# Laplace–Stieltjes transform

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.

Definition

The "Laplace–Stieltjes transform" of a function "g": RR is the function:$\left\{mathcal\left\{L\right\}^*g\right\}\left(s\right) = int_\left\{-infty\right\}^\left\{infty\right\} mathrm\left\{e\right\}^\left\{-sx\right\},dg\left(x\right), quad s in mathbb\left\{C\right\},$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,&infin;) &rarr; R, in which case the we integrate between 0 and &infin;.

Properties

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,:$\left\{mathcal\left\{L\right\}^*\left(g * h\right)\right\}\left(s\right) = \left\{mathcal\left\{L\right\}^*g\right\}\left(s\right)\left\{mathcal\left\{L\right\}^*h\right\}\left(s\right),$(where each of these transforms exists).

Applications

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

ee also

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' ".::$\left\{mathcal\left\{L\right\}^*g\right\}\left(s\right) = \left\{mathcal\left\{L\right\}g\text{'}\right\}\left(s\right),$
* We can obtain the Fourier–Stieltjes transform of "g" (and, by the above note, the Fourier transform of "g' ") by::$\left\{mathcal\left\{F\right\}^*g\right\}\left(s\right) = \left\{mathcal\left\{L\right\}^*g\right\}\left(mathrm\left\{i\right\}s\right), quad s in mathbb\left\{R\right\}.$

Examples

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

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

References

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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Thomas Joannes Stieltjes — This article is about Thomas Joannes Stieltjes (pronounced sti:ltʃəs), the mathematician. For his father, the Dutch engineer and politician, see Thomas Joannes Stieltjes Snr. Infobox Scientist name = Thomas Joannes Stieltjes image width = caption …   Wikipedia

• Fourier transform — Fourier transforms Continuous Fourier transform Fourier series Discrete Fourier transform Discrete time Fourier transform Related transforms The Fourier transform is a mathematical operation that decomposes a function into its constituent… …   Wikipedia

• List of transforms — This is a list of transforms in mathematics.Integral transforms*Abel transform *Fourier transform **Short time Fourier transform *Hankel transform *Hartley transform *Hilbert transform **Hilbert Schmidt integral operator *Laplace transform… …   Wikipedia

• List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

• Fluid queue — In probability theory, a fluid queue is a mathematical model used to describe the fluid level in a reservoir subject to randomly determined periods of filling and emptying. The term dam theory was used in earlier literature for these models. The… …   Wikipedia

• LST — is a three character combination that may refer to:*London School of Theology * Landing Ship, Tank * Local Sidereal Time * Life Support Technician * Least slack time scheduling * Launceston Airport * Linear Stability Theory, a method to… …   Wikipedia

• Zeta function regularization — In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to superficially divergent sums. The technique is now commonly applied to problems in physics, but… …   Wikipedia

• Dirac delta function — Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention… …   Wikipedia

• Time-scale calculus — In mathematics, time scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying …   Wikipedia

• Hilbert space — For the Hilbert space filling curve, see Hilbert curve. Hilbert spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It… …   Wikipedia