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Statistics of first-passage Brownian functionals

We study the distribution of first-passage functionals ${\cal A}= \int_0^{t_f} x^n(t)\, dt$, where $x(t)$ is a Brownian motion (with or without drift) with diffusion constant $D$, starting at $x_0>0$, and $t_f$ is the first-passage time to the origin. In the driftless case, we compute exactly, for all $n>-2$, the probability density $P_n(A|x_0)=\text{Prob}.(\mathcal{A}=A)$. This probability density has an essential singular tail as $A\to 0$ and a power-law tail $\sim A^{-(n+3)/(n+2)}$ as $A\to \infty$. The former is reproduced by the optimal fluctuation method (OFM), which also predicts the optimal paths of the conditioned process for small $A$. For the case with a drift toward the origin, where no exact solution is known for general $n>-1$, the OFM predicts the distribution tails. For $A\to 0$ it predicts the same essential singular tail as in the driftless case. For $A\to \infty$ it predicts a stretched exponential tail $-\ln P_n(A|x_0)\sim A^{1/(n+1)}$ for all $n>0$. In the limit of large Péclet number $\text{Pe}= μx_0/(2D)\gg 1$, where $μ$ is the drift velocity, the OFM predicts a large-deviation scaling for all $A$: $-\ln P_n(A|x_0)\simeq\text{Pe}\, Φ_n\left(z= A/\bar{A}\right)$, where $\bar{A}=x_0^{n+1}/{μ(n+1)}$ is the mean value of $\mathcal{A}$. We compute the rate function $Φ_n(z)$ analytically for all $n>-1$. For $n>0$ $Φ_n(z)$ is analytic for all $z$, but for $-1<n<0$ it is non-analytic at $z=1$, implying a dynamical phase transition. The order of this transition is $2$ for $-1/2<n<0$, while for $-1<n<-1/2$ the order of transition changes continuously with $n$. Finally, we apply the OFM to the case of $μ<0$ (drift away from the origin). We show that, when the process is conditioned on reaching the origin, the distribution of $\mathcal{A}$ coincides with the distribution of $\mathcal{A}$ for $μ>0$ with the same $|μ|$.