On the properties of Laplace transform originating from one-sided Lévy stable laws

Abstract

We consider the conventional Laplace transform of $f(x)$, denoted by $\mathcal{L}[f(x);p]\equiv F(p)={\int }_0^{\infty }{e}^{-px}f(x)dx$ with $\mathfrak{Re}(p)>0$. 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 Lévy 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 $l,k=1,2,\dots$, $l<k$, 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 Lévy integral transformations.

References in corpus (2)

• Exact and explicit probability densities for one-sided Levy stable distributions
• On the Laplace transform of the Fréchet distribution

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• Theory of relativistic heat polynomials and one-sided Lévy distributions
• Operational solutions for the generalized Fokker-Planck and generalized diffusion-wave equations