Sign changes as a universal concept in first-passage-time calculations

First-passage-time problems are ubiquitous across many fields of study, including transport processes in semiconductors and biological synapses, evolutionary game theory and percolation. Despite their prominence, first-passage-time calculations have proven to be particularly challenging. Analytical results to date have often been obtained under strong conditions, leaving most of the exploration of first-passage-time problems to direct numerical computations. Here we present an analytical approach that allows the derivation of first-passage-time distributions for the wide class of nondifferentiable Gaussian processes. We demonstrate that the concept of sign changes naturally generalizes the common practice of counting crossings to determine first-passage events. Our method works across a wide range of time-dependent boundaries and noise strengths, thus alleviating common hurdles in first-passage-time calculations.

Figure 1

An OUP (solid black line) with mean $\overline{v}$ (dashed black line) crosses through time dependent boundaries $S\left(t\right)$ (green dashed line). The boundaries are (a) $S\left(t\right)=-5cos\left(t\right)exp\left\{-{[0.1cos\left(t\right)]}^2\right\}+t$ and (b) $S\left(t\right)={\sum }_n{\alpha }_n{\left[2(t-T_n)\right]}^{4}exp[-2(t-T_n)]H(t-T_n)$ with $\alpha \in \{0.8,0.75\}, T\in \{4,10\}$. Parameter values are (a) $\tau =1.0, \overline{v}=10.0, D=2.0$ and (b) $\tau =2.0, \overline{v}=4.0, D=0.2$. $H$ denotes the Heaviside step function.

Figure 2

Crossing probability ${\mathcal{I}}_1\left(t\right)$ for the two cases shown in Fig. 1 obtained from MC simulations (green circles) and from Eq. (4) (black dashed line).

Figure 3

FPT probability $\mathcal{F}\left(t\right)$ corresponding to Fig. 1 (a, b) and Fig. 1 (c, d) obtained from MC simulations (solid gray line), Eq. (4) (dashed green line), and Eq. (9) (green circles).

Figure 4

Crossing probability for two disjunct intervals ${\mathcal{I}}_2(s,t)$ for the two cases shown in Fig. 1 for a fixed time $t$ indicated by the solid vertical black line at $t=3.1$ (a) and $t=5.75$ (b) obtained from MC simulations (green circles) and Eq. (8) (black line).

Figure 5

(a, b) An OUP (solid black line) with mean $\overline{v}$ (dashed black line) crosses through a time dependent boundary $S\left(t\right)=-40cos\left(t\right)exp\left\{-{[4cos\left(t\right)]}^2\right\}+1.5t$ (green dashed line). Parameter values are $\tau =0.2, \overline{v}=7$ and (a) $D=2.0$, (b) $D=10$. (c, d) FPT probabilities $\mathcal{F}\left(t\right)$ corresponding to (a) and (b) obtained from MC simulations (solid gray line), Eq. (4) (dashed green line), and Eq. (9) (green circles).

Figure 6

(a) ${\mathcal{I}}_1\left(t\right)$ obtained from MC simulations (green circles) and Eq. (4) (red dashed line). (b) FPT distribution obtained from MC simulations (gray lines) and using for $p\left(x_0\right)=\delta (\overline{v}+2-x_0)$ (red). The boundary is given by $S\left(t\right)=-40cos\left(t\right)exp\left\{-{[4cos\left(t\right)]}^2\right\}+1.5t$, and the parameter values of the OUP are $\tau =10.0, \overline{v}=10.0, D=2.0$.