Related papers: Asymptotic behavior of divergences and Cameron-Mar…

In this paper, we aim to interpret the background gravitational effects appearing in quantum field theory on curved space-time by studying the Brownian motion of quantum states along with the Hamilton-Perelman Ricci flow. It has been shown…

We survey recent developments in the field of complexity of pathwise approximation in $p$-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion.…

We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms $G_p: T_pM \to [0,\infty]$ are given. When we consider sub-Riemannian manifolds, our definition coincide with…

For the general class of quasifree fermionic right mover/left mover systems over the infinitely extended two-sided discrete line introduced in [8] within the algebraic framework of quantum statistical mechanics, we study the von Neumann…

We construct, for $p>n$, a concrete example of a complete non-compact $n$-dimensional Riemannian manifold of positive sectional curvature which does not support any $L^p$-Calder\'on-Zygmund inequality: \[ \forall\,\varphi\in…

Rademacher's Theorem can be interpreted as an almost-everywhere \emph{little-$o$ improvement principle}: if a function admits a uniform pointwise first-order Lipschitz control at every point, then this control improves to a vanishing one at…

Let $\mathbb{H}$ be the sub-Riemannian Heisenberg group. That $\mathbb{H}$ supports a rich family of quasiconformal mappings was demonstrated by Kor\'{a}nyi and Reimann using the so-called flow method. Here we supply further evidence of the…

The regularity of limit spaces of Riemannian manifolds with L^p curvature bounds, $p > n/2$, is investigated under no apriori non-collapsing assumption. A regular subset, defined by a local volume growth condition for a limit measure, is…

We develop a quantitative theory of Lipschitz harmonic functions (LHF) on finitely generated groups, with emphasis on the Lipschitz Liouville property, affine rigidity, and quasi-isometric invariance for groups of polynomial growth. On…

A unified treatment is given of some results of H. Donnelly-P. Li and L. Schwartz concerning the behaviour of heat semigroups on open manifolds with given compactifications, on one hand, and the relationship with the behaviour at infinity…

In this paper we introduce a simple space-filtration discretization scheme on Wiener space which allows us to study weak decompositions and smooth explicit approximations for a large class of Wiener functionals. We show that any Wiener…

In 1994, M. M. Popov [On integrability in F-spaces, Studia Math. no 3, 205-220] showed that the fundamental theorem of calculus fails, in general, for functions mapping from a compact interval of the real line into the lp-spaces for 0<p<1,…

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration. The theory provides a differential structure which describes the infinitesimal evolution of Wiener functionals at very small…

We establish an $\mathrm{L}^p$-index theorem for Dolbeault--Dirac operators on compact K\"ahler manifolds with coefficients in a Hermitian holomorphic vector bundle $E$. For every $p \in (1,\infty)$, we prove that the closed…

We show that, under mild assumptions on the limiting curve, a sequence of simple chordal planar curves converges uniformly whenever certain Loewner driving functions converge. We extend this result to random curves. The random version…

The following theorem on the circle group $\mathbb{T}$ is due to Norbert Wiener: If $f\in L^{1}\left( \mathbb{T}\right) $ has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then $f\in…

In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and…

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map $f:M\to N$ between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is…

The usual Gromoll-Meyer's generalized Morse lemma near degenerate critical points on Hilbert spaces, so called splitting lemma, is stated for at least $C^2$-smooth functionals. In this paper we establish a splitting theorem and a shifting…

In [Bailo, Carrillo, Hu. SIAM J. Appl. Math. 2023] the authors introduce a finite-volume method for aggregation-diffusion equations with non-linear mobility. In this paper we prove convergence of this method using an Aubin--Simons…