排序方式: 共有26条查询结果,搜索用时 31 毫秒
1.
Karl-Theodor Sturm 《manuscripta mathematica》1992,76(1):367-395
We investigate the Feynman-Kac semigroupP t V and its densityp V(t,.,.),t>0, associated with the Schrödinger operator ?1/2Δ+V on ?d\{0}.V will be a highly singular, oscillating potential like $V\left( x \right) = k \cdot \left\| x \right\|^{ - 1} \cdot \sin \left( {\left\| x \right\|^{ - m} } \right)$ with arbitraryk, l, m>0. We derive conditions (onk,l,m) which are sufficientand necessary for the existence of constants α, β, γ, ∈ ? such that for allt, x, y p V(t, x, y)≤γ·p(βt, x, y)·eat. On the other hand, also conditions are derived which imply thatp V (t, x, y)≡∞ for allt, x, y. The aim is to see to which extent quick oscillations can lead to annihilations of the singularities ofV. For this purpose, we analyse the above example in great detail. Note that forl≥2 the potential is so singular that none of the usual perturbation techniques applies. 相似文献
2.
Tapio Rajala Karl-Theodor Sturm 《Calculus of Variations and Partial Differential Equations》2014,50(3-4):831-846
We prove that in metric measure spaces where the entropy functional is \(K\) -convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite second moments and this plan is given by a map. The results are applicable in metric measure spaces having Riemannian Ricci curvature bounded below, and in particular they hold also for Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below by some constant. 相似文献
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Karl-Theodor Sturm 《Probability Theory and Related Fields》1991,89(4):387-406
Summary LetV=(V
)0 be a (not necessarily sub-Markovian) resolvent such that the kernelV
for some 0 is compact and irreducible. We prove the following general gauge theorem: If there exists at least oneV-excessive function which is notV-inviriant, thenV
0 is bounded.This result will be applied to resolventsU
M
arising from perturbation of sub-Markovian right resolventsU by multiplicative functionalsM (not necessarily supermartingale), for instance, by Feynman-Kac functionals. Among others, this leads to an extension of the gauge theorem of Chung/Rao and even of one direction of the conditional gauge theorem of Falkner and Zhao. 相似文献
5.
Karl-Theodor Sturm 《Acta Mathematica》2006,196(1):65-131
We introduce and analyze lower (Ricci) curvature bounds
⩾ K for metric measure spaces
. Our definition is based on convexity properties of the relative entropy
regarded as a function on the L
2-Wasserstein space of probability measures on the metric space
. Among others, we show that
⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds,
⩾ K if and only if
⩾ K
for all
.
The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation.
We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence. 相似文献
6.
Shizan Fang Jinghai Shao Karl-Theodor Sturm 《Probability Theory and Related Fields》2010,146(3-4):535-565
The goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the Wiener space, based on and extending fundamental results of Feyel–Üstünel. Carrying out the program of Ambrosio–Gigli–Savaré, we present a complete characterization of the derivative processes for certain class of absolutely continuous curves. We prove existence of the gradient flow curves for the relative entropy w.r.t. the Wiener measure and identify these gradient flow curves with solutions of the Ornstein–Uhlenbeck evolution equation. 相似文献
7.
Karl-Theodor Eisele 《Insurance: Mathematics and Economics》2008,42(1):65-72
We show how to generalize the result given in [Eisele, K.-Th., 2006. Recursions for compound phase distributions. Insurance: Math. Econom. 38, 149-156] to the multivariate case, i.e. we find a Panjer-like recursion principle for the distribution of a multivariate compound phase variable. Recursion formulas and procedures for the bivariate case are given in detail. We give a possible application for agricultural risks and calculate concrete examples via a VB-program. 相似文献
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Shin-ichi?OhtaEmail author Karl-Theodor?Sturm 《Archive for Rational Mechanics and Analysis》2012,204(3):917-944
We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds.
Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from
the usual convexity along geodesics in non-Riemannian Finsler manifolds. As an application, we show that the heat flow on
Minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic Wasserstein distance. 相似文献
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