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We present a study of the galaxy merger and interaction activity within the Hyperion Proto-supercluster at z~2.5 in an effort to assess the occurrence of galaxy mergers and interactions in contrast to the coeval field and their impact on the build up of stellar mass in high density environments at higher-z. For this work, we utilize data from the Charting Cluster Construction with VUDS and ORELSE Survey (C3VO) along with extensive spectroscopic and photometric datasets available for the COSMOS field, including the HST-Hyperion Survey. To evaluate potential merger and interaction activity, we measure the fraction of galaxies with close kinematic companions ($f_{ckc}$) both within Hyperion and the coeval field by means of a Monte Carlo (MC) methodology developed in this work that probabilistically employs our entire combined spectroscopic and photometric dataset. We validate our $f_{ckc}$ MC methodology on a simulated lightcone built from the GAlaxy Evolution and Assembly semi-analytic model, and we determine correction factors that account for the underlying spectroscopic sampling rate of our dataset. We find that galaxies in Hyperion have close kinematic companions $\gtrsim 2\times$ more than galaxies in the field and measure a corrected $f_{ckc}=49_{-7.8}^{+7.4}$% for Hyperion and a corrected $f_{ckc}=23_{-1.3}^{+1.2}$% for the surrounding field; a $>3\sigma$ difference. This increase in $f_{ckc}$ indicates an enhancement in the merger and interaction activity within Hyperion and matches the trend seen in other structures. The rate of merger and interactions within the field implied from our field $f_{ckc}$ measurement is well aligned with values measured from other observations in similar redshift ranges. The enhanced $f_{ckc}$ measured within Hyperion suggests that merger and interaction activity play an important role in the mass growth of galaxies in denser environments at higher z., Comment: 23 pages, 10 figures, 4 tables. Submitted to A&A. Comments welcome

We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way., Comment: arXiv admin note: substantial text overlap with arXiv:2212.05646

We conduct an analysis of a stochastic hydrodynamic pilot-wave theory, which is a Langevin equation with a memory kernel that describes the dynamics of a walking droplet (or "walker") subjected to a repulsive singular potential and random perturbations through additive Gaussian noise. Under suitable assumptions on the singularities, we show that the walker dynamics is exponentially attracted toward the unique invariant probability measure. The proof relies on a combination of the Lyapunov technique and an asymptotic coupling specifically tailored to our setting. We also present examples of invariant measures, as obtained from numerical simulations of the walker in two-dimensional Coulomb potentials. Our results extend previous work on the ergodicity of stochastic pilot-wave dynamics established for smooth confining potentials.

We consider the long time statistics of a one-dimensional stochastic Ginzburg-Landau equation with cubic nonlinearity while being subjected to random perturbations via an additive Gaussian noise. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we find that the dynamics is attractive toward the unique invariant probability measure with a polynomial rate that is independent of the vanishing viscosity. This relies on a coupling technique exploiting a Foias-Prodi argument specifically tailored to the system. Then, in the inviscid regime, we show that the sequence of invariant measures converges toward the invariant measure of the stochastic Schr\"odinger equation in a suitable Wasserstein distance. Together with the uniform polynomial mixing, we obtain the validity of the inviscid limit for the solutions on the infinite time horizon with a log log rate.

This letter introduces a convergence prediction model (CPM) for decentralized market clearing mechanisms. The CPM serves as a tool to detect potential cyber-attacks that affect the convergence of the consensus mechanism during ongoing market clearing operations. In this study, we propose a successively elongating Bayesian logistic regression approach to model the probability of convergence of real-time market mechanisms. The CPM utilizes net-power balance among all the prosumers/market participants as a feature for convergence prediction, enabling a low-dimensional model to operate efficiently for all the prosumers concurrently. The results highlight that the proposed CPM has achieved a net false rate of less than 0.01% for a stressed dataset.

We study the long time statistics of a two-dimensional Hamiltonian system in the presence of Gaussian white noise. While the original dynamics is known to exhibit finite time explosion, we demonstrate that under the impact of the stochastic forcing as well as a deterministic perturbation, the solutions are exponentially attractive toward the unique invariant probability measure. This extends previously established results in which the system is shown to be noise-induced stable in the sense that the solutions are bounded in probability.

