Laplace-Stieltjes transform

The Laplace-Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is a transform similar to the Laplace transform. It is useful in a number of areas of mathematics, including functional analysis, and certain areas of theoretical and applied probability.

[edit] Definition

The Laplace-Stieltjes transform of a function g: R → R is the function

$\{\mathcal{L}^*g\}(s) = \int_{-\infty}^{\infty} \mathrm{e}^{-sx}\,dg(x), \quad s \in \mathbb{C},$

whenever the integral exists. The integral here is the Lebesgue-Stieltjes integral.

Often, s is a real variable, and in some cases we are interested only in a function g: [0,∞) → R, in which case the we integrate between 0 and ∞.

The Laplace-Stieltjes transform shares many properties with the Laplace transform.

One example is convolution: if g and h both map from the reals to the reals,

$\{\mathcal{L}^*(g * h)\}(s) = \{\mathcal{L}^*g\}(s)\{\mathcal{L}^*h\}(s),$

(where each of these transforms exists).

Laplace-Stieltjes transforms are frequently useful in theoretical and applied probability, and stochastic processes contexts. For example, if X is a random variable with cumulative distribution function F, then the Laplace-Stieltjes transform can be expressed in terms of expectation:

$\{\mathcal{L}^*F\}(s) = \mathrm{E}\left[\mathrm{e}^{-sX}\right].$

Specific applications include first passage times of stochastic processes such as Markov chains, and renewal theory. In physics, the transform is sometimes used to regularize sums in quantum field theory by means of heat kernel regularization.

The Laplace-Stieltjes transform is closely related to other integral transforms, including the Fourier transform and the Laplace transform. In particular, note the following:

If g has derivative g' then the Laplace-Stieltjes transform of g is the Laplace transform of g' .

$\{\mathcal{L}^*g\}(s) = \{\mathcal{L}g'\}(s),$

We can obtain the Fourier-Stieltjes transform of g (and, by the above note, the Fourier transform of g' ) by

$\{\mathcal{F}^*g\}(s) = \{\mathcal{L}^*g\}(\mathrm{i}s), \quad s \in \mathbb{R}.$

For an exponentially distributed random variable $Y$ the LST is,

$\tilde F_Y(s) = f_Y^*(s) = \int_0^\infty e^{-st} \lambda e^{-\lambda t} dt = \frac{\lambda}{\lambda+s}.$

Common references for the Laplace-Stieltjes transform include the following,

Apostol, T.M. (1957). Mathematical Analysis. Addison-Wesley, Reading, MA. (For 1974 2nd ed, ISBN 0-201-00288-4).
Apostol, T.M. (1997). Modular Functions and Dirichlet Series in Number Theory, 2nd ed. Springer-Verlag, New York. ISBN 0-387-97127-0.

and in the context of probability theory and applications,

Grimmett, G.R. and Stirzaker, D.R. (2001). Probability and Random Processes, 3nd ed. Oxford University Press, Oxford. ISBN 0-19-857222-0.