Orateur
Prof.
Loïc Chaumont
(Université d'Angers)
Description
We show that any $\mathbb{R}^d$-valued self-similar Markov process $X$ with index $\alpha>0$ absorbed at
0, can be represented as a path transformation of some Markov additive process (MAP) $(\theta,\xi)$ in $S_{d-1}\times\mathbb{R}$.
This result extends the well known Lamperti transformation. Then we prove that the
same transformation of the dual MAP in the weak sense of $(\theta,\xi)$ is itself in weak duality with $X$, with respect to the
measure $\pi(x/\|x\|)\|x\|^{\alpha-d}dx$, if and only if $(\theta,\xi)$ is reversible with respect to the measure $\pi(s)ds$, where $ds$ is
the Lebesgue measure on $S_{d-1}$. Besides, the dual process $\widehat{X}$ has the same law as the inversion
$(X_{\gamma_t}/\|X_{\gamma_t}\|^2,t\ge0)$ of $X$, where $\gamma_t$ is the inverse of $t\mapsto\int_0^t\|X\|_s^{-2\alpha}\,ds$.
As an application, we prove that in some instances, the Kelvin transform of $X$ can be obtained as an $h$-transform of some functional of $X$.
This is a joint work with Larbi Alili, Piotr Graczyk and Tomasz Zak.
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Auteur principal
Prof.
Loïc Chaumont
(Université d'Angers)
