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1. Front location determines convergence rate to traveling waves

   https://doi.org/10.4171/AIHPC/120  

   An, Jing; Henderson, Christopher; Ryzhik, Lenya (May 2024, Annales de l'Institut Henri Poincaré C, Analyse non linéaire)

   We propose a novel method for establishing the convergence rates of solutions to reaction–diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function introduced in An et. al. [Arch. Ration. Mech. Anal. 247 (2023), no. 5, article no. 88]. It turns out that the convergence rate is controlled by the distance between thephantom front locationfor the shape defect function and the true front location of the solution. Curiously, the convergence to a traveling wave has a pulled nature, regardless of whether the traveling wave itself is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach dramatically simplifies the proof in the Fisher–KPP case and gives a unified, succinct explanation for the known algebraic rates of convergence in the Fisher–KPP case and the exponential rates in the pushed case. 

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2. Asymptotics of Yule's nonsense correlation for Ornstein-Uhlenbeck paths: The correlated case.

   Douissi, Soukaina; Ernst, Philip; Viens, Frederi (April 2025, arXiv)

   We study the continuous-time version of the empirical correlation coefficient between the paths of two possibly correlated Ornstein-Uhlenbeck processes, known as Yule's nonsense correlation for these paths. Using sharp tools from the analysis on Wiener chaos, we establish the asymptotic normality of the fluctuations of this correlation coefficient around its long-time limit, which is the mathematical correlation coefficient between the two processes. This asymptotic normality is quantified in Kolmogorov distance, which allows us to establish speeds of convergence in the Type-II error for two simple tests of independence of the paths, based on the empirical correlation, and based on its numerator. An application to independence of two observations of solutions to the stochastic heat equation is given, with excellent asymptotic power properties using merely a small number of the solutions' Fourier modes. 

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3. Convergence and nonconvergence in a nonlocal gradient flow

   https://doi.org/10.1112/jlms.70047  

   Park, Sangmin; Pego, Robert L (January 2025, Journal of the London Mathematical Society)

   Abstract We study the asymptotic convergence as of solutions of , a nonlocal differential equation that is formally a gradient flow in a constant‐mass subspace of arising from simplified models of phase transitions. In case the solution takes finitely many values, we provide a new proof of stabilization that uses a Łojasiewicz‐type gradient inequality near a degenerate curve of equilibria. Solutions with infinitely many values in generalneed notconverge to equilibrium, however, which we demonstrate by providing counterexamples for piecewise linear and cubic functions . Curiously, the exponentialrateof convergence in the finite‐value case can jump from order to arbitrarily small values upon perturbation of parameters. 

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4. Structure aware Runge–Kutta time stepping for spacetime tents

   https://doi.org/10.1007/s42985-020-00020-4  

   Gopalakrishnan, Jay; Schöberl, Joachim; Wintersteiger, Christoph (August 2020, SN Partial Differential Equations and Applications)

   null (Ed.)

   Abstract We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge–Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations. 

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5. Birth–death dynamics for sampling: global convergence, approximations and their asymptotics

   https://doi.org/10.1088/1361-6544/acf988  

   Lu, Yulong; Slepčev, Dejan; Wang, Lihan (September 2023, Nonlinearity)

   Abstract Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth–death dynamics. We improve results in previous works (Liuet al2023Appl. Math. Optim.8748; Luet al2019 arXiv:1905.09863) and provide weaker hypotheses under which the probability density of the birth–death governed by Kullback–Leibler divergence or byχ²divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth–death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker–Planck equation and relies on kernel-based approximations of the measure. Using the technique of Γ-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelised dynamics converge on finite time intervals, to the pure birth–death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimisers of the energy corresponding to the kernelised dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelised dynamics towards the Gibbs measure. 

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