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

$\int _{A}e^{-\varphi (x)}\,\mathrm {d} x<+\infty .$

Then

$\lim _{\theta \to +\infty }{\frac {1}{\theta }}\log \int _{A}e^{-\theta \varphi (x)}\,\mathrm {d} x=-\mathop {\mathrm {ess\,inf} } _{x\in A}\varphi (x),$

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

$\int _{A}e^{-\theta \varphi (x)}\,\mathrm {d} x\approx \exp \left(-\theta \mathop {\mathrm {ess\,inf} } _{x\in A}\varphi (x)\right).$

Application

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

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

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

$\lim _{\varepsilon \downarrow 0}\varepsilon \log \mathbf {P} {\big [}{\sqrt {\varepsilon }}X\in A{\big ]}=-\mathop {\mathrm {ess\,inf} } _{x\in A}{\frac {x^{2}}{2}}$

for every measurable set A.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Template:MathSciNet