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The metric flow is introduced and extensively studied by Bamler [Compactness theory of the space of super Ricci flows, preprint (2020), arXiv:2008.09298; Structure theory of non-collapsed limits of Ricci flows, preprint (2020), arXiv:2009.03243], especially as an 픽-limit of a sequence of smooth Ricci flows with uniformly bounded Nash entropy, in which case each regular point on the limit is a point of smooth convergence. In this note, we shall consider the 픽-convergence of a sequence of 픽-limit flows, and, like Bamler, show that each regular point on the limit is also a point of smooth convergence. The main result will be applied in another work of the authors [P.-Y. Chan, Z. Ma and Y. Zhang, Dimension reduction for positively curved steady solitons, preprint (2023), arXiv:2310.14020]. [ABSTRACT FROM AUTHOR]

We give a multi-valued version of M iterative process [K. Ullah and M. Arshad, Numerical reckoning fixed points Suzuki's generalized nonexpansive mappings via new iteration process, Filomat 32 (2018) 187–196] in the setting of Banach spaces. After this, we prove some weak and strong convergence results for the class of multi-valued quasi-nonexpansive mappings (which contains the class of multi-valued nonexpansive mappings) with proper numerical examples using multi-valued version of M-iteration process. These results generalize several well-known comparable results in the literature. [ABSTRACT FROM AUTHOR]

This paper derives an approximation for a generalized Langevin equation driven by a force with random oscillation in time and periodic oscillation in space. By a diffusion approximation and the weak convergence of periodic oscillation function, the solution of the generalized Langevin equation is shown to converge in distribution to the solution of a stochastic partial differential equations (SPDEs) driven by time white noise. [ABSTRACT FROM AUTHOR]

This paper aims to establish the central limit theorem and moderate deviation principle for the stochastic Kuramoto–Sivashinsky equation driven by multiplicative noise on a bounded domain. The moderate deviation principle is investigated using the weak convergence approach based on a variational representation for expected values of positive functionals of the Brownian motion. The approach relies on proving basic qualitative properties of controlled versions of the original stochastic partial differential equation which is under consideration. [ABSTRACT FROM AUTHOR]

The stochastic two-dimensional Cahn–Hilliard–Navier–Stokes equations under non-Lipschitz conditions are considered. This model consists of the Navier–Stokes equations controlling the velocity and the Cahn–Hilliard model controlling the phase parameters. By iterative techniques, a priori estimates and weak convergence method, the existence and uniqueness of an energy weak solution to the equations under non-Lipschitz conditions have been obtained. [ABSTRACT FROM AUTHOR]

A Freidlin–Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist. 20 (2000) 39–61]. [ABSTRACT FROM AUTHOR]

In this paper, we prove some weak and strong convergence results for generalized α -nonexpansive mappings using M iteration process in the framework of Banach spaces. This generalizes former results proved by Ullah and Arshad [Numerical reckoning fixed points for Suzuki's generalized nonexpansive mappings via new iteration process, Filomat 32(1) (2018) 187–196]. [ABSTRACT FROM AUTHOR]

We consider on the 2D torus the modified Surface Quasi-Geostrophic (mSQG) equation with L 2 -initial data and perturbed by multiplicative transport noise. Under a suitable scaling of the noises, we show that the solutions converge weakly to the unique solution of the dissipative mSQG equation. [ABSTRACT FROM AUTHOR]

Let ξ 0 , ξ 1 , ... be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form f (z) = ∑ k = 0 ∞ ξ k c k z k , where c 0 , c 1 , ... is a real sequence such that c n 2 is regularly varying with index γ − 1 , where γ > 0. We prove that 𝔼 N [ 0 , 1 − 𝜖 ] ∼ γ 2 π | log 𝜖 | as 𝜀 ↓ 0 , where N [ 0 , r ] denotes the number of real zeroes of f in the interval [ 0 , r ]. [ABSTRACT FROM AUTHOR]

Let (X , d) be a locally compact separable ultrametric space. Given a measure m on X and a function C (B) defined on the set of all non-singleton balls B of X , we consider the hierarchical Laplacian L = L C . The operator L acts in ℒ 2 (X , m) , is essentially self-adjoint and has a purely point spectrum. Choosing a sequence { 𝜀 (B) } of i.i.d. random variables, we consider the perturbed function C (B , ω) and the perturbed hierarchical Laplacian L ω = L C (ω). Under certain conditions, the density of states 𝔪 exists and it is a continuous function. We choose a point λ such that 𝔪 (λ) > 0 and build a sequence of point processes defined by the eigenvalues of L ω located in the vicinity of λ. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity 𝔪 (λ). [ABSTRACT FROM AUTHOR]

