Ben Arous, Fribergh and Sidoravicius \cite{ GAV2014} proved that speed of biased random walk $RW_\lambda$ on a Galton-Watson tree without leaves is strictly decreasing for $\lambda\leq \frac{ m_1}{ 1160},$ where $m_1$ is minimal degree of the Galton-Watson tree. And A\"{ \i}d\'{ e}kon \cite{ EA2013} improved this result to $\lambda\leq \frac{ 1}{ 2}.$ In this paper, we prove that for the $RW_{ \lambda}$ on a Galton-Watson tree without leaves, its speed is strictly decreasing for $\lambda\in \left[0,\frac{ m_1}{ 1+\sqrt{ 1-\frac{ 1}{ m_1}}}\right]$ when $m_1\geq 2;$ and we owe the proof to A\"{ \i}d\'{ e}kon \cite{ EA2013}.
Categories: math.PR