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Extreme Mass Ratio Inspirals (EMRIs) are key observational targets for the Laser Interferometer Space Antenna (LISA) mission. Unresolvable EMRI signals contribute to forming a gravitational wave background (GWB). Characterizing the statistical features of the GWB from EMRIs is of great importance, as EMRIs will ubiquitously affect large segments of the inference scheme. In this work, we apply a frequentist test for GWB Gaussianity and stationarity, exploring three astrophysically-motivated EMRI populations. We construct the resulting signal by combining state-of-the-art EMRI waveforms and a detailed description of the LISA response with time-delay interferometric variables. Depending on the brightness of the GWB, our analysis demonstrates that the resultant EMRI foregrounds show varying degrees of departure from the usual statistical assumptions that the GWBs are both Gaussian and Stationary. If the GWB is non-stationary with non-Gaussian features, this will challenge the robustness of Gaussian-likelihood model, when applied to global inference results, e.g. foreground estimation, background detection, and individual-source parameters reconstruction., Comment: 12 pages, 5 figures, 3 tables

One of the primary sources of gravitational waves (GWs) anticipated to be detected by the Laser Interferometer Space Antenna (LISA) are Galactic double white dwarf binaries (DWDs). However, most of these binaries will be unresolved, and their GWs will overlap incoherently, creating a stochastic noise known as the Galactic foreground. Similarly, the population of unresolved systems in the Milky Way's (MW) satellites is expected to contribute to a stochastic gravitational wave background (SGWB). Due to their anisotropy and the annual motion of the LISA constellation, both the Galactic foreground and the satellite SGWB fall into the category of cyclostationary processes. Leveraging this property, we develop a purely frequency-based method to study LISA's capability to detect the MW foreground and SGWBs from the most promising MW satellites. We analyze both mock data generated by an astrophysically motivated SGWB spectrum, and realistic ones from a DWD population generated via binary population synthesis. We are able to recover or put constrains on the candidate foregrounds, reconstructing -- in the presence of noise uncertainties -- their sky distribution and spectrum. Our findings highlight the significance of the interplay between the astrophysical spectrum and LISA's sensitivity to detect the satellites' SGWB. Considering an astrophysically motivated prior on the satellite positions improves their detectability, which becomes otherwise challenging in the presence of the Galactic foreground. Furthermore, we explore the potential to observe a hypothetical satellite located behind the Galactic disk. Our results suggest that a Large Magellanic Cloud-like satellite could indeed be observable by LISA., Comment: 19 pages, 13 figures, 2 tables

In this work, we establish a link between the small-time approximate controllability of bilinear Schr\"odinger PDEs (posed on a boundaryless Riemannian manifold $M$) and the control in the group ${\rm Diff}_c^0(M)$ of the diffeomorphisms, isotopic to the identity and with compact support, of the underlying manifold $M$. More precisely, under a density assumption on the Lie algebra generated by the control potential and the Laplacian, we show that compositions $|J_P|^{1/2}(\psi_0\circ P)$ of the initial wavefunction $\psi_0\in L^2(M,\mathbb{C})$ with any diffeomorphism $P\in{\rm Diff}_c^0(M)$ can be approximately reached, in arbitrarily small times, by controlled solutions of the Schr\"odinger equation (here, $|J_P|$ denotes the determinant of the Jacobian of $P$). We illustrate this property on two examples, posed respectively on the torus $\mathbb{T}^d$ and on the euclidean space $\mathbb{R}^d$. As a physical application, we obtain in particular the small-time approximate control of the quantum particle's averaged positions. This yields also new small-time approximate controllability properties between families of eigenstates on $\mathbb{T}^d$. To prove the result, we first construct solutions of the Schr\"odinger equation that approximately evolve, arbitrarily fast, along any unitary transport flow on $L^2(M,\mathbb{C})$. In this way, we control the composition with any diffeomorphism that can be decomposed as a product of flows on $M$. We then combine this property with a result of Thurston on the simplicity of the group ${\rm Diff}_c^0(M)$ to conclude., Comment: 33 pages

