Related papers: Generators of some non-commutative stochastic proc…

We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cutoff at time $N\ln(N)$. Then, we study the induced classical process on the real line and compute its atoms and density. This…

In this article we consider the Levy processes and the corresponding semigroup. We represent the generator of this semigroup in a convolution form. Using the obtained convolution form and the theory of integral equations we investigate the…

We investigate the characterization of generators $\mathcal{L}$ of L\'evy processes satisfying the Liouville theorem: Bounded functions $u$ solving $\mathcal{L}[u]=0$ are constant. These operators are degenerate elliptic of the form…

The problem of appropriately matching items subject to compatibility constraints arises in a number of important applications. While most of the literature on matching theory focuses on a static setting with a fixed number of items, several…

We study the process of $2K-B$, where $B$ is a standard one-dimensional Brownian motion and $K$ is its concave majorant. In light of Pitman's $2M-B$ theorem, it was recently conjectured by Ouaki and Pitman \cite{OP} that $2K-B$ has the law…

Using heat kernel estimates, we prove the pathwise uniqueness for strong solutions of irregular stochastic differential equation driven by a family of Markov process, whose generator is a non-local and non-symmetric L\'evy type operator.…

A functional representation of free L\'evy processes is established via an ensemble of unitarily invariant Hermitian matrix-valued L\'evy processes. This is accomplished by proving functional asymptotics of their empirical spectral…

We introduce a class of Gaussian processes with stationary increments which exhibit long-range dependence. The class includes fractional Brownian motion with Hurst parameter H>1/2 as a typical example. We establish infinite and finite past…

We propose a novel stochastic method to generate paths conditioned to start in an initial state and end in a given final state during a certain time $t_{f}$. These paths are weighted with a probability given by the overdamped Langevin…

We introduce the concept evolutionary semigroups on path spaces, generalizing the notion of transition semigroups to possibly non-Markovian stochastic processes. We study the basic properties of evolutionary semigroups and, in particular,…

For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(\tau(f)+t \mod1)+f(1)1_{\{t+\tau(f) \geq 1\}}-f(\tau(f))$, where $\tau(f)$ corresponds to the first time at which the minimum of $f$ is attained.…

Let $W_t$ be a standard Brownian motion. It is well-known that the Langevin equation $d U_t = -\theta U_td t + d W_t$ defines a stationary process called Ornstein-Uhlenbeck process. Furthermore, Langevin equation can be used to construct…

This paper provides a multivariate extension of Bertoin's pathwise construction of a L\'evy process conditioned to stay positive/negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original…

We study the relation between L\'evy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one relation between nonlinear L\'evy processes and nonlinear Markovian convolution…

We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability $1/2$, respectively. The theorem entails that the law of every one-dimensional regular…

We provide a probabilistic description of the stationary measures for the open KPZ on the spatial interval $[0,1]$ in terms of a Markov process $Y$, which is a Doob's $h$ transform of the Brownian motion killed at an exponential rate. Our…

We present a detailed study of a simple quantum stochastic process, the quantum phase space Brownian motion, which we obtain as the Markovian limit of a simple model of open quantum system. We show that this physical description of the…

Nowadays many tools, e.g. fluctuation relations, are available to characterize the statistical properties of non-equilibrium systems. However, most of these tools rely on the assumption that the driving noise is normally distributed. Here…

This paper deals with the properties of the stochastic generators of the effective (driven) processes associated with atypical values of transition-dependent time-integrated currents with Gallavotti-Cohen symmetry in Markov jump processes.…

We consider a class of piecewise-deterministic Markov processes where the state evolves according to a linear dynamical system. This continuous time evolution is interspersed by discrete events that occur at random times and change (reset)…