Related papers: On small balls problem for stable measures in a Hi…
The paper is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The paper also indicates a special case when an additional integral and…
We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive…
Stability results for the Helmholtz equations in both deterministic and random periodic structures are proved in this paper. Under the assumption of excluding resonances, by a variational method and Fourier analysis in the energy space, the…
Although quantitative stability for critical points of the Sobolev and fractional Sobolev inequalities has been extensively studied, the corresponding stability theory for critical points of the Hardy--Littlewood--Sobolev (HLS) inequality…
Let $S=\sum_{i=1}^{+\infty}\lambda_{i}Z_{i}$ where the $Z_{i}$'s are i.d.d. positive with $\mathbb{E}\| Z\| ^{3}<+\infty$ and $(\lambda_{i})_{i\in\mathbb{N}}$ a positive nonincreasing sequence such that $\sum\lambda_{i}<+\infty$. We study…
This paper proves the existence of unstable shocks of the Burgers-Hilbert equation conjectured in arXiv:2006.05568. More precisely, we construct smooth initial data with finite $H^9$-norm such that the solution in self-similar coordinates…
In recent years, Bayesian inference in large-scale inverse problems found in science, engineering and machine learning has gained significant attention. This paper examines the robustness of the Bayesian approach by analyzing the stability…
We study in this paper a control problem in a space of random variables. We show that its Hamilton Jacobi Bellman equation is related to the Master equation in Mean field theory. P.L. Lions in [14,15] introduced the Hilbert space of square…
In this note, we highlight some properties of the metric projection onto a closed convex in a Hilbert space. In particular, we use some recent results on fixed points of nonexpansive potential operators.
The dilation equation arises naturally when using a multiresolution analysis to construct a wavelet basis. We consider solutions in the space of signed measures, which, after normalization, can be viewed as pseudo-probability measures.…
In this paper, we are concerned with the stochastic time-fractional diffusion-wave equations in a Hilbert space. The main objective of this paper is to establish properties of the stochastic weak solutions of the initial-boundary value…
As physics searches for invariants in observations, this paper looks for invariants of probabilistic observation without assuming physical structure. Structure emerges from the basic assumption of science that new information shall lead to…
We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models sub jected to local transformations. Such systems arise in the study of a stochastic time-evolution of…
We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate…
The general problem is studied on a simple example. A quantum particle in an infinite one-dimensional well potential is considered. Let the boundaries of well changes in a finite time $T$. The standard methods for calculating probability of…
We show that small perturbations of the metric of a ball in Euclidean n-space to metrics with nonpositive curvature do not reduce the isoperimetric ratio. Furthermore, the isoperimetric ratio is preserved only if the perturbation…
Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries.…
In this paper, we study the existence of SRB measures and their properties for infinite dimensional dynamical systems in a Hilbert space. We show several results including (i) if the system has a partially hyperbolic attractor with…
In this paper, we study the Hilbert$-$Schmidt frame (HS-frame) theory for separable Hilbert spaces. We first present some characterizations of HS-frames and prove that HS-frames share many important properties with frames. Then, we show how…
There are several known constructions of equilibrium states for H\"older continuous potentials in the context of both subshifts of finite type and uniformly hyperbolic systems. In this article we present another method of building such…