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\noindent Consider an infinite collection of particles on the real line moving according to independent Brownian motions and such that the $i$-th particle from the left gets the drift $g_{i-1}$. The case where $g_0=1$ and $g_{i}=0$ for all $i \in \mathbb{N}$ corresponds to the well studied infinite Atlas model. Under conditions on the drift vector $\boldsymbol{g} = (g_0, g_1, \ldots)'$ it is known that the Markov process corresponding to the gap sequence of the associated ranked particles has a continuum of product form stationary distributions $\{\pi_a^{\boldsymbol{g}}, a \in S^{\boldsymbol{g}}\}$ where $S^{\boldsymbol{g}}$ is a semi-infinite interval of the real line. In this work we show that all of these stationary distributions are extremal and ergodic. We also prove that any product form stationary distribution of this Markov process that satisfies a mild integrability condition must be $\pi_a^{\boldsymbol{g}}$ for some $a \in S^{\boldsymbol{g}}$. These results are new even for the infinite Atlas model. The work makes progress on the open problem of characterizing all the invariant distributions of general competing Brownian particle systems interacting through their relative ranks. Proofs rely on synchronous and mirror coupling of Brownian particles and properties of the intersection local times of the various particles in the infinite system., Comment: 29 pages. To appear in Probability Theory and Related Fields

We consider parabolic PDEs associated with fractional type operators drifted by non-linear singular first order terms. When the drift enjoys some boundedness properties in appropriate Lebesgue and Besov spaces, we establish by exploiting a priori Besov-type estimates, the H{\"o}lder continuity of the solutions. In particular, we handle the almost critical case in whole generality.

We study stochastic differential equations(SDEs) with a small perturbation parameter. Under the dissipative condition on the drift coefficient and the local Lipschitz condition on the drift and diffusion coefficients we prove the existence and uniqueness result for the perturbed SDE, also the convergence result for the solution of the perturbed system to the solution of the unperturbed system when the perturbation parameter approaches zero.We consider the application of the above-mentioned results to the Cauchy problem and the transport equations.

In-situ, two-dimensional (2D) Langmuir probe measurements across a large part of the TCV divertor are reported in L-mode discharges with and without divertor baffles. This provides detailed insights into time averaged profiles, particle fluxes, and fluctuations behavior in different divertor regimes. The presence of the baffles is shown to substantially increase the divertor neutral pressure for a given upstream density and to facilitate the access to detachment, an effect that increases with plasma current. The detailed, 2D probe measurements allow for a divertor particle balance, including ion flux contributions from parallel flows and ExB drifts. The poloidal flux contribution from the latter is often comparable or even larger than the former, such that the divertor parallel flow direction reverses in some conditions, pointing away from the target. In most conditions, the integrated particle flux at the outer target can be predominantly ascribed to ionization along the outer divertor leg, consistent with a closed-box approximation of the divertor. The exception is a strongly detached divertor, achieved here only with baffles, where the total poloidal ion flux even decreases towards the outer target, indicative of significant plasma recombination. The most striking observation from relative density fluctuation measurements along the outer divertor leg is the transition from poloidally uniform fluctuation levels in attached conditions to fluctuations strongly peaking near the X-point when approaching detachment., Comment: Added reference

Multiple medical institutions collaboratively training a model using federated learning (FL) has become a promising solution for maximizing the potential of data-driven models, yet the non-independent and identically distributed (non-iid) data in medical images is still an outstanding challenge in real-world practice. The feature heterogeneity caused by diverse scanners or protocols introduces a drift in the learning process, in both local (client) and global (server) optimizations, which harms the convergence as well as model performance. Many previous works have attempted to address the non-iid issue by tackling the drift locally or globally, but how to jointly solve the two essentially coupled drifts is still unclear. In this work, we concentrate on handling both local and global drifts and introduce a new harmonizing framework called HarmoFL. First, we propose to mitigate the local update drift by normalizing amplitudes of images transformed into the frequency domain to mimic a unified imaging setting, in order to generate a harmonized feature space across local clients. Second, based on harmonized features, we design a client weight perturbation guiding each local model to reach a flat optimum, where a neighborhood area of the local optimal solution has a uniformly low loss. Without any extra communication cost, the perturbation assists the global model to optimize towards a converged optimal solution by aggregating several local flat optima. We have theoretically analyzed the proposed method and empirically conducted extensive experiments on three medical image classification and segmentation tasks, showing that HarmoFL outperforms a set of recent state-of-the-art methods with promising convergence behavior. Code is available at https://github.com/med-air/HarmoFL., Comment: Accepted at AAAI 2022

We study the spectrum of the semiclassical Witten Laplacian $\Delta_{f}$ associated to a smooth function $f$ on ${\mathbb R}^d$. We assume that $f$ is a confining Morse--Bott function. Under this assumption we show that $\Delta_{f}$ admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring-Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds., Comment: 28 pages, 6 figures

Particle drifts perpendicular to the background magnetic field have been proposed by some authors as an explanation for the very efficient perpendicular transport of solar energetic particles (SEPs). This process, however, competes with perpendicular diffusion caused by magnetic turbulence, which can also disrupt the drift patterns and reduce the efficiency of drift effects. The latter phenomenon is well known in cosmic ray studies, but not yet considered in SEP models. Additionally, SEP models which do not include drifts, especially for electrons, use turbulent drift reduction as a justification of this omission, without critically evaluating or testing this assumption. This article presents the first theoretical step for a theory of drift suppression in SEP transport. This is done by deriving the turbulence-dependent drift reduction function with a pitch-angle dependence, as applicable for anisotropic particle distributions, and by investigating to what extent drifts will be reduced in the inner heliosphere for realistic turbulence conditions and different pitch-angle dependencies of the perpendicular diffusion coefficient. The influence of the derived turbulent drift reduction factors on the transport of SEPs are tested, using a state-of-the-art SEP transport code, for several expressions of theoretically-derived perpendicular diffusion coefficients. It is found, for realistic turbulence conditions in the inner heliosphere, that cross-field diffusion will have the largest influence on the perpendicular transport of SEPs, as opposed to particle drifts., Comment: This is a pre-print of an article accepted for publication in The Astrophysical Journal

