A variation on the Donsker–Varadhan inequality for the principal eigenvalue
2017 ◽
Vol 473
(2204)
◽
pp. 20160877
Keyword(s):
The purpose of this short paper is to give a variation on the classical Donsker–Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain Ω by the largest mean first exit time of the associated drift–diffusion process via λ 1 ≥ 1 sup x ∈ Ω E x τ Ω c . Instead of looking at the mean of the first exit time, we study quantiles: let d p , ∂ Ω : Ω → R ≥ 0 be the smallest time t such that the likelihood of exiting within that time is p , then λ 1 ≥ log ( 1 / p ) sup x ∈ Ω d p , ∂ Ω ( x ) . Moreover, as p → 0 , this lower bound converges to λ 1 .
1968 ◽
Vol 19
(2)
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pp. 292-292
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2015 ◽
Vol 52
(3)
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pp. 649-664
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Keyword(s):
2015 ◽
Vol 52
(03)
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pp. 649-664
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1974 ◽
Vol 25
(3)
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pp. 422-424
1983 ◽
Vol 94
(2)
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pp. 328-337
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1982 ◽
Vol 17
(2)
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pp. 233-238
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1984 ◽
Vol 17
(1)
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pp. 31-44
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