This paper is concerned with optimal control problems for systems governed by mean-field stochastic differential equation, in which the control enters both the drift and the diffusion coefficient. We prove that the relaxed state process, associated with measure valued controls, is governed by an orthogonal martingale measure rather than a Brownian motion. In particular, we show by a counter example that replacing the drift and diffusion coefficient by their relaxed counterparts does not define a true relaxed control problem. We establish the existence of an optimal relaxed control, which can be approximated by a sequence of strict controls. Moreover, under some convexity conditions, we show that the optimal control is realized by a strict control. [ABSTRACT FROM AUTHOR]

In this paper, assuming an appropriate condition, we prove that -generalized hybrid mappings are demiclosed in Hilbert spaces. Using this fact, we prove a weak convergence theorem of Ishikawa type for these nonlinear mappings. Also, a strong convergence theorem of Halpern-Ishikawa type and a nonlinear ergodic theorem for -generalized hybrid mappings have been proven in Hilbert spaces. [ABSTRACT FROM AUTHOR]

We study the one-sided limit order book corresponding to limit sell orders and model it as a measure-valued process. Limit orders arrive to the book according to a Poisson process and are placed on the book according to a distribution which varies depending on the current best price. Market orders to buy periodically arrive to the book according to a second, independent Poisson process and remove from the book the order corresponding to the current best price. We consider the above described limit order book in a high frequency regime in which the rate of incoming limit and market orders is large and traders place their limit sell orders close to the current best price. Our first set of results provide weak limits for the unscaled price process and the properly scaled measure-valued limit order book process in the high frequency regime. In particular, we characterize the limiting measure-valued limit order book process as the solution to a measure-valued stochastic differential equation. We then provide an analysis of both the transient and long-run behavior of the limiting limit order book process. [ABSTRACT FROM AUTHOR]

Second-order stochastic dominance plays an important role in reliability and various branches of economics such as finance and decision-making under risk, and statistical testing for the stochastic dominance is often useful in practice. In this paper, we present a test of stochastic equality under the constraint of second-order stochastic dominance based on the theory of empirical processes. The asymptotic distribution of the test statistic is obtained, and a simple method to compute the critical value is derived. Simulation results and real data examples are presented to illustrate the proposed test method. [ABSTRACT FROM AUTHOR]

The Lie-Trotter-Kato product formula has been recently extended into Hadamard spaces by Stojkovic [Approximation for convex functionals on non-positively curved spaces and the Trotter-Kato product formula, Adv. Calc. Var. 5 (2012) 77-216]. The aim of our short note is to give a simpler proof relying upon weak convergence instead of an ultrapower technique. [ABSTRACT FROM AUTHOR]

In this paper, we consider a quasi-linear stochastic heat equation in one dimension on [0,1], with Dirichlet boundary conditions driven by an additive fractional white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter n ε N which can approximate the fractional noise in some sense. Then, we provide sufficient conditions ensuring that the real-valued mild solution of the SPDE perturbed by this family of noises converges in law, in the space C([0,T] X [0,1]) of continuous functions, to the solution of the fractional noise driven SPDE. [ABSTRACT FROM AUTHOR]

We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet Wα, β with Hurst parameter (α, β) ∈ (0, 1)2. When $0 \lt \alpha \leq 1-\frac{1}{2q}$ or $0 \lt \beta \leq 1-\frac{1}{2q}$ a central limit theorem holds for the renormalized Hermite variations of order q ≥ 2, while for $1-\frac{1}{2q} \lt \alpha, \beta \lt 1$ we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time (1, 1). [ABSTRACT FROM AUTHOR]

In this paper, we obtain a large deviation principle for the flow of homeomorphisms of the stochastic differential equations on the sphere Sd associated with the critical Sobolev Brownian vector fields. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a large deviation principle for the flow of homeomorphisms of the stochastic differential equations on the sphere Sd associated with the critical Sobolev Brownian vector fields. [ABSTRACT FROM AUTHOR]

We study a family of memory-based persistent random walks and we prove weak convergences after space-time rescaling. The limit processes are not only Brownian motions with drift. We have obtained a continuous but non-Markov process (Zt) which can be easily expressed in terms of a counting process (Nt). In a particular case the counting process is a Poisson process, and (Zt) permits to represent the solution of the telegraph equation. We study in detail the Markov process ((Zt, Nt); t ≥ 0). [ABSTRACT FROM AUTHOR]

We focus on the problem of learning equilibria in a particular routing game similar to the Wardrop traffic model. We describe a routing game played by a large number of players and present a distributed learning algorithm that we prove to converge weakly to equilibria for the system. The proof of convergence is based on a differential equation governing the global evolution of the system that is inferred from all the local evolutions of the agents in play. We prove that the differential equation converges with the help of Lyapunov techniques. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of global mild solutions to a class of stochastic Cahn–Hilliard equations driven by fractional noises (fractional in time and white in space), through a weak convergence argument. [ABSTRACT FROM AUTHOR]