Continuous valuations on the space of Lipschitz functions on the sphere

We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere $S^{n?1}$. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere.     

Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates

This article proposes a global, chaos-based procedure for the discretization of functionals of Brownian motion into functionals of a Poisson process with intensity $\lambda >0$. Under this discretization we study the weak convergence, as the intensity of the underlying Poisson process goes to infinity, of Poisson functionals and their corresponding Malliavin-type derivatives to their Wiener counterparts. In addition, we derive a convergence rate of $O\left({\lambda }^{?1∕4}\right)$ for the Poisson discretization of Wiener functionals by combining the multivariate Chen–Stein method with the Malliavin calculus. Our proposed sufficient condition for establishing the mentioned convergence rate involves the kernel functions in the Wiener chaos, yet we provide examples, especially the discretization of some common path dependent Wiener functionals, to which our results apply without committing the explicit computations of such kernels. To the best our knowledge, these are the first results in the literature on the universal convergence rate of a global discretization of general Wiener functionals.     

An explicit compact universal space for real flows

The Kakutani–Bebutov Theorem (1968) states that any compact metric real flow whose fixed point set is homeomorphic to a subset of $\mathbb{R}$ embeds into the Bebutov flow, the $\mathbb{R}$-shift on $C(\mathbb{R},[0,1])$. An interesting fact is that this universal space is a function space. However, it is not compact, nor locally compact. We construct an explicit countable product of compact subspaces of the Bebutov flow which is a universal space for all compact metric real flows, with no restriction; namely, into which any compact metric real flow embeds. The result is compared to previously known universal spaces.     

Limit theorems for multivariate Bessel processes in the freezing regime

Multivariate Bessel processes describe the stochastic dynamics of interacting particle systems of Calogero–Moser–Sutherland type and are related with $\beta$-Hermite and Laguerre ensembles. It was shown by Andraus, Katori, and Miyashita that for fixed starting points, these processes admit interesting limit laws when the multiplicities $k$ tend to $\infty$, where in some cases the limits are described by the zeros of classical Hermite and Laguerre polynomials. In this paper we use SDEs to derive corresponding limit laws for starting points of the form $\sqrt{k}?x$ for $k\to \infty$ with $x$ in the interior of the corresponding Weyl chambers. Our limit results are a.s. locally uniform in time. Moreover, in some cases we present associated central limit theorems.     

Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group

Let $G$ be a second countable locally compact abelian group. The aim of this paper is to characterize the left translates on the Heisenberg group $\mathbb{H}\left(G\right)$ to be frames and Riesz bases in terms of the group Fourier transform.     

Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

The asymmetric multitype contact process

We study the multitype contact process on ${\mathbb{Z}}^d$ under the assumption that one of the types has a birth rate that is larger than that of the other type, and larger than the critical value of the standard contact process. We prove that, if initially present, the stronger type has a positive probability of never going extinct. Conditionally on this event, it takes over a ball of radius growing linearly in time. We also completely characterize the set of stationary distributions of the process and prove a complete convergence theorem.     

The tamed unadjusted Langevin algorithm

In this article, we consider the problem of sampling from a probability measure $\pi$ having a density on ${\mathbb{R}}^d$ proportional to $x?{\mathrm{e}}^{?U\left(x\right)}$. The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential $U$ is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in $V$-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and $\pi$, as well as weak error bounds. Numerical experiments are presented which support our findings.     

Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n?1$ in ${\mathbb{R}}^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in ${\mathbb{R}}^n$, $d<n?1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ${\omega }_L$ is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ${\omega }_L$ and the Hausdorff measure.     

Hölder continuity of quasiconformal harmonic mappings from the unit ball to a spatial domain with C1 boundary

We prove that every quasiconformal mapping from the harmonic $\beta$-Bloch space between the unit ball and a spatial domain with $C^1$ boundary is globally $\alpha$-Hölder continuous for $\alpha <1?\beta$, with the Hölder coefficient that does not depend neither on the mapping nor on $\beta$. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for $\alpha <1$. This is an approach towards the extension of some results from the complex plane obtained by Warschawski (1951) for conformal mappings and Kalaj (2022) for quasiconformal harmonic mappings.     

Absolute continuity of the law for the two dimensional stochastic Navier–Stokes equations

We consider the two dimensional Navier–Stokes equations in vorticity form with a stochastic forcing term given by a gaussian noise, white in time and colored in space. First, we prove existence and uniqueness of a weak (in the Walsh sense) solution process $\xi$ and we show that, if the initial vorticity ${\xi }_0$ is continuous in space, then there exists a space–time continuous version of the solution. In addition we show that the solution $\xi (t,x)$ (evaluated at fixed points in time and space) is locally differentiable in the Malliavin calculus sense and that its image law is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.     

On the controllability and stabilization of the Benjamin equation on a periodic domain

The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain $\mathbb{T}$. We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in $H_p^s(\mathbb{T})$, with $s\ge 0$. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz., propagation of compactness and propagation of regularity in Bourgain's spaces. The global exponential stability of the system combined with a local controllability result yields the global controllability as well. Using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate. The results obtained here extend the ones we proved for the linearized Benjamin equation in [32].     

Uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings

This paper is concerned with the mixed initial–boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects. We successfully establish uniform exponential and polynomial decay rates for the solutions to this initial–boundary value problem under much weak conditions concerning memory effects. More specifically, we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memory-effect region includes a part of the system boundary, while the condition is a necessity in the previous literature; moreover, for the polynomial decay rates we only assume minimal conditions on the memory kernel function g, without the usual assumption of $g^{\prime }$ controlled by g.