Heat Kernel and Analysis on Manifolds

By Alexander Grigorʼyan. 2012

959 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/95dc0361244bff8fbe0e10b3fa84ac295cceecad

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

Citations in the corpus

A formula for the time derivative of the entropic cost and applications (2019, background): “Within our framework it is also well known (see for instance [20]) that there exists a unique kernel (transition probability) pt(x, y) representing Pt in the following sense: Ptu(x) = ∫ u(y)pt(x, y)m(dy), ∀u ∈ L∞(m).”
Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincar\'e inequalities (2012, background): “quent applications. Notice that it is well-known, in the weighted case, that non-uniqueness issues may arise even for bounded initial data, this being true even for the linear heat equation (see e.g. [28] for examples on this latter fact in the manifold setting – this is related to stochastic comple”
Heat Kernel Asymptotics for the Measurable Riemannian Structure on the Sierpinski Gasket (2012, background): “3) is known to hold for Riemannian manifolds which are either compact or complete with non-negative Ricci curvature; see [8,15,30,33,35,36] and references therein.”
Obata’s Rigidity Theorem for Metric Measure Spaces (2014, neutral): “2) that yields ultracontractivity (see [12]).”
Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds (2022, background): “The fact that the isoperimetric proﬁle controls the whole scale of p -isoperimetric proﬁles is classical [39, 49].”
Analysis and Geometry of Markov Diffusion Operators (2013, background): “looking at the way a state space reacts to an evolution, leads to a very efficient description of its geometry (see [1, 3]).”
Embeddings of Riemannian Manifolds with Heat Kernels and Eigenfunctions (2013, background): “ive estimate on the harmonic radius rh. 6 JACOBUS W. PORTEGIES 3. Someheat kernel estimates We will now recall some properties of the decay of the heat kernel, following mostly the book by Grigor’yan [26]. We will show how these estimates imply heat kernel decay in coordinates. Moreover, we will see”
Parabolicity and stochastic completeness of manifolds in terms of the Green formula (2013, background): “, [24]), but we give an alternative proof for the sake of the completeness.”
Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries (2011, methodology): “stic completeness of the heat semigroup is classically equivalent to the uniqueness in the Cauchy problem for initial data in L∞(M). Following the classical approach (see for instance Theorem 8.18 in [33]), we in fact obtain: Proposition 4.4. Suppose that Msatisfy Hypothesis 1.1, Hypothesis 1.4. The”
Uniqueness of blowups and Łojasiewicz inequalities (2013, methodology): “tion by parts. The second follows from [BE] since Σ has positive Bakry-Emery Ricci curvature and ﬁnite weighted volume. Finally, the last´ claim is a consequence of the ﬁrst two (cf. theorem 10.20 in [Gr]). We will also use the following Gaussian Poincar´e inequality on Σ = Sk√ 2k × Rn−k. The middle”
The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour (2016, background): “o be complete, simply connected Riemannian manifolds with nonpositive sectional curvatures. It is known that they are dieomorphic to Rn and that the cut locus of any point ois empty (see, e.g. [10], [12]). One then denes polar coordinates around any given pole o. Set indeed, given any x2Mnfog, r(x) ”