A variation on the Donsker-Varadhan inequality for the principal eigenvalue

Area of Science:

• Probability Theory
• Stochastic Analysis
• Partial Differential Equations

Background:

• The Donsker-Varadhan inequality relates eigenvalues of elliptic operators to mean first exit times.
• Classical bounds rely on the average exit time, which may not fully capture process behavior.

Purpose of the Study:

• To present a variation of the Donsker-Varadhan inequality.
• To establish a new lower bound for the first eigenvalue of a second-order elliptic operator.
• To utilize quantiles of first exit times instead of their mean.

Main Methods:

• Analysis of a drift-diffusion process on a bounded domain Ω.
• Introduction of p-th quantiles of first exit times, denoted as T_p.
• Derivation of a novel inequality relating the first eigenvalue (λ₁) to these quantiles.

Main Results:

• A new inequality is derived: λ₁ ≥ (1/T_p) log(1/p).
• This inequality provides a lower bound for the first eigenvalue using exit time quantiles.
• As p approaches 0, the derived bound converges to the classical eigenvalue λ₁.

Conclusions:

• The study successfully extends the Donsker-Varadhan inequality by incorporating exit time quantiles.
• The novel bound offers an alternative perspective for analyzing elliptic operators.
• This approach provides a tighter relationship between stochastic process exit times and spectral properties.