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori-Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard-Jones and Coulomb functions.

We consider a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. Our main study is the long time statistics of the system in the singular regime as the memory kernel collapses to a Dirac function. Specifically, we show that provided that sufficiently many directions in the phase space are stochastically forced, there is a unique invariant probability measure to which the system converges, with respect to a suitable Wasserstein-type topology, and at an exponential rate which is independent of the decay rate of the memory kernel. We then prove the convergence of this unique statistically steady state to the unique invariant probability measure of the classical stochastic reaction-diffusion equation in the zero-memory limit. Consequently, we establish the global-in-time validity of the short memory approximation.

We study the long time statistics of a walker in a hydrodynamic pilot-wave system, which is a stochastic Langevin dynamics with an external potential and memory kernel. While prior experiments and numerical simulations have indicated that the system may reach a statistically steady state, its long-time behavior has not been studied rigorously. For a broad class of external potentials and pilot-wave forces, we construct the solutions as a dynamics evolving on suitable path spaces. Then, under the assumption that the pilot-wave force is dominated by the potential, we demonstrate that the walker possesses a unique statistical steady state. We conclude by presenting an example of such an invariant measure, as obtained from a numerical simulation of a walker in a harmonic potential.

We study the long time statistics of a class of semi--linear wave equations modeling the motions of a particle suspended in continuous media while being subjected to random perturbations via an additive Gaussian noise. By comparison with the nonlinear reaction settings, of which the solutions are known to possess geometric ergodicity, we find that, under the impact of nonlinear dissipative damping, the mixing rate is at least polynomial of any order. This relies on a combination of Lyapunov conditions, the contracting property of the Markov transition semigroup as well as the notion of $d$--small sets.

We study the long time statistics of a class of semi--linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase space are stochastically forced, the system is exponentially attractive toward its unique invariant measure with a convergent rate that is uniform with respect to the mass. Then, in the small mass limit, we prove the convergence of the first marginal of the invariant measures in a suitable Wasserstein distance toward the unique invariant measure of a stochastic reaction--diffusion equation. This together with uniform geometric ergodcity implies the validity of the small mass limit for the solutions on the infinite time horizon $[0,\infty)$, thereby extending previously known results established for the damped wave equations under Lipschitz nonlinearities.

We consider a class of semi-linear differential Volterra equations with memory terms, polynomial nonlinearities and random perturbation. For a broad class of nonlinearities, we study statistically steady states of the system and find that they possess regularities compatible with those of the weak solutions. Moreover, if sufficiently many directions in the phase space are stochastically forced, we employ the \emph{generalized coupling} approach to establish the existence and uniqueness of the invariant probability measure to which the system is exponentially attractive. This extends ergodicity results previously established in [Bonaccorsi et al., SIAM J. Math. Anal., 44 (2012)].

We study the statistically invariant structures of the nonlinear generalized Langevin equation (GLE) with a power-law memory kernel. For a broad class of memory kernels, including those in the subdiffusive regime, we construct solutions of the GLE using a Gibbsian framework, which does not rely on existing Markovian approximations. Moreover, we provide conditions on the decay of the memory to ensure uniqueness of statistically steady states, generalizing previous known results for the GLE under particular kernels as a sum of exponentials.

We study a system of Skorokhod stochastic differential equations (SDEs) modeling the pairwise dispersion (in spatial dimension $d=2$) of heavy particles transported by a rough self-similar, turbulent flow with H\"{o}lder exponent $h\in (0,1)$. Under the assumption that $h>0$ is sufficiently small, we use Lyapunov methods and control theory to show that the Markovian system is nonexplosive and has a unique, exponentially attractive invariant probability measure. Furthermore, our Lyapunov construction is radially sharp and gives partial confirmation on a predicted asymptotic behavior with respect to the H\"{o}lder exponent $h$ of the invariant probability measure. A physical interpretation of the asymptotics is that intermittent clustering is weakened when the carrier flow is sufficiently rough.

