We consider high-order stochastic processes $x(t)$ described by the Langevin equation $\frac{ { { d^m}x\left( t \right)}}{ { d{ t^m}}}= \sqrt{ 2D} \xi(t)$, where $\xi(t)$ is a delta-correlated Gaussian noise with zero mean, and $D$ is the strength of noise. We focus on the short-time statistics of the first-passage functionals $A=\int_{ 0}^{ T} \left[ x(t)\right] ^n dt$ along the trajectories starting from $x(0)=L$ and terminating whenever passing through the origin for the first-time at $t=T$. Using the optimal fluctuation method, we analytically obtain the most likely realizations of the first-passage processes for a given constraint $A$ with $n=0$ and 1, corresponding to the first-passage time itself and the area swept by the first-passage trajectory, respectively. The tail of the distribution of $A$ shows an essential singularity at $A \to 0$, $P_{ m,n}(A |L) \sim \exp\left(-\frac{ \alpha_{ m,n}L^{ 2mn-n+2}}{ D A^{ 2m-1}} \right)$, where the explicit expressions for the exponents $\alpha_{ m,0}$ and $\alpha_{ m,1}$ for arbitrary $m$ are obtained.
Categories: cond-mat.stat-mech