We present a proof of strong consistency of a Ziv-Merhav-type estimator of the cross entropy rate for pairs of hidden-Markov processes. Our proof strategy has two novel aspects: the focus on decoupling properties of the laws and the use of tools from the thermodynamic formalism., Comment: Short note prepared for the IEEE International Symposium on Information Theory 2024 on a special case of arXiv:2312.02098

In this work we analyse the small-time reachability properties of a nonlinear parabolic equation, by means of a bilinear control, posed on a torus of arbitrary dimension $d$. Under a saturation hypothesis on the control operators, we show the small-time approximate controllability between states sharing the same sign. Moreover, in the one-dimensional case $d=1$, we combine this property with a local exact controllability result, and prove the small-time exact controllability of any positive states towards the ground state of the evolution operator.

A variety of physically relevant bilinear Schr\"odinger equations are known to be globally approximately controllable in large times. There are however examples which are globally approximately controllable in large times, but not in small times. This obstruction happens e.g. in the presence of (sub)quadratic potentials, because Gaussian states are preserved, at least for small times. In this work, we provide the first examples of small-time globally approximately controllable bilinear Schr\"odinger equations. In particular, we show that a control on the frequency of a quadratic potential permits to construct approximate solutions that evolve arbitrarily fast along specific transport flows, namely, space-dilations. Once we have access to space-dilations, we can exploit them to generate time-contractions. In this way, we build on previous results of large-time control, to obtain global approximate controllability in small times., Comment: 24 pages

We derive a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable distributions by proving a uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The main advantage of this method is that it can be easily applied to cases that are beyond the reach of the techniques currently used in the literature, such as random walks with long-ranged memory kernels or randomly stopped sums where the random time is not concentrated around its expectation. The method reveals the role of the characteristic function when Cram\'er's condition is violated and provides a unified approach within regular variation., Comment: 25 pages

We prove asymptotic results for a modification of the cross-entropy estimator originally introduced by Ziv and Merhav in the Markovian setting in 1993. Our results concern a more general class of decoupled measures on shift spaces over a finite alphabet and in particular imply strong asymptotic consistency of the modified estimator for all pairs of functions of stationary, irreducible, finite-state Markov chains satisfying a mild decay condition. Our approach is based on the study of a rescaled cumulant-generating function called the cross-entropic pressure, importing to information theory some techniques from the study of large deviations within the thermodynamic formalism.

Detecting stochastic gravitational wave backgrounds (SGWBs) with The Laser Interferometer Space Antenna (LISA) is among the mission science objectives. Disentangling SGWBs of astrophysical and cosmological origin is a challenging task, further complicated by the noise level uncertainties. In this study, we introduce a Bayesian methodology to infer upon SGWBs, taking inspiration from Gaussian stochastic processes. We investigate the suitability of the approach for signal of unknown spectral shape. We do by discretely exploring the model hyperparameters, a first step towards a more efficient transdimensional exploration. We apply the proposed method to a representative astrophysical scenario: the inference on the astrophysical foreground of Extreme Mass Ratio Inspirals, recently estimated in~\cite{Pozzoli2023}. We find the algorithm to be capable of recovering the injected signal even with large priors, while simultaneously providing estimate of the noise level., Comment: 16 pages, 10 figures, 0 tables

Achiral molecules can be made temporarily chiral by excitation with electric fields, in the sense that an average over molecular orientations displays a net chiral signal [Tikhonov et al., Sci. Adv. 8, eade0311 (2022)]. Here, we go beyond the assumption of molecular orientations to remain fixed during the excitation process. Treating both rotations and vibrations quantum mechanically, we identify conditions for the creation of chiral vibrational wavepackets -- with net chiral signals -- in ensembles of achiral molecules which are initially randomly oriented. Based on the analysis of symmetry and controllability, we derive excitation schemes for the creation of chiral wavepackets using a combination of (a) microwave and IR pulses and (b) a static field and a sequence of IR pulses. These protocols leverage quantum rotational dynamics for pump-probe spectroscopy of chiral vibrational dynamics, extending the latter to regions of the electromagnetic spectrum other than the UV., Comment: 16 pages, 8 figures

We generalize to a broader class of decoupled measures a result of Ziv and Merhav on universal estimation of the specific cross (or relative) entropy for a pair of multi-level Markov measures. The result covers pairs of suitably regular g-measures and pairs of equilibrium measures arising from the small space of interactions in mathematical statistical mechanics.

We address the small-time controllability problem for a nonlinear Schr\"odinger equation (NLS) on $\mathbb{R}^N$ in the presence of magnetic and electric external fields. We choose a particular framework where the equation becomes $i\partial_t \psi = [-\Delta+u_0(t)h_{\vec{0}}+\langle u(t), P\rangle +\kappa|\psi|^{2p}]\psi$. Here, the control operators are defined by the zeroth Hermite function $h_{\vec{0}}(x)$ and the momentum operator $P=i\nabla$. In detail, we study when it is possible to control the dynamics of (NLS) as fast as desired via sufficiently large control signals $u_0$ and $u$. We first show the existence of a family of quantum states for which this property is verified. Secondly, by considering some specific states belonging to this family, as a physical consequence we show the capability of controlling arbitrary changes of energy in bounded regions of the quantum system, in time zero. Our results are proved by exploiting the idea that the nonlinear term in (NLS) is only a perturbation of the linear problem when the time is as small as desired. The core of the proof, then, is the controllability of the bilinear equation which is tackled by using specific non-commutativity properties of infinite-dimensional propagators., Comment: 15 pages

In this work, we investigate detection rates and parameter estimation of strongly-lensed extreme mass-ratio inspirals (LEMRIs) in the context of the Laser Interferometer Space Antenna (LISA). Our results indicate that LEMRIs constitute a new gravitational-wave target signal for LISA, with detection rates ranging from zero to $\sim 40$ events over a four year-observation, and that it is possible to reveal and characterize LEMRIs at redshift $z \gtrsim 1$. We finally show that one LEMRI observation with identified host galaxy may yield percent constraints or better on the Hubble constant., Comment: accepted for publication on PRD

We consider a bilinear control problem for the wave equation on a torus of arbitrary dimension. We show that the system is globally approximately controllable in arbitrarily small times from a dense family of initial states. The control strategy is explicit, and based on a small-time limit of conjugated dynamics to move along non-directly accessible directions (a.k.a. Lie brackets of the generators)., Comment: 12 pages, 1 figure

We consider the evolution of a free quantum particle on the Grushin cylinder, under different type of quantizations. In particular we are interested to understand if the particle can cross the singular set, i.e., the set where the structure is not Riemannian. We consider intrinsic and extrinsic quantizations, where the latter are obtained by embedding the Grushin structure isometrically in ${\bf R}^3$ (with singularities). As a byproduct we provide formulas to embed the Grushin cylinder in ${\bf R^3}$ that could be useful for other purposes. Such formulas are not global, but permit to study the embedding arbitrarily close to the singular set. We extend these results to the case of $\alpha $-Grushin cylinders., Comment: 16 pages, 8 figures

Extreme mass ratio inspirals (EMRIs) are among the primary targets for the Laser Interferometer Space Antenna (LISA). The extreme mass ratios of these systems result in relatively weak GW signals, that can be individually resolved only for cosmologically nearby sources (up to $z\approx2$). The incoherent piling up of the signal emitted by unresolved EMRIs generate a confusion noise, that can be formally treated as a stochastic GW background (GWB). In this paper, we estimate the level of this background considering a collection of astrophysically motivated EMRI models, spanning the range of uncertainties affecting EMRI formation. To this end, we employed the innovative Augmented Analytic Kludge waveforms and used the full LISA response function. For each model, we compute the GWB SNR and the number of resolvable sources. Compared to simplified computations of the EMRI signals from the literature, we find that for a given model the GWB SNR is lower by a factor of $\approx 2$ whereas the number of resolvable sources drops by a factor 3-to-5. Nonetheless, the vast majority of the models result in potentially detectable GWB which can also significantly contribute to the overall LISA noise budget in the 1-10 mHz frequency range.