We consider the generalized Langevin equation (GLE) in a harmonic potential with power law decay memory. We study the anomalous diffusion of the particle's displacement and velocity. By comparison with the free particle situation in which the velocity was previously shown to be either diffusive or subdiffusive, we find that, when trapped in a harmonic potential, the particle's displacement may either be diffusive or superdiffusive. Under slightly stronger assumptions on the memory kernel, namely, for kernels related to the broad class of completely monotonic functions, we show that both the free particle and the harmonically bounded GLE satisfy the equipartition of energy condition. This generalizes previously known results for the GLE under particular kernel instances such as the generalized Rouse kernel or (exactly) a power law function.

We present a new mass function of galaxy clusters and groups using optical/near-infrared wavelength spectroscopic and photometric data from the Observations of Redshift Evolution in Large-Scale Environments (ORELSE) survey. At $z\sim1$, cluster mass function studies are rare regardless of wavelength and have never been attempted from an optical/near-infrared perspective. This work serves as a proof of concept that $z\sim1$ cluster mass functions are achievable without supplemental X-ray or Sunyaev-Zel'dovich (SZ) data. Measurements of the cluster mass function provide important contraints on cosmological parameters and are complementary to other probes. With ORELSE, a new cluster finding technique based on Voronoi tessellation Monte-Carlo (VMC) mapping, and rigorous purity and completeness testing, we have obtained $\sim$240 galaxy overdensity candidates in the redshift range $0.55

For a Demand Response (DR) program with internet data centers (IDC), the Price-Amount curve that estimates how the potential DR amount depends on the DR price determined by power systems is crucial. Constructing this curve is challenging mainly due to the uncertainty in IDCs' operation. A novel Gaussian Process Regression-based estimation method is thus proposed. The variance of resulting curve reflecting the IDC operational uncertainty is also calculated., Comment: 3 pages, 4 figures, submitted to TSG

Bilevel optimization problems can be used to represent the collaborative interaction between a power system and grid-connected entities, called the followers, such as data centers. Most existing approaches assume that such followers' response behaviors are made available to the power system in the operation decision-making, which may be untenable in reality. This work presents a novel idea of solving bilevel optimization problems without assuming power systems' omniscience. The followers' responses will be represented by a function of the power system's decisions using Gaussian Process Regression. Then the two layers in the bilevel problem can be solved separately by the power system and its followers. This not only avoids the omniscience assumption, but also significantly increases the computational efficiency without compromising accuracy, especially for the problems with a complex lower layer. Moreover, a bilevel critical load restoration model is developed to test the proposed technique. Compared to the conventional methods, the proposed restoration model considers the load-side operation and the varying load marginal value, and can accurately estimate load-side loss and achieve better restoration solutions. Two case studies validate the advantages of the proposed approaches from different perspectives., Comment: 13 pages, 9 figures, submitted to IEEE

Battery energy storage systems are widely adopted in grid-connected applications to mitigate the impact of intermittent renewable generations and enhance power system resiliency. Degradation of the battery during its service time is one of the major concerns in the deployment that strongly affects the long-term lifetime. Apart from environmental factors, this intrinsic property of a battery depends on the daily operating conditions. Thus, optimally engaging the daily operation of the battery based on its current status in order to meet the required remaining useful life becomes a practical and demanding need. To address this issue, this paper proposes a health-informed RUL-constrained optimal power flow framework to characterize the corresponding optimal feasible operation space. The targeted service lifespan is achieved if the battery's working condition is confined within this feasible domain. Equivalent box constraints are then constructed for better computational efficiency in solving the optimization problem. In this framework, a Monte Carlo-based data-driven approach and a health indicator (HI) representing the battery's current states are introduced. The performance of the proposed method is illustrated with the IEEE 39-bus system., Comment: 10 pages, 7 figures, submitted to PESGM 2021

