Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

Anker, Jean-Philippe; Papageorgiou, Effie; Zhang, Hongwei

doi:10.1016/j.jfa.2022.109828

Cited by 7 publications

(14 citation statements)

References 16 publications

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“…Let us comment about (1.5) and (1.6). Firstly, the L 1 convergence (1.5) holds without the bi-K-invariance restriction on initial data, as in the case of the distinguished Laplacian and in contrast to the case of the Laplace-Beltrami operator, [APZ23]. Also, the sup norm estimate (1.6) resembles the Euclidean setting and the case of the distinguished Laplacian ∆ d .…”

Section: Introductionmentioning

confidence: 96%

“…Recall that in hyperbolic spaces Brownian motion X t tends to escape to ∞ along geodesics, which means that it "remembers" at least the direction of the starting point x 0 . For full generalizations of the previous result in Riemannian symmetric spaces of noncompact type, we refer to the recent paper [APZ23], where some arguments in [Váz19] are also clarified. Note that these spaces have nonpositive sectional curvature.…”

Section: Introductionmentioning

confidence: 99%

“…Note that these spaces have nonpositive sectional curvature. Besides the Laplace-Beltrami operator, a second Laplacian is considered in [APZ23], the socalled distinguished Laplacian. In this paper we consider the Doob transform of the Brownian motion arising by the unique bi-K-invariant ground spherical function.…”

Section: Introductionmentioning

confidence: 99%

“…The change of measure (2.18) satisfying (2.19) and the associated opeartor change (2.20) are referred to as Doob's h-transform, which is a useful tool to describe Markov processes conditioned upon certain events.Remark 2.3. In[APZ23] the long-time asymptotics of the distinguished Laplacian were studied.…”

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confidence: 99%

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Asymptotics for the infinite Brownian loop on noncompact symmetric spaces

Papageorgiou¹

2023

Preprint

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The infinite Brownian loop on a Riemannian manifold is the limit in distribution of the Brownian bridge of length T around a fixed origin when T → +∞. The aim of this note is to study its long-time asymptotics on Riemannian symmetric spaces G/K of noncompact type and of general rank. This amounts to the behavior of solutions to the heat equation subject to the Doob transform induced by the ground spherical function. Unlike the standard Brownian motion, we observe in this case phenomena which are similar to the Euclidean setting, namely L 1 asymptotic convergence without requiring bi-K-invariance for initial data, and strong L ∞ convergence.where dµ(y) denotes the Riemannian measure.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Asymptotics for the infinite Brownian loop on noncompact symmetric spaces

Papageorgiou¹

2023

Preprint

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“…The connection between the long-time behavior of the solution u(t, x) of (1.1) for initial data u 0 ∈ L 1 (M, μ) (where μ is the Riemannian measure on M) and that of the heat kernel h t (x, y) has recently been the subject of extensive studies, see, for example, [7,19,30] or see [1,2,7,23] for variants and related questions. Denote by M = M dμ(x) u 0 (x) the mass of the initial data.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

Papageorgiou

2024

J. Evol. Equ.

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This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for $$L^1$$ L 1 initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as $$L^1$$ L 1 asymptotic convergence without the assumption of bi-K-invariance.

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