Bernstein-gamma functions and exponential functionals of Lévy processes
A talk in the Keine Reihe series by
Mladen Savov
| Abstract: | For any negative definite function $\Psi$ we consider a recurrent equation of the type $f(z+1)= \frac{-z}{\Psi(-z)}f(z)$. Using the Wiener-Hopf factorization of $\Psi$ we solve this equation in a three term product involving the solutions of $W_{\phi}(z+1) = \phi(z)W_{\phi}(z)$ on $\{ z \in \mathbb{C} : \mathcal{R}(z) > 0 \},$ where $\phi$ is any Bernstein function. We call $W_{\phi}(z)$ a Bernstein-gamma function and note that $W_{\phi}(z)$ has appeared in
more restricted context in some earlier studies. The Bernstein-gamma functions
are characterized as meromorphic functions on an identifiable complex strip.
This is achieved in terms of parameters depending on the input function $\phi$. Moreover, we establish universal, explicit Stirling type asymptotic of $W_{\phi}(z)$ . This
allows the thorough understanding of the decay of $| f(z) | $ at least along the
imaginary lines $z=a+i \mathbb{R}$, $a \in (0,1)$, and an access to quantities relevant
for some theoretical and applied studies in probability theory and other areas.
The foremost motivation for the aforementioned results is their application to
the study of an important class of non-self-adjoint Markov processes. However,
in this talk, as an application, we present some general results on the law of
the exponential functional of Lévy processes, that is $\int_0^{\infty} e^{-\xi_s} ds$, which are a
consequence of the understanding of $f$ and the fact that the Mellin transform of
the exponential functional satisfies the recurrent equation $f(z+1) = \frac{-z}{\Psi(-z)}f(z)$. We discuss results such as smoothness, large and small asymptotic, expansions,
bounds and Mellin Barnes representations. When $\int_0^{\infty} e^{-\xi_s} ds = \infty$ we study
under the Spitzer’s condition the weak convergence of the measures induced
from $\int_0^{\infty} e^{-\xi_s} ds$, as $t \rightarrow \infty$. The derivation of our results relies on analytical,
complex-analytical and probabilistic techniques. $$ $$
This talk is based on joint work with Pierre Patie, Cornell, USA. $$ $$
References:
[1] Patie, P. and Savov, M. (2018)"Bernstein-gamma functions and exponential
functionals of Lévy processes ", Electronic Journal of Probability, v.
75, p.1–101. |