Short-time large deviations of first-passage functionals for high-order stochastic processes

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.