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arXiv:2309.06820 (Published 2023-09-13)
Liouville theorem for $V$-harmonic maps under non-negative $(m, V)$-Ricci curvature for non-positive $m$
Let $V$ be a $C^1$-vector field on an $n$-dimensional complete Riemannian manifold $(M, g)$. We prove a Liouville theorem for $V$-harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative $(m, V)$-Ricci curvature for $m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$ into Cartan-Hadam\-ard manifolds, which extends Cheng's Liouville theorem proved S.~Y.~Cheng for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan-Hadamard manifolds. We also prove a Liouville theorem for $V$-harmonic maps from complete Riemannian manifolds with non-negative $(m, V)$-Ricci curvature for $m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$ into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt-Jost-Wideman and Choi. Our probabilistic proof of Liouville theorem for several growth $V$-harmonic maps into Hadamard manifolds enhances an incomplete argument by Stafford. Our results extend the results due to Chen-Jost-Qiu\cite{ChenJostQiu} and Qiu\cite{Qiu} in the case of $m=+\infty$ on the Liouville theorem for bounded $V$-harmonic maps from complete Riemannian manifolds with non-negative $(\infty, V)$-Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of $V$-harmonic maps and the recurrence property of $\Delta_V$-diffusion processes on manifolds. Our results are new even in the case $V=\nabla f$ for $f\in C^2(M)$.
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arXiv:2001.02528 (Published 2020-01-08)
A Liouville theorem for Lévy generators
Under mild assumptions, we establish a Liouville theorem for the "Laplace" equation $Au=0$ associated with the infinitesimal generator $A$ of a L\'evy process: If $u$ is a weak solution to $Au=0$ which is at most of (suitable) polynomial growth, then $u$ is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with L\'evy processes.
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arXiv:1702.00961 (Published 2017-02-03)
Liouville theorem for bounded harmonic functions on graphs satisfying non-negative curvature dimension condition
Comments: 8 pages, any comment is welcomeIn a general setting, we prove that any bounded harmonic function on a graph satisfying the curvature dimension condition CD(0, \infty) is constant.
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arXiv:1409.5648 (Published 2014-09-19)
On bounded continuous solutions of the archetypical functional equation with rescaling
We study the "archetypical" functional equation $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, where $\mathbb{E}$ denotes expectation and $(\alpha,\beta)$ is random with distribution $\mu$. Particular cases include: (i) $y(x)=\sum_{i} p_{i}\, y(a_i(x-b_i))$ and (ii) $y'(x)+y(x) =\sum_{i} p_{i}\,y(a_i(x-c_i))$ (pantograph equation), both subject to the balance condition $\sum_{i} p_{i}=1$ (${p_{i}>0}$). Solutions $y(x)$ admit interpretation as harmonic functions of an associated Markov chain $(X_n)$ with jumps of the form $x\rightsquigarrow\alpha(x-\beta)$. The paper concerns Liouville-type results asserting that any bounded continuous harmonic function is constant. The problem is essentially governed by the value $K:=\iint_{\mathbb{R}^2}\ln|a|\,\mu(\mathrm{d}a,\mathrm{d}b)=\mathbb{E}\{\ln |\alpha|\}$. In the critical case $K=0$, we prove a Liouville theorem subject to the uniform continuity of $y(x)$. The latter is guaranteed under a mild regularity assumption on the distribution of $\beta$, which is satisfied for a large class of examples including the pantograph equation (ii). Functional equation (i) is considered with $a_i=q^{m_i}$ ($q>1$, $m_i\in\mathbb{Z}$), whereby a Liouville theorem for $K=0$ can be established without the uniform continuity assumption. Our results also include a generalization of the classical Choquet--Deny theorem to the case $|\alpha|\equiv1$, and a surprising Liouville theorem in the resonance case $\alpha(c-\beta)\equiv c$. The proofs systematically employ Doob's Optional Stopping Theorem (with suitably chosen stopping times) applied to the martingale $y(X_n)$.
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arXiv:1012.5687 (Published 2010-12-28)
Coupling and Applications
Comments: 16 pagesCategories: math.PRThis paper presents a self-contained account for coupling arguments and applications in the context of Markov processes. We first use coupling to describe the transport problem, which leads to the concepts of optimal coupling and probability distance (or transportation-cost), then introduce applications of coupling to the study of ergodicity, Liouville theorem, convergence rate, gradient estimate, and Harnack inequality for Markov processes.