On Kelvin transformation
The first main result of the paper states that an appropriately time-changed inversion of a rotationally invariant symmetric \(\alpha\)-stable process is equal in law to a Doob's \(h\)-transform of the process itself. More precisely, let \(0<\alpha \leq 2\), \(d\geq 1\), and let \((X_t: t\geq 0)\) be the rotationally invariant \(\alpha\)-stable process in \({\mathbb R}^d\), killed when it hits the origin (in case the process hits points). Define the additive functional \(A_t=\int_0^t | X_s| ^{-2\alpha}\), and denote by \(\tau_t\) its inverse. Let \(Tx=x/| x| ^2\) be the inversion in \({\mathbb R}^d\). Then the process \((TX_{\tau_t}: t\geq 0)\) has the same law as the process \((X_t:\, t\geq 0)\) conditioned in Doob's sense by the function \(h(x)=| x| ^{\alpha -d}\). As a consequence of this, the authors prove the second main result, a transformation rule for the Green function \(G_D\) of the process \(X_t\): If \(D\) is an open subset of \({\mathbb R}^d\setminus \{0\}\), then \(G_{TD}(x,y)=| x| ^{\alpha-d} | y| ^{\alpha -d} G_D(Tx,Ty)\).
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