A new approach to the Martin boundary via diffusions conditioned to hit a compact set
Let \(L\) be a diffusion generator on an exterior domain. Let \(h\) be the smallest positive solution of \(Lh=0\) with boundary conditions \(h\equiv 1\) on \(\partial D\). Then the harmonic measure boundary at infinity for the \(h\)-transformed diffusion generator \(L^ h\) coincides with the Martin boundary for \(\widetilde {L}\) on \(\mathbb{R}^ d\), the formal adjoint of \(L\). Further the harmonic measure boundary at infinity for the \(\widetilde{h}\)-transformed diffusion generator \(\widetilde{L}^{\widetilde {h}}\) coincides with the Martin boundary for \(L\) on \(\mathbb{R}^ d\). The boundaries are independent of the domain. The impact of these theoretical results is shown by calculation of the boundaries for rotational invariant diffusion generators.
diffusion generator on an exterior domain
rotational invariant diffusion generators
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