The Laplace–Stieltjes transform, named for
Pierre-Simon Laplace
Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
and
Thomas Joannes Stieltjes
Thomas Joannes Stieltjes ( , ; 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics ...
, is an
integral transform
In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily charac ...
similar to the
Laplace transform
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
. For
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real ...
s, it is the Laplace transform of a
Stieltjes measure, however it is often defined for functions with values in a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
. It is useful in a number of areas of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, including
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, and certain areas of
theoretical
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
and
applied probability
Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains.
Scope
Much research involving probability is done under the auspices of applied probability. However, while such re ...
.
Real-valued functions
The Laplace–Stieltjes transform of a real-valued function ''g'' is given by a
Lebesgue–Stieltjes integral of the form
:
for ''s'' a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
. 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
In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
on the region of integration. The most common are:
* The bilateral (or two-sided) Laplace–Stieltjes transform is given by
* The unilateral (one-sided) Laplace–Stieltjes transform is given by
The limit is necessary to ensure the transform captures a possible jump in at , as is needed to make sense of the Laplace transform of the
Dirac delta function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
.
* More general transforms can be considered by integrating over a contour in the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
; 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
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
of a
Stieltjes measure. To wit,
:
In particular, it shares many properties with the usual Laplace transform. For instance, the
convolution theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time dom ...
holds:
:
Often only real values of the variable ''s'' are considered, although if the integral exists as a proper
Lebesgue integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
for a given real value , then it also exists for all complex with .
The Laplace–Stieltjes transform appears naturally in the following context. If ''X'' is a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
with
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ever ...
''F'', then the Laplace–Stieltjes transform is given by the
expectation:
:
The Laplace-Stieltjes transform of a real random variable's cumulative distribution function is therefore equal to the random variable's
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
, but with the sign of the argument reversed.
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
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
. These are, however, important in connection with the study of
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
s that arise in
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how ...
,
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
, and
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
. The most important semigroups are, respectively, the
heat semigroup,
Riemann-Liouville semigroup, and
Brownian motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
and other
infinitely divisible process
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol.
From the time of the ancient Greeks, the philosophical nature of infinity has been the subject of man ...
es.
Let ''g'' be a function from
,∞) to a Banach space ''X'' of strongly bounded variation over every finite interval. This means that, for every fixed subinterval [0,''T''one has
:
where the
supremum is taken over all partitions of
:
The Stieltjes integral with respect to the vector measure ''dg''
:
is defined as a Riemann–Stieltjes integral">,''T''
:
The Stieltjes integral with respect to the vector measure ''dg''
:
is defined as a Riemann–Stieltjes integral. Indeed, if π is the tagged partition of the interval [0,''T''] with subdivision , distinguished points
and mesh size
the Riemann–Stieltjes integral is defined as the value of the limit
: