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We prove the existence of a solution to an equation governing the number density within a compact domain of a discrete particle system for a prescribed class of particle interactions taking into account the effects of the diffusion and…
Consider multiple sums $S_n$ on the $d$-dimensional integer grid,which are generated by i.i.d.\ random variables with a positive expectation. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional…
We study the convergence of random function iterations for finding an invariant measure of the corresponding Markov operator. We call the problem of finding such an invariant measure the stochastic fixed point problem. This generalizes…
The objective of this paper is to examine the restriction of a right process on a Radon topological space, excluding a negligible set, and investigate whether the restricted object can induce a Markov process with desirable properties. We…
Topological phases of matter are often understood and predicted with the help of crystal symmetries, although they don't rely on them to exist. In this chapter we review how topological phases have been recently shown to emerge in amorphous…
In the paper we pursue the analysis from the section 5 of the Talagrand's paper "Sample boundedness of stochastic processes under increment conditions." Ann. Probab. 18, No. 1, 1-49. In particular we give the proof of some Sobolev…
We study black-box testing for stochastic systems and arbitrary $\omega$-regular specifications, explicitly including liveness properties. We are given a finite-state probabilistic system that we can only execute from the initial state. We…
We study random graphs with latent geometric structure, where the probability of each edge depends on the underlying random positions corresponding to the two endpoints. We focus on the setting where this conditional probability is a…
We consider a branching random walk with immigration in a random environment, where the environment is a stationary and ergodic sequence indexed by time. We focus on the asymptotic properties of the sequence of measures $(Z_n)$ that count…
We study Pippenger's model of Boolean networks with unreliable gates. In this model, the conditional probability that a particular gate fails, given the failure status of any subset of gates preceding it in the network, is bounded from…
In this article, we introduce \textit{Mallows processes}, defined to be continuous-time c\`adl\`ag processes with Mallows distributed marginals. We show that such processes exist and that they can be restricted to have certain natural…
We study a Weiner process that is conditioned to pass through a finite set of points and consider the dynamics generated by iterating a sample path from this process. Using topological techniques we are able to characterize the global…
The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit…
Iterative load balancing algorithms for indivisible tokens have been studied intensively in the past. Complementing previous worst-case analyses, we study an average-case scenario where the load inputs are drawn from a fixed probability…
We study a random dynamical system such that one transformation is randomly selected from a family of transformations and then applied on each iteration. For such random dynamical systems, we consider estimates of absolutely continuous…
For a family of random intermittent dynamical systems with a superattracting fixed point we prove that a phase transition occurs between the existence of an absolutely continuous invariant probability measure and infinite measure depending…
This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a…
Let $\sigma$ be a non-atomic, infinite Radon measure on $\mathbb R^d$, for example, $d\sigma(x)=z\,dx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots,x_N$ in a bounded set $\Lambda\subset\mathbb R^d$, where…
Computer science theory provides many different measures of complexity of a system including Kolmogorov complexity, logical depth, computational depth, and Levin complexity. However, these measures are all defined only for deterministic…
We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the…