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In this paper, we provide new results about an invariance of $p$-harmonicfunctions under boundary perturbations by using tug-of-war with noise; aprobabilistic interpretation of $p$-harmonic functions introduced byPeres-Sheffield in \cite{ps}. As a main result, when $E\subset\bo $ iscountable and $f\in C(\bo)$, we provide a necessary and sufficient conditionfor $E$ to guarantee that $H_g=H_f$ whenever $g=f$ on $\bo\setminus E$. Here$H_f$ and $H_g$ denote the Perron solutions of $f$ and $g$. It turns out that$E$ should be of $p$-harmonic measure zero with respect to $\Omega$. As aconsequence, we analyze a structure of a countable set of $p$-harmonic measurezero. In particular, we give some results for the subadditivity of $p$-harmonicmeasures and an invariance result for $p$-harmonic measures. In addition, theresults in this paper solve the problem regarding a perturbation point Bj\"orn\cite{Bjorn} suggested for the case of unweighted $\R^n$.

Tug-of-war with noise and an invariance of p-harmonic functions under boundary perturbations