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 $A={\int }_0^T{[x(t)]}^{n}dt$ 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 $A\to 0$, $P_{m,n}(A\mathrm{\mid }L)\sim \exp (-\frac{{\alpha }_{m,n}{L}^{2mn-n+2}}{D{A}^{2m-1}})$, where the explicit expressions for the exponents ${\alpha }_{m,0}$ and ${\alpha }_{m,1}$ for arbitrary $m$ are obtained.