Abstract:In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset $U$ of a Brelot $\mathcal{P}$-harmonic space $\Omega$ with countable base of open subsets and satisfying the axiom~$D$. When $\Omega$ satisfies the hypothesis of uniqueness, we define the Martin boundary of $U$ and the Martin kernel $K$ and we obtain the integral representation of invariant functions by using the kernel $K$. As an application of the integral representation we extend to the cone $\mathcal{S(U)}$ of nonnegative finely superharmonic functions in $U$ a partition theorem of Brelot. We also establish an approximation result of invariant functions by finely harmonic functions in the case where the minimal invariant functions are finely harmonic.
Keywords: finely harmonic function; finely superharmonic function; fine potential; fine Green kernel; integral representation; Martin boundary; fine Riesz-Martin kernel
DOI: DOI 10.14712/1213-7243.2019.019
AMS Subject Classification: 31D05 31C35 31C40