The Laplace–Stieltjes transform of a real-valued functiong is given by a Lebesgue–Stieltjes integral of the form for s a complex number. As with the usual Laplace transform, one gets a slightly different transform depending on the domain of integration, and for the integral to be defined, one also needs to require that g be of bounded variation on the region of integration. The most common are:
The bilateral Laplace–Stieltjes transform is given by
The unilateral Laplace–Stieltjes transform is given by
More general transforms can be considered by integrating over a contour in the complex plane; see.
The Laplace–Stieltjes transform in the case of a scalar-valued function is thus seen to be a special case of the Laplace transform of a Stieltjes measure. To wit, In particular, it shares many properties with the usual Laplace transform. For instance, the convolution theorem holds: Often only real values of the variables are considered, although if the integral exists as a proper Lebesgue integral for a given real values = σ, then it also exists for all complex s with re ≥ σ. The Laplace–Stieltjes transform appears naturally in the following context. If X is a random variable with cumulative distribution functionF, then the Laplace–Stieltjes transform is given by the expectation:
Vector measures
Whereas the Laplace–Stieltjes transform of a real-valued function is a special case of the Laplace transform of a measure applied to the associated Stieltjes measure, the conventional Laplace transform cannot handle vector measures: measures with values in a Banach space. These are, however, important in connection with the study of semigroups that arise in partial differential equations, harmonic analysis, and probability theory. The most important semigroups are, respectively, the heat semigroup, Riemann-Liouville semigroup, and Brownian motion and other infinitely divisible processes. Let g be a function from one has where the supremum is taken over all partitions of The Stieltjes integralwith respect to the vector measuredg is defined as a Riemann–Stieltjes integral. Indeed, if π is the tagged partition of the interval with subdivision, distinguished points and mesh size the Riemann–Stieltjes integral is defined as the value of the limit taken in the topology on X. The hypothesis of strong bounded variation guarantees convergence. If in the topology of X the limit exists, then the value of this limit is the Laplace–Stieltjes transform of g.
Related transforms
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' .
We can obtain the Fourier–Stieltjes transform of g by
For an exponentially distributed random variable Y with rate parameterλ the LST is, from which the first three moments can be computed as 1/λ, 2/λ2 and 6/λ3.