In this paper, we propose a novel Health-Focused Optimal Power Flow (HF-OPF) to take into account the equipment health in operational and physical constraints. The health condition index is estimated based on the possible fault characteristics for generators and batteries. The paper addresses the need for understanding the relationship between health condition index and the operational constraints in OPF problems. Such a relationship is established through Finite Element Method (FEM)-based simulations. The simulations for generator and battery faults indicate the presence of an inflection point after which effect of fault becomes severe. A microgrid system is used to illustrate the performance of the proposed HF-OPF. The results show that health condition inflicts high cost of generation and can lead to infeasibility even with less critical faults., Comment: 5 Pages

Battery energy storage is an essential component of a microgrid. The working temperature of the battery is an important factor as a high-temperature condition generally increases losses, reduces useful life, and can even lead to fire hazards. Hence, it is indispensable to regulate the temperature profile of the battery modules/packs properly in the battery energy storage during the operation. In view of this, a two-layer optimal control and operation scheme is proposed for a microgrid with energy storage. In the first layer, an optimal control model is formed to derive the optimal control policy that minimizes the control efforts, consisting of the fan speed and battery current magnitude, in order to achieve a temperature distribution reference over the battery modules. In the second layer, the system operator of the microgrid performs an optimal power flow to search for the optimal temperature distribution reference used in the first stage and the corresponding operating current of the battery that minimize the operation cost of the entire microgrid system. This two-layer scheme offers a great computational benefit that allows for large-scale integration of batteries. A case study is performed on the proposed two-layer model to illustrate its performance., Comment: 5 pages, 5 figures, submitted to PESGM 2020

In light of recent work on particles fluctuating in linear viscoelastic fluids, we study a linear stochastic partial-integro-differential equation with memory that is driven by a stationary noise on a bounded, smooth domain. Using the framework of generalized stationary solutions introduced in~\cite{mckinley2018anomalous}, we provide sufficient conditions on the differential operator and the noise to obtain the existence as well as H\"older regularity of the stationary solutions for the concerned equation. As an application of the regularity results, we compare to analogous classical results for the stochastic heat equation. When the 1d stochastic heat equation is driven by white noise, solutions are continuous with space and time regularity that is H\"older $(1/2-\ep)$ and $(1/4-\ep)$ respectively. When driven by colored-in-space noise, solutions can have a range of regularity properties depending on the structure of the noise. Here, we show that the particular form of colored-in-time memory that arises in viscoelastic diffusion applications, satisfying what is called the Fluctuation--Dissipation relationship, yields sample paths that are H\"older $(1/2-\ep)$ and $(1/2-\ep)$ in space and time.

Utilizing spectroscopic observations taken for the VIMOS Ultra-Deep Survey (VUDS), new observations from Keck/DEIMOS, and publicly available observations of large samples of star-forming galaxies, we report here on the relationship between the star formation rate (SFR) and the local environment ($\delta_{gal}$) of galaxies in the early universe ($2

In this letter, we present a novel Gaussian Process Learning-based Probabilistic Optimal Power Flow (GP-POPF) for solving POPF under renewable and load uncertainties of arbitrary distribution. The proposed method relies on a non-parametric Bayesian inference-based uncertainty propagation approach, called Gaussian Process (GP). We also suggest a new type of sensitivity called Subspace-wise Sensitivity, using observations on the interpretability of GP-POPF hyperparameters. The simulation results on 14-bus and 30-bus systems show that the proposed method provides reasonably accurate solutions when compared with Monte-Carlo Simulations (MCS) solutions at different levels of uncertain renewable penetration as well as load uncertainties, while requiring much less number of samples and elapsed time.

Underwater vehicles are employed in the exploration of dynamic environments where tuning of a specific controller for each task would be time-consuming and unreliable as the controller depends on calculated mathematical coefficients in idealised conditions. For such a case, learning task from experience can be a useful alternative. This paper explores the capability of probabilistic inference learning to control autonomous underwater vehicles that can be used for different tasks without re-programming the controller. Probabilistic inference learning uses a Gaussian process model of the real vehicle to learn the correct policy with a small number of real field experiments. The use of probabilistic reinforced learning looks for a simple implementation of controllers without the burden of coefficients calculation, controller tuning or system identification. A series of computational simulations were employed to test the applicability of model-based reinforced learning in underwater vehicles. Three simulation scenarios were evaluated: waypoint tracking, depth control and 3D path tracking control. The 3D path tracking is done by coupling together a line-of-sight law with probabilistic inference for learning control. As a comparison study LOS-PILCO algorithm can perform better than a robust LOS-PID. The results shows that probabilistic model based reinforced learning is a possible solution to motion control of underactuated AUVs as can generate capable policies with minimum quantity of episodes.