Quantum gates are typically vulnerable to imperfections in the classical control fields applied to physical qubits to drive the gates. One approach to reduce this source of error is to break the gate into parts, known as composite pulses (CPs), that typically leverage the constancy of the error over time to mitigate its impact on gate fidelity. Here we extend this technique to suppress secular drifts in Rabi frequency by regarding them as sums of power-law drifts whose first-order effects on over- or under-rotation of the state vector add linearly. Power-law drifts have the form $t^p$ where $t$ is time and the constant $p$ is its power. We show that composite pulses that suppress all power-law drifts with $p \leq n$ are also high-pass filters of filter order $n+1$ arXiv:1410.1624. We present sequences that satisfy our proposed power-law amplitude criteria, $\text{PLA}(n)$, obtained with this technique, and compare their simulated performance under time-dependent amplitude errors to some traditional composite pulse sequences. We find that there is a range of noise frequencies for which the $\text{PLA}(n)$ sequences provide more error suppression than the traditional sequences, but in the low frequency limit, non-linear effects become more important for gate fidelity than frequency roll-off. As a result, the previously known $F_1$ sequence, which is one of the two solutions to the $\text{PLA}(1)$ criteria and furnishes suppression of both linear secular drift and the first order nonlinear effects, is a sharper noise filter than any of the other $\text{PLA}(n)$ sequences in the low frequency limit.

We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection-diffusion equation \[ \partial_t \theta - \Delta \theta + b \cdot \nabla \theta = 0 \] to satisfy local boundedness, a single-scale Harnack inequality, and upper bounds on fundamental solutions. We demonstrate these properties for drifts $b$ belonging to $L^q_t L^p_x$, where $\frac{2}{q} + \frac{n}{p} < 2$, or $L^p_x L^q_t$, where $\frac{3}{q} + \frac{n-1}{p} < 2$. For steady drifts, the condition reduces to $b \in L^{\frac{n-1}{2}+}$. The space $L^1_t L^\infty_x$ of drifts with `bounded total speed' is a borderline case and plays a special role in the theory. To demonstrate sharpness, we construct counterexamples whose goal is to transport anomalous singularities into the domain `before' they can be dissipated., Comment: 33 pages, 3 figures

A failure detection system is the first step towards predictive maintenance strategies. A popular data-driven method to detect incipient failures and anomalies is the training of normal behaviour models by applying a machine learning technique like feed-forward neural networks (FFNN) or extreme learning machines (ELM). However, the performance of any of these modelling techniques can be deteriorated by the unexpected rise of non-stationarities in the dynamic environment in which industrial assets operate. This unpredictable statistical change in the measured variable is known as concept drift. In this article a wind turbine maintenance case is presented, where non-stationarities of various kinds can happen unexpectedly. Such concept drift events are desired to be detected by means of statistical detectors and window-based approaches. However, in real complex systems, concept drifts are not as clear and evident as in artificially generated datasets. In order to evaluate the effectiveness of current drift detectors and also to design an appropriate novel technique for this specific industrial application, it is essential to dispose beforehand of a characterization of the existent drifts. Under the lack of information in this regard, a methodology for labelling concept drift events in the lifetime of wind turbines is proposed. This methodology will facilitate the creation of a drift database that will serve both as a training ground for concept drift detectors and as a valuable information to enhance the knowledge about maintenance of complex systems., Comment: 11 pages, 2 figures, 1 table

For second-order elliptic or parabolic equations with subcritical or critical drifts, it is well-known that the Harnack inequality holds and their bounded weak solutions are H\"older continuous. We construct time-independent supercritical drifts in $L^{n-\lambda}(\mathbb{R}^n)$ with arbitrarily small $\lambda>0$ such that the Harnack inequality and the H\"older continuity fail in both the elliptic and the parabolic cases, thus confirming a conjecture by Seregin, Silvestre, Sverak and Zlatos. These results are sharp, and they also apply to a toy model of the axi-symmetric Navier-Stokes equations in space dimension $3$.

In this paper, by utilizing Wang's Harnack inequality with power and the Banach fixed point theorem, the weak well-posedness for McKean-Vlasov SDEs with integrable drift is investigated. In addition, using the decoupled method, some regularity such as relative entropy and Sobolev's estimate of invariant probability measure are proved. Finally, by Banach's fixed theorem, the existence and uniqueness of invariant probability measure for symmetric McKean-Vlasov SDEs and stochastic Hamiltonian system with integrable drifts are obtained., Comment: 15 pages

In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by $\alpha$-stable processes with $\alpha \in(1,2)$. The existence part is based on Euler's approximation as \cite{HRZ20}, while, the uniqueness is based on the Schauder estimates in Besov spaces for nonlocal Fokker-Planck equations. For the existence, we only assume the drift being continuous in the density variable. For the weak uniqueness, the drift is assumed to be Lipschitz in the density variable, while for the strong uniqueness, we also need to assume the drift being $\beta_0$-order H\"older continuous in the spatial variable, where $\beta_0\in(1-\alpha/2,1)$., Comment: 27 pages