scholarly journals On parabolic convergence of positive solutions of the heat equation

2021 ◽  
pp. 1-17
Author(s):  
Jayanta Sarkar
Keyword(s):  
Heat Equation ◽  
Positive Solutions

Gradient and Harnack-type estimates for PageRank

2020 ◽  
pp. 1-19
Author(s):  
Paul Horn ◽  
Lauren M. Nelsen
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Gradient Estimate ◽  
Personalized Pagerank ◽  
Geometric Properties ◽  
Algorithmic Design

Abstract Personalized PageRank has found many uses in not only the ranking of webpages, but also algorithmic design, due to its ability to capture certain geometric properties of networks. In this paper, we study the diffusion of PageRank: how varying the jumping (or teleportation) constant affects PageRank values. To this end, we prove a gradient estimate for PageRank, akin to the Li–Yau inequality for positive solutions to the heat equation (for manifolds, with later versions adapted to graphs).

scholarly journals Initial trace of positive solutions of a class of degenerate heat equation with absorption

2013 ◽  
Vol 33 (5) ◽  
pp. 2033-2063 ◽  
Author(s):  
Tai Nguyen Phuoc ◽  
◽  
Laurent Véron ◽  
Keyword(s):  
Heat Equation ◽  
Positive Solutions

Tail estimates for positive solutions of stochastic heat equation

Stochastics ◽  
2007 ◽  
Vol 79 (5) ◽  
pp. 407-417
Author(s):  
Habib Ouerdiane ◽  
José Luis da Silva
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Stochastic Heat Equation

scholarly journals The persistence of logconcavity for positive solutions of the one dimensional heat equation

1990 ◽  
Vol 48 (2) ◽  
pp. 246-263 ◽  
Author(s):  
G. Keady
Keyword(s):  
Maximum Principle ◽  
Heat Equation ◽  
Positive Solutions ◽  
Space Variable ◽  
Time Variable ◽  
One Dimensional ◽  
The Maximum Principle ◽  
Proof Techniques ◽  
The One

AbstractConsider positive solutions of the one dimensional heat equation. The space variable x lies in (–a, a): the time variable t in (0,∞). When the solution u satisfies (i) u (±a, t) = 0, and (ii) u(·, 0) is logconcave, we give a new proof based on the Maximum Principle, that, for any fixed t > 0, u(·, t) remains logconcave. The same proof techniques are used to establish several new results related to this, including results concerning joint concavity in (x, t) similar to those considered in Kennington [15].

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