Related papers: On continuity equations in infinite dimensions wit…

We consider the space of probability measures on a discrete set $X$, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset $Y \subseteq X$, it is natural to ask whether they can be connected…

In the paper we study closures of classes of log--concave measures under taking weak limits, linear transformations and tensor products. We consider what uniform measures on convex bodies can one obtain starting from some class…

We establish the stability of metric viscosity solutions to first-order Hamilton--Jacobi equations under Gromov--Hausdorff convergence. Our proof combines a characterization of metric viscosity solutions via quadratic distance functions…

Motivated by recent developments on calculus in metric measure spaces $(X,\mathsf d,\mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,\mathsf d)$ and…

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$. We prove that for every symmetric convex sets $K,L$ in $\mathbb{R}^n$ and every $\lambda\in(0,1)$, $$\gamma_n(\lambda K+(1-\lambda)L)^{\frac{1}{n}} \geq \lambda…

Let $\Gamma$ be a Zariski dense discrete subgroup of a connected simple real algebraic group $G_1$. We discuss a rigidity problem for discrete faithful representations $\rho:\Gamma\to G_2$ and a surprising role played by higher rank…

Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria…

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an…

Let $\Lambda$ be an isolated non-trival transitive set of a $C^1$ generic diffeomorphism $f\in\Diff(M)$. We show that the space of invariant measures supported on $\Lambda$ coincides with the space of accumulation measures of time averages…

Given positive measures $\nu,\mu$ on an arbitrary measurable space $(\Omega, \mathcal F)$, we construct a sequence of finite partitions $(\pi_n)_n$ of $(\Omega, \mathcal F)$ s.t. $$ \sum_{A\in \pi_n: \mu(A)>0} 1_{A} \frac{\nu(A)}{\mu(A)}…

Motivated by the fundamental theorem of calculus, and based on the works of Feller as well as Kac and Kre\u{\i}n, given an atomless Borel probability measure $\eta$ supported on a compact subset of $\mathbb{R}$, Freiberg and Z\"{a}hle…

We study the Gaussian sequence model, i.e. $X \sim N(\mathbf{\theta}, I_\infty)$, where $\mathbf{\theta} \in \Gamma \subset \ell_2$ is assumed to be convex and compact. We show that goodness-of-fit testing sample complexity is lower bounded…

We consider a gas whose each particle is characterised by a pair $(x,v_x)$ with the position $x\in \mathbb R^d$ and the velocity $v_x\in \mathbb R^d_0= \mathbb R^d\setminus \{0\}$. We define Gibbs measures on the cone of vector-valued…

The convective Brinkman-Forchheimer equations (CBFEs) \[ \frac{\partial \boldsymbol{X}}{\partial t} - \mu \Delta\boldsymbol{X} + (\boldsymbol{X}\cdot\nabla)\boldsymbol{X} + \alpha\boldsymbol{X} + \beta|\boldsymbol{X}|^{r-1}\boldsymbol{X} +…

Methods for measuring convexity defects of compacts in R^n abound. However, none of the those measures seems to take into account continuity. Continuity in convexity measure is essential for optimization, stability analysis, global…

We prove a log-Sobolev inequality for a certain class of log-concave measures in high dimension. These are the probability measures supported on the unit cube in R^n whose density takes the form exp(-H) where the function H is assumed to be…

We show that in any complete metric space the probability measures $\mu$ with compact and connected support are the ones having the property that the optimal tranportation distance to any other probability measure $\nu$ living on the…

We study the connection between the concavity properties of a measure $\nu$ and the convexity properties of the associated relative entropy $D(\cdot \Vert \nu)$ along optimal transport. As a corollary we prove a new dimensional…

In the article, we address the problem of absolute continuity of translated Rosenblatt measures on the path space. In [\v{C}oupek, P., K\v{r}\'i\v{z}, P., Maslowski, B., Stoch. Proc. Appl. 179 (2025) art. no. 104499], it is shown that there…

We establish multi-scale convergence theory for a class of Hamilton-Jacobi PDEs in space of probability measures. They arise from context of hydrodynamic limit of N-particle deterministic action minimizing (global) Lagrangian dynamics. From…