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            <title xml:lang="en">On inversions and Doob $h$-transforms of linear diffusions</title>
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                <term xml:lang="it">Diffusion process</term>
                <term xml:lang="it">Dual process</term>
                <term xml:lang="it">Doob conditional process</term>
                <term xml:lang="it">Inversion</term>
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              <p>Let $X$ be a regular linear diffusion whose state space is an open interval $E\subseteq\mathbb{R}$. We consider a diffusion $X^*$ which probability law is obtained as a Doob $h$-transform of the law of $X$, where $h$ is a positive harmonic function for the infinitesimal generator of $X$ on $E$. This is the dual of $X$ with respect to $h(x)m(dx)$ where $m(dx)$ is the speed measure of $X$. Examples include the case where $X^*$ is $X$ conditioned to stay above some fixed level. We provide a construction of $X^*$ as a deterministic inversion of $X$, time changed with some random clock. The study involves the construction of some inversions which generalize the Euclidean inversions. Brownian motion with drift and Bessel processes are considered in details.</p>
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