Potential theory in Lipschitz domains (Q2753434)

The author provides comparison theorems of life probability in a Lipschitz domain between Brownian motion and random walks. The main result is as follows: Let \(D\) be some globally Lipschitz domain. Then for all \(0< \varepsilon\leq 4/5\) there exists \(B> 0\), a positive constant, that only depends on \(d\) and \(A\), \(\varepsilon\) such that for all \(\mu\in \mathbb{P}(\mathbb{R}^d)\) that satisfy NEWLINE\[NEWLINE\int x d\mu(x)= 0;\quad \int x_ix_jd\mu(x)= \delta_{ij},\;i,j= 1,\dots, d;\quad \int|x|^B d\mu(x)\leq M_B<+\infty,NEWLINE\]NEWLINE we have NEWLINE\[NEWLINE|P(t,x)- P^\mu(t,x)|\leq C{P(t, x)\over\delta(t, x)^\varepsilon},\quad \delta(x)\geq C,\quad t\geq C,NEWLINE\]NEWLINE where \(C>0\) only depends on \(d\), \(A\), \(B\), \(M_B\) and \(\varepsilon\). If the measure is compactly supported we can take \(0< \varepsilon< 1\).NEWLINENEWLINENEWLINEThe paper is didactically written and self-contained.