In this paper, a novel convexification approach for Small-Signal Stability Constraint Optimal Power Flow (SSSC-OPF) has been presented that does not rely on eigenvalue analysis. The proposed methodology is based on the sufficient condition for the small-signal stability, developed as a Bilinear Matrix Inequality (BMI), and uses network structure-preserving Differential Algebraic Equation (DAE) modeling of the power system. The proposed formulation is based on Semi-definite Programming (SDP) and objective penalization that has been proposed for feasible solution recovery, making the method computationally efficient for large-scale systems. A vector-norm based objective penalty function has also been proposed for feasible solution recovery while working over large and dense BMIs with matrix variables. An effectiveness study carried out on WECC 9-bus, New England 39-bus, and IEEE 118-bus test systems show that the proposed method is capable of achieving a stable equilibrium point without inflicting a high stability-induced additional cost.

In this work, we propose a non-parametric probabilistic load flow (NP-PLF) technique based on the Gaussian Process (GP) learning to understand the power system behavior under uncertainty for better operational decisions. The technique can provide "semi-explicit" power flow solutions by implementing the learning and testing steps which map control variables to inputs. The proposed NP-PLF leverages upon GP upper confidence bound (GP-UCB) sampling algorithm. The salient features of this NP-PLF method are: i) applicable for power flow problem having power injection uncertainty with an unknown class of distribution; ii) providing probabilistic learning bound (PLB) which further provides control over the error and convergence; iii) capable of handling intermittent distributed generation as well as load uncertainties, and iv) applicable to both balanced and unbalanced power flow with different type and size of power systems. The simulation results performed on the IEEE 30-bus and IEEE 118-bus system show that the proposed method can learn the voltage function over the power injection subspace using a small number of training samples. Further, the testing with different input uncertainty distributions indicates that complete statistical information can be obtained for the probabilistic load flow problem with average percentage relative error of order $10^{-3}$\% on 50000 test points.

While most power system small-signal stability assessments rely on the reduced Jacobian, which depends non-linearly on the states, uncertain operating points introduce nontrivial hurdles in certifying the system's stability. In this paper, a novel probabilistic robust small-signal stability (PRS) framework is developed for a power system based on Gaussian process (GP) learning. The proposed PRS assessment provides a robust stability certificate for a state subspace, such as that specified by the error bounds of the state estimation, with a given probability. With such a PRS certificate, all inner points of the concerned subspace will be stable with at least the corresponding confidence level. To this end, the behavior of the critical eigenvalue of the reduced Jacobian with state points in a state subspace is learned using GP. The proposed PRS certificate along with the Subspace-based Search and Confidence-based Search mechanisms constitute a holistic framework catering to all scenarios. The proposed framework is a powerful approach to assess the stability under uncertainty because it does not require input uncertainty distributions and other state-specific input-to-output approximations. Further, the critical eigenvalue behavior in a state subspace is analyzed using an upper bound of the eigenvalue variations and their inferences are discussed in detail. The results on three-machine nine-bus WSCC system show that the proposed certificate can find the robust stable state subspace with a given probability.

This paper introduces a novel framework to construct the region of attraction (ROA) of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics. Most existing works on estimating ROA rely on analytical Lyapunov functions, which are subject to two limitations: the analytic Lyapunov functions may not be always readily available, and the resulting ROA may be overly conservative. This work overcomes these two limitations by leveraging the converse Lyapunov theorem in control theory to eliminate the need of an analytic Lyapunov function and learning the unknown Lyapunov function with the Gaussian Process (GP) approach. In addition, a Gaussian Process Upper Confidence Bound (GP-UCB) based sampling algorithm is designed to reconcile the trade-off between the exploitation for enlarging the ROA and the exploration for reducing the uncertainty of sampling region. Within the constructed ROA, it is guaranteed in probability that the system state will converge to the stable equilibrium with a confidence level. Numerical simulations are also conducted to validate the assessment approach for the ROA of the single machine infinite bus system and the New England $39$-bus system. Numerical results demonstrate that our approach can significantly enlarge the estimated ROA compared to that of the analytic Lyapunov counterpart.

