Fluctuation Moments for Regular Functions of Wigner Matrices

We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of [Male, Mingo, Pech\'e, Speicher 2022], showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem obtained recently in the companion paper [Reker 2023].Comment: 52 pages (including appendix), 20 figure

On the operator norm of a Hermitian random matrix with correlated entries

We consider a correlated $N\times N$ Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one

Multi-Point Functional Central Limit Theorem for Wigner Matrices

Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic matrices $A_1,\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$, thus extending the results of [Cipolloni, Erd\H{o}s, Schr\"oder 2023] to products of arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i} tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.Comment: 48 pages (including appendix

Short-time behavior of solutions to L\'evy-driven SDEs

We consider solutions of L\'evy-driven stochastic differential equations of the form $\mathrm{d} X_t=\sigma(X_{t-})\mathrm{d} L_t$, $X_0=x$ where the function $\sigma$ is twice continuously differentiable and maximal of linear growth and the driving L\'evy process $L=(L_t)_{t\geq0}$ is either vector or matrix-valued. While the almost sure short-time behavior of L\'evy processes is well-known and can be characterized in terms of the characteristic triplet, there is no complete characterization of the behavior of the process $X$. Using methods from stochastic calculus, we derive limiting results for stochastic integrals of the from $\smash{t^{-p}\int_{0+}^t\sigma(X_{t-})\mathrm{d} L_t}$ to show that the behavior of the quantity $t^{-p}(X_t-X_0)$ for $t\downarrow0$ almost surely mirrors the behavior of $t^{-p}L_t$. Generalizing $t^p$ to a suitable function $f:[0,\infty)\rightarrow\mathbb{R}$ then yields a tool to derive explicit LIL-type results for the solution from the behavior of the driving L\'evy process

On the law of killed exponential functionals

For two independent L\'{e}vy processes $\xi$ and $\eta$ and an exponentially distributed random variable $\tau$ with parameter $q>0$ that is independent of $\xi$ and $\eta$, the killed exponential functional is given by $V_{q,\xi,\eta} := \int_0^\tau \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$. With the killed exponential functional arising as the stationary distribution of a Markov process, we calculate the infinitesimal generator of the process and use it to derive different distributional equations describing the law of $V_{q,\xi,\eta}$, as well as functional equations for its Lebesgue density in the absolutely continuous case. Various special cases and examples are considered, yielding more explicit information on the law of the killed exponential functional and illustrating the applications of the equations obtained. Interpreting the case $q=0$ as $\tau=\infty$ leads to the classical exponential functional $\int_0^\infty \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$, allowing to extend many previous results to include killing

Continuity properties and the support of killed exponential functionals

For two independent L\'evy processes $\xi$ and $\eta$ and an exponentially distributed random variable $\tau$ with parameter $q>0$, independent of $\xi$ and $\eta$, the killed exponential functional is given by $V_{q,\xi,\eta} := \int_0^\tau \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$. Interpreting the case $q=0$ as $\tau=\infty$, the random variable $V_{q,\xi,\eta}$ is a natural generalization of the exponential functional $\int_0^\infty \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$, the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein-Uhlenbeck process. In this paper, the support and continuity of the law of killed exponential functionals is characterized, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of $\smash{\int_0^te^{-\xi_{s-}}\mathrm{d}\eta_s}$ for fixed $t\geq0$, as well as for integrals of the form $\smash{\int_0^\infty f(s) \, \mathrm{d}\eta_s}$ for deterministic functions $f$. Furthermore, applying the same techniques to the case $q=0$, new results on the absolute continuity of the improper integral $\int_0^\infty \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$ are derived. We also show that the law of the killed exponential functional $V_{q,\xi,\eta}$ arises as a stationary distribution of a solution to a certain stochastic differential equation, thus establishing a close connection to generalised Ornstein-Uhlenbeck processes

Distributional properties of killed exponential functionals and short-time behavior of Lévy-driven stochastic differential equations

This thesis covers different properties of solutions to Lévy-driven stochastic differential equations. In Chapter 2, we consider various distributional properties of killed exponential functionals of Lévy processes. Similar to the case without killing, there are two ways to view the distribution of this random variable, leading to two main approaches to studying its properties. First, we consider the killed exponential functional as a stopped stochastic integral. Here, using tools from probability theory and infinitely divisible distributions, the support and continuity of the law of the killed exponential functional are characterized, and many sufficient conditions for absolute continuity are given. Additionally, sufficient conditions for absolute continuity of different conditioned integrals and in the case without killing are obtained. Further, it can be shown that the law of the killed exponential functional arises as the stationary distribution of a solution to a stochastic differential equation. Since the solution is closely related to the generalized Ornstein-Uhlenbeck process and, in particular, a Markov process, tools from functional analysis become applicable. In the second part of Chapter 2, the infinitesimal generator of the process is calculated and used to derive different distributional equations describing the law of the killed exponential functional, as well as functional equations for its Lebesgue density in the absolutely continuous case. We then consider different special cases and examples to obtain more explicit information on the law of the killed exponential functional and to illustrate some applications of the equations, which cover the case without killing as well. In Chapter 3, we consider solutions to more general Lévy-driven stochastic differential equations. While the almost sure short-time behavior of Lévy processes is well-known and can be characterized in terms of the generating triplet, there is no complete characterization of the behavior of the solution X. Using methods from stochastic calculus, we derive limiting results for stochastic integrals to show that the behavior of X(t)-X(0) almost surely mirrors the behavior of L in terms of power law functions. Generalizing to suitable f : [0, ∞) → R then yields a tool to derive explicit law of the iterated logarithm-type results for the solution also allows to give statements for convergence in probability or in distribution

Prethermalization for Deformed Wigner Matrices

We prove that a class of weakly perturbed Hamiltonians of the form $H_\lambda = H_0 + \lambda W$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by $H_\lambda$ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order $\lambda^{-2}$. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix $H_\lambda$.Comment: 31 pages (including appendix), 3 figure