Analytical Survival Analysis of the Non-autonomous Ornstein-Uhlenbeck Process

TL;DR

This work addresses the survival probability of a periodic non-autonomous Ornstein–Uhlenbeck process by deriving analytical approximations to the escape rate and survival probability in the presence of an absorbing boundary at $\beta$. It develops two complementary approaches: (i) a matched asymptotic expansions method that partitions the domain into an interior region and a boundary layer, yielding a uniformly valid survival solution and a time-dependent escape rate $r(t)$; (ii) an integral-method approach based on first-passage relations that provides an alternative form for $r(t)$. Numerical comparisons show the asymptotic matched-expansions method to be more accurate across parameter regimes, especially for large $\beta$ and substantial time-periodic forcing. The resulting closed-form survival probability, involving $\operatorname{erfc}$ and exponential terms, offers a practical tool for predicting exit events in periodically driven stochastic systems with broad applications in climate science and engineering.

Abstract

The survival probability for a periodic non-autonomous Ornstein-Uhlenbeck process is calculated analytically using two different methods. The first uses an asymptotic approach. We treat the associated Kolmogorov Backward Equation with an absorbing boundary by dividing the domain into an interior region, centered around the origin, and a "boundary layer" near the absorbing boundary. In each region we determine the leading-order analytical solutions, and construct a uniformly valid solution over the entire domain using asymptotic matching. In the second method we examine the integral relationship between the probability density function and the mean first passage time probability density function. These allow us to determine approximate analytical forms for the exit rate. The validity of the solutions derived from both methods is assessed numerically, and we find the asymptotic method to be superior.

Figures (6)

• Figure 1: Factorization of $\rho(y,t)$ with an absorbing boundary at $y=\beta$. The black curve is $\rho_S(y,t)$ while the red curve is $\phi(y,t)$.
• Figure 2: Schematic of the non-autonomous Ornstein-Uhlenbeck problem addressed. We divide the time-dependent potential $U(X,t) = \frac{1}{2} a(t) X^2-f(t) X$ into two regions: a broad $O(1)$ region ($I$) that contains the minimum of the potential $X = 0$, and a narrow $O(1/\beta)$ boundary layer region ($II$) near $X = \beta$, where $\beta \gg 1$. In each region we find the asymptotically dominant solutions ($\phi^{out}$ in $I$ and $\phi^{in}$ in $II$) and then match them as described in Section \ref{['sec3']}.
• Figure 3: The root mean square error between the numerical ($S_{\textrm{num}}(0,t)$) and the analytical ($S_{\textrm{an}}(0,t)$) solutions for the survival probability for different values of $\beta$ and $K$. From left to right: $S_{\textrm{an}}(0,t)$ is obtained from the first method ($\textrm{RMSE}_1$), from Section \ref{['sec3']}, the second method ($\textrm{RMSE}_2$), from Section \ref{['sec4']}, and their difference (Difference).
• Figure 4: Plot of $S_{\textrm{num}}(x,t)$, $S_{\textrm{an}}(x,t)$ and their difference in function of $x$ and $t$. In each row, we used a different value of $\beta$ and different coefficients as reported in Appendix \ref{['secA3']}.
• Figure 5: First panel: Comparison of $h_1(t)$ and $h_2(t)$ as derived in Sections \ref{['sec3']} and \ref{['sec4']} respectively, using the time dependent coefficients of Eq. (\ref{['neg_params']}). Second panel: Comparison between the survival probabilities computed numerically ($S_{num}(t)$: blue lines), and the approximate analytical results described in Section \ref{['sec3']} ($S_1(t)$: orange lines), and in Section \ref{['sec4']} ($S_2(t)$: green lines).