Summary
Let μ be a probability measure on R dwith density c(exp(-2U(x)), where U∈C 2(Rd), \(\left| {\nabla U(x)} \right|^2 - \Delta U(x) \to \infty \) and U(x)→∞ as |x|→∞. By using stochastic analysis and theorems in Schrödinger operators we have the following result: there exists a constant c>0 such that
for any f∈L 1(μ) with a well-defined distributional gradient ∇f. Under our condition, the operator \( - \frac{1}{2}\Delta + \nabla U \cdot \nabla \) in L 2(μ) has discrete spectrum 0 = λ1 < λ2 = ... = λm < λm + 1 ≦ ... with corresponding eigenfunctions {φ n} which form a C.O.N.S. (complete orthonormal system). If the R.H.S. of (1) is finite then equality holds iff \(f = \sum\limits_{i = 1}^m {b_i \phi _i } \) for some b 1,...,bm∈R. Moreover, the constant c can be taken as (2λ2)−1.
When U is a quadratic form, (1) is the Chernoff inequality (Chernoff 1981; Chen 1982). The approach here can be generalized to infinite dimensional Gaussian measures, or the case with μ being a probability measure in a bounded domain of R dor some discrete cases.
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This research was partially supported by the National Science Council of the Republic of China
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Hwang, CR., Sheu, SJ. A generalization of Chernoff inequality via stochastic analysis. Probab. Th. Rel. Fields 75, 149–157 (1987). https://doi.org/10.1007/BF00320088
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DOI: https://doi.org/10.1007/BF00320088