### Summary

Let μ be a probability measure on *R*^{d}with density *c*(exp(-2*U(x)*), where *U∈C*^{2}(R^{d}),
$$\left| {\nabla U(x)} \right|^2 - \Delta U(x) \to \infty $$
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
(1)
$$Var_\mu f\underline \leqslant c E_\mu \left| {\nabla f} \right|^2 $$
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 = λ_{1} < λ_{2} = ... = λ_{m} < λ_{m + 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},...,b_{m}∈R. Moreover, the constant *c* can be taken as (2λ_{2})^{−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*^{d}or some discrete cases.