In this presentation, we focus on the decay rate of the expected signature of a stopped Brownian motion; more specifically we consider two types of the stopping time: the first one is the Brownian motion up to the first exit time from a bounded domain $\Gamma$, denoted by $\tau_{\Gamma}$, and the other one is the Brownian motion up to $min(t, \tau_{\Gamma\})$. For the first case, we use the Sobolev theorem to show that its expected signature is geometrically bounded while for the second case we use the result in paper (Integrability and tail estimates for Gaussian rough differential equation by Thomas Cass, Christian Litterer and Terry Lyons) to show that each term of the expected signature has the decay rate like 1/ \sqrt((n/p)!) where p>2. The result for the second case can imply that its expected signature determines the law of the signature according to the paper (Unitary representations of geometric rough paths by Ilya Chevyrev)
