I'm reading a book on stochastic processes at the moment, and I have come across the following definitions.
- The Laplace transform of a function $X(t)$ is $E[\exp(-\lambda X(t))]$.
- The Laplace functional of the point process $N$ is the non-negative function given by $\Psi_N(f) = E[\exp(-N(f))]$.
These definitions look practically the same to me (are they basically just moment generating functions), could someone please explain the difference?
A Laplace transform is often used to simplify differential equations that would be normally be complex (e.g. step/Heaviside functions, unit impulses) into simple algebra, and then transform back to the original.
A Laplace functional may be the probability analog to the Laplace transform, but as I'm not familiar with probability, that's my best estimate.
You are correct they are generating functions, however.