Wireless dynamic charging technologies are becoming a promising alternative solution to plug-in ones as they allow on-the-move charging for electric vehicles. From a power network point of view, this type of charging makes electric vehicles a new type of loads--the moving loads. Such moving loads are different from the traditional loads as they may change their locations constantly in the grids. To study the effect of these moving loads on power distribution grids, this work focuses on the steady-state analysis of electrified roads equipped with wireless dynamic charging. In particular, the voltage profile and the long-term voltage stability of the electrified roads are considered. Unusual shapes of the voltage profile are observed such as the half-leaf veins for a one-way road and the harp-like shape for a two-way road. Voltage swings are also detected while the vehicles move in the two-way road configuration. As for the long-term voltage stability, continuation power flow is used to characterize the maximum length of a road as well as the maximum number of vehicles that the road can accommodate., Comment: 6 pages, 8 figures, accepted to present at eenergy19, ACM19, Phoenix, AZ, USA, June 2019

The Observations of Redshift Evolution in Large Scale Environments (ORELSE) survey is an ongoing imaging and spectroscopic campaign initially designed to study the effects of environment on galaxy evolution in high-redshift ($z\sim1$) large-scale structures. We use its rich data in combination with a powerful new technique, Voronoi tessellation Monte-Carlo (VMC) mapping, to search for serendipitous galaxy overdensities at $0.55 < z < 1.37$ within 15 ORELSE fields, a combined spectroscopic footprint of $\sim$1.4 square degrees. Through extensive tests with both observational data and our own mock galaxy catalogs, we optimize the method's many free parameters to maximize its efficacy for general overdensity searches. Our overdensity search yielded 402 new overdensity candidates with precisely measured redshifts and an unprecedented sensitivity down to low total overdensity masses ($\mathcal{M}_{tot} \gtrsim 5\times 10^{13} M_{\odot}$). Using the mock catalogs, we estimated the purity and completeness of our overdensity catalog as a function of redshift, total mass, and spectroscopic redshift fraction, finding impressive levels of both 0.92/0.83 and 0.60/0.49 for purity/completeness at $z=0.8$ and $z=1.2$, respectively, for all overdensity masses at spectroscopic fractions of $\sim$20%. With VMC mapping, we are able to measure precise systemic redshifts, provide an estimate of the total gravitating mass, and maintain high levels of purity and completeness at $z\sim1$ even with only moderate levels of spectroscopy. Other methods (e.g., red-sequence overdensities and hot medium reliant detections) begin to fail at similar redshifts, which attests to VMC mapping's potential to be a powerful tool for current and future wide-field galaxy evolution surveys at $z\sim1$ and beyond., Comment: 39 pages, 18 figures, 6 tables, accepted by MNRAS

The generalized Langevin equation (GLE) is a universal model for particle velocity in a viscoelastic medium. In this paper, we consider the GLE family with fractional memory kernels. We show that, in the critical regime where the memory kernel decays like $1/t$ for large $t$, the mean squared displacement (MSD) of particle motion grows linearly in time up to a slowly varying (logarithm) term. Moreover, we establish the well-posedness of the GLE in this regime. This solves an open question from [Mckinley 2018 Anomalous] and completes the answer to the conjecture put forward in [Morgado 2002 Relation] on the relationship between memory kernel decay and anomalously diffusive behavior. Under slightly stronger assumptions on the memory kernel, we construct an Abelian-Tauberian framework that leads to robust bounds on the deviation of the MSD around its asymptotic trend. This bridges the gap between the GLE memory kernel and the spectral density of anomalously diffusive particle motion characterized in [Didier 2017 Asymptotic].