arXiv subjects

Mathias Braun

Publications and source records attributed to Mathias Braun.

21 records · Page 2Linked to original sources

arXiv2021

Heat kernel bounds and Ricci curvature for Lipschitz manifolds

Given any $d$-dimensional Lipschitz Riemannian manifold $(M,g)$ with heat kernel $\mathsf{p}$, we establish uniform upper bounds on $\mathsf{p}$ which can always be decoupled in space and time. More precisely, we prove the existence of a constant $C>0$ and a bounded Lipschitz function $R\colon M \to (0,\infty)$ such that for every $x\in M$ and every $t>0$, \begin{align*} \sup_{y\in M} \mathsf{p}(t,x,y) \leq C\min\{t, R^2(x)\}^{-d/2}. \end{align*} This allows us to identify suitable weighted Lebesgue spaces w.r.t. the given volume measure as subsets of the Kato class induced by $(M,g)$. In the case $\partial M \neq \emptyset$, we also provide an analogous inclusion for Lebesgue spaces w.r.t. the surface measure on $\partial M$. We use these insights to give sufficient conditions for a possibly noncomplete Lipschitz Riemannian manifold to be tamed, i.e. to admit a measure-valued lower bound on the Ricci curvature, formulated in a synthetic sense.

math.DG↗

arXiv2021

Heat flow regularity, Bismut-Elworthy-Li's derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature

We prove that if the Ricci tensor $\mathrm{Ric}$ of a geodesically complete Riemannian manifold $M$, endowed with the Riemannian distance $\mathsf{d}$ and the Riemannian measure $\mathfrak{m}$, is bounded from below by a continuous function $k\colon M\to\mathbb{R}$ whose negative part $k^-$ satisfies, for every $t>0$, the exponential integrability condition \begin{equation*} \sup_{x\in M} \mathbb{E}\big[\mathrm{e}^{\int_0^t k^-(\mathsf{b}_r^x)/2\,\mathrm{d} r}\,1_{\{t < ζ^x\}}\big] < \infty, \end{equation*} then the lifetime $ζ^x$ of Brownian motion $\mathsf{b}^x$ on $M$ starting in any $x\in M$ is a.s. infinite. This assumption on $k$ holds if $k^-$ belongs to the Kato class of $M$. We also derive a Bismut-Elworthy-Li derivative formula for $\nabla \mathsf{P}_tf$ for every $f\in L^\infty(M)$ and $t>0$ along the heat flow $(\mathsf{P}_t)_{t\geq 0}$ with generator $Δ/2$, yielding its $L^\infty$-$\mathrm{Lip}$-regularization as a corollary. Moreover, given the stochastic completeness of $M$, but without any assumption on $k$ except continuity, we prove the equivalence of lower boundedness of $\mathrm{Ric}$ by $k$ to the existence, given any $x,y\in M$, of a coupling $(\mathsf{b}^x,\mathsf{b}^y)$ of Brownian motions on $M$ starting in $(x,y)$ such that a.s., \begin{equation*} \mathsf{d}\big(\mathsf{b}_t^x,\mathsf{b}_t^y\big) \leq \mathrm{e}^{-\int_s^t \underline{k}(\mathsf{b}_r^x,\mathsf{b}_r^y)/2\,\mathrm{d} r}\,\mathsf{d}\big(\mathsf{b}_s^x,\mathsf{b}_s^y\big) \end{equation*} holds for every $s,t\geq 0$ with $s\leq t$, involving the "average" $\underline{k}(u,v) := \inf_γ\int_0^1 k(γ_r)\,\mathrm{d} r$ of $k$ along geodesics from $u$ to $v$. Our results generalize to weighted Riemannian manifolds, where the Ricci curvature is replaced by the corresponding Bakry-Émery Ricci tensor.

math.PR↗

arXiv2019

Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds

Given a metric measure space $(X,\mathsf{d},\mathfrak{m})$ and a lower semicontinuous, lower bounded function $k\colon X\to\mathbb{R}$, we prove the equivalence of the synthetic approaches to Ricci curvature at $x\in X$ being bounded from below by $k(x)$ in terms of $\bullet$ the Bakry-Émery estimate $ΔΓ(f)/2 - Γ(f,Δf) \geq k\,Γ(f)$ in an appropriate weak formulation, and $\bullet$ the curvature-dimension condition $\mathrm{CD}(k,\infty)$ in the sense Lott-Sturm-Villani with variable $k$. Moreover, for all $p\in(1,\infty)$, these properties hold if and only if the perturbed $p$-transport cost \begin{equation*} W_p^{\underline{k}}(μ_1,μ_2,t):=\inf_{(\mathsf{b}^1,\mathsf{b}^2)} \mathbb{E}\Big[\mathrm{e}^{\int_0^{2t} p \underline{k}\left(\mathsf{b}^1_{r}, \mathsf{b}^2_{r}\right)/2\,\mathrm{d} r} \mathsf{d}^p\!\left(\mathsf{b}^1_{2t},\mathsf{b}^2_{2t} \right)\!\Big]^{1/p} \end{equation*} is nonincreasing in $t$. The infimum here is taken over pairs of coupled Brownian motions $\mathsf{b}^1$ and $\mathsf{b}^2$ on $X$ with given initial distributions $μ_1$ and $μ_2$, respectively, and $\underline{k}(x,y) := \inf_γ\int_0^1 k(γ_s)\,\mathrm{d} s$ denotes the "average" of $k$ along geodesics $γ$ connecting $x$ and $y$. Furthermore, for any pair of initial distributions $μ_1$ and $μ_2$ on $X$, we prove the existence of a pair of coupled Brownian motions $\mathsf{b}^1$ and $\mathsf{b}^2$ such that a.s. for every $s,t\in[0,\infty)$ with $s\leq t$, we have \begin{equation*} \mathsf{d}\!\left(\mathsf{b}_t^1,\mathsf{b}_t^2\right)\leq \mathrm{e}^{-\int_s^t \underline{k}\left(\mathsf{b}_r^1,\mathsf{b}_r^2\right)/2\,\mathrm{d} r} \mathsf{d}\!\left(\mathsf{b}_s^1,\mathsf{b}_s^2\right)\!. \end{equation*}

math.FA↗