Conference Agenda

Markovian dynamics of an open quantum system is determined by a family of super operators $(\phi_{s,t})_{s\leq t}$ that satisfy the \emph{composition law} \[ \phi_{r,t} = \phi_{s,t}\circ \phi_{r,s} \quad \forall \quad s \leq r \leq t\, . \] These super-operators are called \emph{dynamical maps} or \emph{dynamic propagators} are generated by a time-dependent Lindbladian $\mathcal L$ (also known as a GKLS generator): \[ \phi_{s,t} = \mathcal T\left\{\text{exp} \left( \int_s^t \mathcal L(\tau) \ d\tau\right)\right\}\, . \] Here $\mathcal T$ denotes the time-ordering. We now consider the case where $\mathcal L$ is time-dependent but stationary in time. As such, we are interested in asymptotics of time-inhomogeneous Markovian dynamics obtained from a random Lindbladian process $(\mathcal L_t)_{t\in\mathbb R}$ that is strictly stationary.

Under certain irreducibility criteria we obtain stationary processes of random full-rank matrices $(Z_t)_{t\in\mathbb R}$, $(Z'_t)_{t\in\mathbb R}$ and a family of rank-one super operators $\left( ^s{\Xi}{_t}\right)_{s,t}$ that approximate the dynamics of the open quantum system exponentially fast almost surely and super-polynomial fast in mean. In the discrete time-parameter such propagators describes quantum dynamics of a random repeated interaction (random collision) model and in discrete time-parameter we obtain a Law of Large Numbers (LLN) and a Central Limit Theorem (CLT) involving the top Lyapunov exponent of the product \[ \widetilde\Phi^{(n)} = \phi_{n-1, n}\circ \ldots\circ \phi_{0,1} \, .\]

The Quickselect algorithm (also called FIND) is a fundamental algorithm to select ranks or quantiles within a set of data. It was shown by Grübel and Rösler that the number of key comparisons required by Quickselect as a process of the quantiles $\alpha\in[0,1]$ in a natural probabilistic model converges after normalization in distribution within the càdlàg space $D[0,1]$ endowed with the Skorokhod metric.

We show that the process of the residuals in the latter convergence after normalization converges in distribution to a mixture of Gaussian processes in $D[0,1]$ and identify the limit's conditional covariance functions. A similar result holds for the related algorithm QuickVal. Our method extends to other cost measures such as the number of swaps (key exchanges) required by QuickSelect or cost measures which are based on key comparisons but take into account that the cost of a comparison between two keys may depend on their values, an example being the number of bit comparisons needed to compare keys given by their bit expansions.

It is well-established that Patterson-Sullivan measures on the boundary of a hyperbolic space, along with the associated Bowen-Margulis-Sullivan measure, provide valuable insights into the action of a group of isometries on the space's boundary through analysis of the geodesic flow. Given that paths of a random walk on a hyperbolic groups lie close to the group's quasi-geodesics, it is natural to ask whether similar behaviour can also be seen in the flow along bi-infinite random walk paths.

In this talk, I will show how studying bi-infinite random walks on a discrete hyperbolic group $G$ leads to an analogue of the Patterson-Sullivan measure on $\partial^2G$. This measure can be constructed in multiple measure-equivalent ways, each giving distinct perspectives on its intrinsic structure. Moreover, as in the classical case, the action $G \curvearrowright \partial^2 G$ is ergodic with respect to this measure. Central to the construction of these measure is the almost sure convergence of the random walk to the boundary $\partial G$ and the study of the distribution of the hitting points. This talk is based on joint work with Mahan Mj and Chiranjib Mukherjee.

There exists a continuing interest in establishing consistency and asymptotic normality of estimators for homogenized SDE models, which emerge from the respective multiscale SDE model when the scale parameter $\epsilon$ goes to zero. However, most of these asymptotic results are proved when limits are taken sequentially, that is, the time horizon $T$ goes to infinity and then the scale parameter $\epsilon$ goes to zero, or vice versa. In this talk, we want to present a first attempt at answering the questions of taking the simultaneous limit, i.e. $T=T_\epsilon$ depends explicitly on $\epsilon$ such that $T_\epsilon$ goes to infinity when $\epsilon$ goes to zero.

We present two limit theorems, a mean ergodic theorem and a central limit theorem, for a specific class of one-dimensional ergodic diffusion processes that depend on the small scale parameter $\epsilon$. In these results, we not only allow for the time horizon $T_\epsilon$ to depend explicitly on epsilon but also for the test function $\phi_\epsilon$. The novelty of the results arises from the circumstance that many quantities are unbounded for $\epsilon \rightarrow 0$, so that formerly established theory is not directly applicable here and a careful investigation of all relevant $\epsilon$-dependent terms is required.

As a mathematical application, we then use these limit theorems to prove robustness and asymptotic normality of a minimum distance estimator for parameters in homogenized Langevin equations subject to multiscale observations.