannot vanish on a set of positive Lebesgue measure if $\mu$ has certain decay. We prove a several variable…

Let $\mu$ be a compactly supported probability measure on the positive half-line and let $\mu^{\boxtimes t}$ be the free multiplicative convolution semigroup. We show that the support of $\mu^{\boxtimes t}$ varies continuously as $t$…

Let $\mu$ be a probability measure of compact support on the set $\mathbb{P}_n$ of all positive definite matrices, let $t\in(0,1]$, and let $P_t(\mu)$ be the unique positive solution of $X=\int_{\mathbb{P}_n}X\sharp_t Z d\mu(Z)$. In this…

We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In…

For a singular measure $\mu$, Ahlfors regular of order $\alpha>0,$ with compact support in $\mathbb{R}^{\mathbf{N}}$ and a pseudodifferential operator $\mathbf{A}$ of order $-l=-\mathbf{N}/2$ we consider the compact operator…

Let $K$ be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map $T: K\rightarrow K$ induced by the shift. Let $\mu$ be the natural…

Let $\mu$ be a self-affine measure on $\mathbb{R}^{d}$ associated to a self-affine IFS $\{\varphi_{\lambda}(x) = A_{\lambda}x + v_{\lambda}\}_{\lambda\in\Lambda}$ and a probability vector $p=(p_{\lambda})_{\lambda}>0$. Assume the strong…

We derive explicit integrability conditions for stochastic integrals taken over time and space driven by a random measure. Our main tool is a canonical decomposition of a random measure which extends the results from the purely temporal…

The histories-based framework of Quantum Measure Theory assigns a generalized probability or measure $\mu(E)$ to every (suitably regular) set $E$ of histories. Even though $\mu(E)$ cannot in general be interpreted as the expectation value…

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. For $\alpha>0$, the generalized Hankel matrix $\mathcal{H}_{\mu, \alpha}=(\mu_{n, k, \alpha})_{n, k \geq 0}$ with entries $\mu_{n, k, \alpha}=\int_{[0,1)}…