Multipliers and centralizers (Banach spaces)

In mathematics, Laplace's principle is a basic theorem in large deviations theory, similar to Varadhan's lemma. It gives an asymptotic expression for the Lebesgue integral of exp(−θφ(x)) over a fixed set A as θ becomes large. Such expressions can be used, for example, in statistical mechanics to determining the limiting behaviour of a system as the temperature tends to absolute zero.

Statement of the result

Let A be a Lebesgue-measurable subset of d-dimensional Euclidean space R^d and let φ : R^d → R be a measurable function with

$\underset{A}{\int }{e}^{-\phi (x)}\mathrm{d}x<+\mathrm{\infty }.$

Then

$\underset{\theta \to +\mathrm{\infty }}{\lim }\frac{1}{\theta }\log \underset{A}{\int }{e}^{-\theta \phi (x)}\mathrm{d}x=-{\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{f}}_{x\in A}\phi (x),$

where ess inf denotes the essential infimum. Heuristically, this may be read as saying that for large θ,

$\underset{A}{\int }{e}^{-\theta \phi (x)}\mathrm{d}x\approx \exp (-\theta {\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{f}}_{x\in A}\phi (x)).$

Application

The Laplace principle can be applied to the family of probability measures P_θ given by

${\mathbf{P}}_{\theta }(A)=\left(\underset{A}{\int }{e}^{-\theta \phi (x)}\mathrm{d}x\right)/\left(\underset{{\mathbf{R}}^d}{\int }{e}^{-\theta \phi (y)}\mathrm{d}y\right)$

to give an asymptotic expression for the probability of some set/event A as θ becomes large. For example, if X is a standard normally distributed random variable on R, then

$\underset{\epsilon \downarrow 0}{\lim }\epsilon \log \mathbf{P}[\sqrt{\epsilon }X\in A]=-{\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{f}}_{x\in A}\frac{x^2}{2}$

for every measurable set A.

References

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