Analyticity of the entropy for some random walks (Q426207)

The author considers the entropy \(h_p\) and the linear drift \(l_p\) associated with a finitely supported probability measure \(p\), on a free group \(F\) with a finite number of generators. Let \(B\subset F\) be such a finite set that \(\bigcup_n B^n=F\). Denote by \(\mathcal P(B)\) the set of such probability measures \(p\) that \(p(x)>0\) if and only if \(x\in B\). It is shown that the functions \(p\mapsto h_p\) and \(p\mapsto l_p\) are real analytic on \(\mathcal P(B)\).