We study the first-passage time distribution (FPTD) $F\left(t_f\right|x_0,L)$ for a freely diffusing particle starting at $x_0$ in one dimension, to a target located at $L$, averaged over the initial position $x_0$ drawn from a normalized distribution $(1/\sigma )g(x_0/\sigma )$ of finite width $\sigma$. We show the averaged FPTD undergoes a sharp dynamical phase transition from a two-peak structure for $b=L/\sigma >b_c$ to a single-peak structure for $b<b_c$. This transition is generated by the competition of two characteristic timescales ${\sigma }^2/D$ and $L^2/D$, where $D$ is the diffusion coefficient. A very good agreement is found between theoretical predictions and experimental results obtained with a Brownian bead whose diffusion is initialized by an optical trap which determines the initial distribution $g(x_0/\sigma )$. We show that this transition is robust: It is present for all initial conditions with a finite $\sigma$, in all dimensions, and also exists for more general stochastic processes going beyond free diffusion.

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• Brownian motion
• Diffusion
• Fluctuations & noise
• Random walks