On the properties of Laplace transform originating from one-sided L\'evy stable laws

TL;DR

This paper derives explicit inverse Laplace transform formulas involving one-sided Lévy stable distributions for rational stability parameters, extending known cases and analyzing the properties of these transformations.

Contribution

It provides closed-form inverse Laplace transforms for functions involving fractional powers, expressed through Lévy stable distributions for rational lpha, expanding previous results beyond lpha=1/2.

Findings

Explicit inverse Laplace transforms involving Lévy stable distributions for rational lpha.

Extension of known lpha=1/2 case to all rational lpha in (0,1).

Analysis of the integral kernels and Lévy integral transformations.

Abstract

We consider the conventional Laplace transform of $f(x)$, denoted by $\mathcal{L}[f(x); p]~\equiv~F(p)=\int_{0}^{\infty} e^{-p x} f(x) dx$ with ${\rm \mathfrak{Re}}(p) > 0$. For $0 < \alpha < 1$ we furnish the closed form expressions for the inverse Laplace transforms $\mathcal{L}^{-1}[F(p^{\alpha}); x]$ and $\mathcal{L}^{-1}[p^{\alpha-1}F(p^{\alpha}); x]$. In both cases they involve definite integration with kernels which are appropriately rescaled one-sided L\'{e}vy stable probability distribution functions $g_{\alpha}(x)$, $0 < \alpha < 1$, $x > 0$. Since $g_{\alpha}(x)$ are exactly and explicitly known for rational $\alpha$, \textit{i.e.} for $\alpha = l/k$ with $l, k=1, 2, \ldots$, $l < k$, our results extend the known and tabulated case of $\alpha = 1/2$ to any rational $0 < \alpha < 1$. We examine the integral kernels of this procedure as well as the resulting two kinds of L\'{e}vy integral transformations.