Over the years, the increasingly complex and interconnected vehicles raised the need for effective and efficient Intrusion Detection Systems against on-board networks. In light of the stringent domain requirements and the heterogeneity of information transmitted on Controller Area Network, multiple approaches have been proposed, which work at different abstraction levels and granularities. Among these, RNN-based solutions received the attention of the research community for their performances and promising results. In this paper, we improve CANnolo, an RNN-based state-of-the-art IDS for CAN, by proposing CANdito, an unsupervised IDS that exploits Long Short-Term Memory autoencoders to detect anomalies through a signal reconstruction process. We evaluate CANdito by measuring its effectiveness against a comprehensive set of synthetic attacks injected in a real-world CAN dataset. We demonstrate the improvement of CANdito with respect to CANnolo on a real-world dataset injected with a comprehensive set of attacks, both in terms of detection and temporal performances.

We consider the extreme value statistics of centrally-biased random walks with asymptotically-zero drift in the ergodic regime. We fully characterize the asymptotic distribution of the maximum for this class of Markov chains lacking translational invariance, with a particular emphasis on the relation between the time scaling of the expected value of the maximum and the stationary distribution of the process., Comment: 26 pages, 6 figures

We consider a one-dimensional Brownian motion with diffusion coefficient $D$ in the presence of $n$ partially absorbing traps with intensity $\beta$, separated by a distance $L$ and evenly spaced around the initial position of the particle. We study the transport properties of the process conditioned to survive up to time $t$. We find that the surviving particle first diffuses normally, before it encounters the traps, then undergoes a period of transient anomalous diffusion, after which it reaches a final diffusive regime. The asymptotic regime is governed by an effective diffusion coefficient $D_\text{eff}$, which is induced by the trapping environment and is typically different from the original one. We show that when the number of traps is \emph{finite}, the environment enhances diffusion and induces an effective diffusion coefficient that is systematically equal to $D_\text{eff}=2D$, independently of the number of the traps, the trapping intensity $\beta$ and the distance $L$. On the contrary, when the number of traps is \emph{infinite}, we find that the environment inhibits diffusion with an effective diffusion coefficient that depends on the traps intensity $\beta$ and the distance $L$ through a non-trivial scaling function $D_\text{eff}=D \mathcal{F}(\beta L/D)$, for which we obtain a closed-form. Moreover, we provide a rejection-free algorithm to generate surviving trajectories by deriving an effective Langevin equation with an effective repulsive potential induced by the traps. Finally, we extend our results to other trapping environments., Comment: 28 pages, 10 figures

In [14] Duca and Nersesyan proved a small-time controllability property of nonlinear Schr\"odinger equations on a d-dimensional torus $\mathbb{T}^d$. In this paper we study a similar property, in the linear setting, starting from a closed Riemannian manifold. We then focus on the 2-dimensional sphere $S^2$, which models the bilinear control of a rotating linear top: as a corollary, we obtain the approximate controllability in arbitrarily small times among particular eigenfunctions of the Laplacian of $S^2$., Comment: 10 pages, 1 figure

We consider a random walk $Y$ moving on a \emph{L\'evy random medium}, namely a one-dimensional renewal point process with inter-distances between points that are in the domain of attraction of a stable law. The focus is on the characterization of the law of the first-ladder height $Y_{\mathcal{T}}$ and length $L_{\mathcal{T}}(Y)$, where $\mathcal{T}$ is the first-passage time of $Y$ in $\mathbb{R}^+$. The study relies on the construction of a broader class of processes, denoted \emph{Random Walks in Random Scenery on Bonds} (RWRSB) that we briefly describe. The scenery is constructed by associating two random variables with each bond of $\mathbb{Z}$, corresponding to the two possible crossing directions of that bond. A random walk $S$ on $\mathbb{Z}$ with i.i.d increments collects the scenery values of the bond it traverses: we denote this composite process the RWRSB. Under suitable assumptions, we characterize the tail distribution of the sum of scenery values collected up to the first exit time $\mathcal{T}$. This setting will be applied to obtain results for the laws of the first-ladder length and height of $Y$. The main tools of investigation are a generalized Spitzer-Baxter identity, that we derive along the proof, and a suitable representation of the RWRSB in terms of local times of the random walk $S$. All these results are easily generalized to the entire sequence of ladder variables., Comment: 26 pages

We consider the Schr\"odinger partial differential equation of a rotating symmetric rigid molecule (symmetric rotor) driven by a z-linearly polarized electric field, as prototype of degenerate infinite-dimensional bilinear control system. By introducing an abstract perturbative criterium, we classify its simultaneous approximate controllability; based on this insight, we numerically perform an orientational selective transfer of rotational population., Comment: Additional material attached. It consists in two files: Resonnant14.m and Control.m used for the numerical simulations of the preprint. These two Octave scripts should be loaded in the Octave editor

The heat equation with inverse square potential on both half-lines of $\mathbb{R}$ is discussed in the presence of \emph{bridging} boundary conditions at the origin. The problem is the lowest energy (zero-momentum) mode of the transmission of the heat flow across a Grushin-type cylinder, a generalisation of an almost Riemannian structure with compact singularity set. This and related models are reviewed, and the issue is posed of the analysis of the dispersive properties for the heat kernel generated by the underlying positive self-adjoint operator. Numerical integration is shown that provides a first insight and relevant qualitative features of the solution at later times.

We consider one-dimensional discrete-time random walks (RWs) in the presence of finite size traps of length $\ell$ over which the RWs can jump. We study the survival probability of such RWs when the traps are periodically distributed and separated by a distance $L$. We obtain exact results for the mean first-passage time and the survival probability in the special case of a double-sided exponential jump distribution. While such RWs typically survive longer than if they could not leap over traps, their survival probability still decreases exponentially with the number of steps. The decay rate of the survival probability depends in a non-trivial way on the trap length $\ell$ and exhibits an interesting regime when $\ell\rightarrow 0$ as it tends to the ratio $\ell/L$, which is reminiscent of strongly chaotic deterministic systems. We generalize our model to continuous-time RWs, where we introduce a power-law distributed waiting time before each jump. In this case, we find that the survival probability decays algebraically with an exponent that is independent of the trap length. Finally, we derive the diffusive limit of our model and show that, depending on the chosen scaling, we obtain either diffusion with uniform absorption, or diffusion with periodically distributed point absorbers., Comment: 29 pages, 5 figures

We present an analytical approach to construct the Lie algebra of finite-dimensional subsystems of the driven asymmetric top rotor. Each rotational level is degenerate due to the isotropy of space, and the degeneracy increases with rotational excitation. For a given rotational excitation, we determine the nested commutators between drift and drive Hamiltonians using a graph representation. We then generate the Lie algebra for subsystems with arbitrary rotational excitation using an inductive argument., Comment: 10 pages, 7 figures

We study both the classical and quantum rotational dynamics of an asymmetric top molecule, controlled through three orthogonal electric fields that interact with its dipole moment. The main difficulties in studying the controllability of these infinite-dimensional quantum systems are the presence of severe spectral degeneracies in the drift Hamiltonian and the nonsolvability of the stationary free Schr\"odinger equation, which lead us to apply a perturbative Lie algebraic approach. In this paper we show that, while the classical equations given by the Hamiltonian system on ${\rm SO}(3)\times \mathbb{R}^3$ are controllable for all values of the rotational constants and all dipole configurations, the Sch\"odinger equation for the quantum evolution on $L^2({\rm SO}(3))$ is approximately controllable for almost all values of the rotational constants if and only if the dipole is not parallel to any of the principal axes of inertia of the asymmetric rigid body., Comment: 23 pages, 6 figures

We consider here a family of singular Laplace-Beltrami operators, focussing our attention on the problem of so-called quantum confinement on the half-plane equipped with Riemannian metrics of Grushin type degenerate at the boundary. By introducing a costant-fiber direct integral scheme we are able to rigorously characterize the presence or absence of self-adjointness of these operators. We also compare our technique and results with the already studied problem of quantum confinement on the half-cylinder.