Dynamical phase transition in the first-passage probability of a Brownian motion

We study the first-passage time distribution (FPTD) $F({t}_{f}|{x}_{0},L)$ 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 $(1/\ensuremath{\sigma})\phantom{\rule{0.16em}{0ex}}g({x}_{0}/\ensuremath{\sigma})$ of finite width $\ensuremath{\sigma}$. We show the averaged FPTD undergoes a sharp dynamical phase transition from a two-peak structure for $b=L/\ensuremath{\sigma}g{b}_{c}$ to a single-peak structure for $bl{b}_{c}$. This transition is generated by the competition of two characteristic timescales ${\ensuremath{\sigma}}^{2}/D$ 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 $g({x}_{0}/\ensuremath{\sigma})$. We show that this transition is robust: It is present for all initial conditions with a finite $\ensuremath{\sigma}$, in all dimensions, and also exists for more general stochastic processes going beyond free diffusion.