One type of Laplace functional,[1][2] also known as a characteristic functional[a] is defined in relation to a point process, which can be interpreted as random counting measures, and has applications in characterizing and deriving results on point processes.
[5] Its definition is analogous to a characteristic function for a random variable.
The other Laplace functional is for probability spaces equipped with metrics and is used to study the concentration of measure properties of the space.
The Laplace functional characterizes a point process, and if it is known for a point process, it can be used to prove various results.
[2][6] For some metric probability space (X, d, μ), where (X, d) is a metric space and μ is a probability measure on the Borel sets of (X, d), the Laplace functional: The Laplace functional maps from the positive real line to the positive (extended) real line, or in mathematical notation: The Laplace functional of (X, d, μ) can be used to bound the concentration function of (X, d, μ), which is defined for r > 0 by where The Laplace functional of (X, d, μ) then gives leads to the upper bound: