相关论文: Laplacian cut-offs, porous and fast diffusion on m…

We exhibit a large class of Lyapunov functionals for nonlinear drift-diffusion equations with non-homogeneous Dirichlet boundary conditions. These are generalizations of large deviation functionals for underlying stochastic many-particle…

This work deals with the Entire solutions of a nonlinear equation. The first part of this paper is devoted to investigation of the Liouville property on compact manifolds, which extends a result by Castorina-Mantegazza [4] for positive f.…

Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is…

In this paper, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in…

Let $L_t:=\Delta_t +Z_t $, $t\in [0,T_c)$ on a differential manifold equipped with time-depending complete Riemannian metric $(g_t)_{t\in [0,T_c)}$, where $\Delta_t$ is the Laplacian induced by $g_t$ and $(Z_t)_{t\in [0,T_c)}$ is a family…

After establishing some new global facts (like a measure theoretic structure theorem and approximation results) about complex-valued functions with bounded variation on arbitrary noncompact Riemannian manifolds, we extend results of…

We prove an abstract and general version of the Lewy-Stampacchia inequality. We then show how to reproduce more classical versions of it and, more importantly, how it can be used in conjunction with Laplacian comparison estimates to produce…

This paper studies the problem of distributed Riemannian optimization over a network of agents whose cost functions are geodesically smooth but possibly geodesically non-convex. Extending a well-known distributed optimization strategy…

Let $v \ne 0$ be a vector in $\R^n$. Consider the Laplacian on $\R^n$ with drift $\Delta_{v} = \Delta + 2v\cdot \nabla$ and the measure $d\mu(x) = e^{2 \langle v, x \rangle} dx$, with respect to which $\Delta_{v}$ is self-adjoint. This…

We define horizontal diffusion in $C^1$ path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be…

Diffusion-driven patterns appear on curved surfaces in many settings, initiated by unstable modes of an underlying Laplacian operator. On a flat surface or perfect sphere, the patterns are degenerate, reflecting translational/rotational…

Some analysis on the Lorentzian distance in a spacetime with controlled sectional (or Ricci) curvatures is done. In particular, we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the Omori-Yau…

We construct many new examples of complete Calabi-Yau metrics of maximal volume growth on certain smoothings of Cartesian products of Calabi-Yau cones with smooth cross-sections. A detailed description of the geometry at infinity of these…

Occupied diffusions offer a Markovian framework for path-dependent dynamics by lifting the state space with a flow of occupation measures. Because this additional feature is infinite-dimensional, the simulation of these processes remains…

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic properties of test functions and…

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=\Delta u^m$$ where $m>1$, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li-Yau type for…

Distributed optimization has gained substantial interest in recent years due to its wide applications in machine learning. However, most of existing algorithms are designed for Euclidean spaces, leaving composite optimization on Riemannian…

We generalize the splitting theorem of Cai-Galloway for complete Riemannian manifolds with $\Ric\geq-(n-1)$ admitting a family of compact hypersurfaces tending to infinity with mean curvatures tending to $n-1$ sufficiently fast to the…

We consider overdamped Langevin diffusions in Euclidean space, with curvature equal to the spectral gap. This includes the Ornstein-Uhlenbeck process as well as non-Gaussian and non-product extensions with convex interaction, such as the…

We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. In doing this we also discuss some geometrical properties of the cutlocus of a point and we compare some…