Editing Fokker–Planck equation (section)

== Analytical Theory of Mean First Passage Time ==
In the theory of stochastic processes, the [[First-hitting-time model|mean first passage time]] (MFPT) is the expected time for a stochastic trajectory to reach a specified boundary or target region for the first time. For a diffusion process governed by the stochastic differential equation (SDE)

:<math>dX_t = b(X_t)\,dt + \sqrt{2\varepsilon}\,\sigma(X_t)\,dW_t,</math>

the evolution of the probability density <math>p(x,t)</math> is described by the Fokker–Planck equation:

:<math> \frac{\partial p}{\partial t} = -\nabla \cdot (b(x)p) + \varepsilon \nabla \cdot \left( D(x) \nabla p \right), </math>
where <math>D(x) = \sigma(x)\sigma(x)^T</math> is the diffusion tensor, and <math>\varepsilon \ll 1</math> is the noise intensity.

To compute the MFPT <math>u(x) = \mathbb{E}_x[\tau]</math>, where <math>\tau</math> is the first exit time from a domain <math>\Omega</math>, one solves the backward Kolmogorov equation, also known as the Dynkin equation:

:<math> \mathcal{L} u(x) = -1, \quad x \in \Omega; \qquad u(x) = 0, \quad x \in \partial\Omega, </math> 

with generator

:<math> \mathcal{L} = b(x) \cdot \nabla + \varepsilon \sum_{i,j} D_{ij}(x) \frac{\partial^2}{\partial x_i \partial x_j}. </math>

===Boundary Layers and WKB Asymptotics===

In the small noise regime (<math>\varepsilon \to 0</math>), solutions typically exhibit boundary layers near <math>\partial\Omega</math>, [[Narrow escape problem|where escape occurs]]. The MFPT can be approximated using the WKB (Wentzel–Kramers–Brillouin) ansatz:

:<math> u(x) \sim A(x) \exp\left( \frac{S(x)}{\varepsilon} \right), </math>

where <math>S(x)</math> is the quasi-potential or minimum action required for escape, and <math>A(x)</math> is a transport coefficient. The function <math>S(x)</math> solves a Hamilton–Jacobi equation and represents the most likely escape path under small random perturbations. These techniques were developed in the analytical framework of Zeev Schuss.<ref>{{Cite journal |last=Holcman |first=David |last2=Schuss |first2=Zeev |date=2018 |title=Asymptotics of Elliptic and Parabolic PDEs |url=https://link.springer.com/book/10.1007/978-3-319-76895-3 |journal=Applied Mathematical Sciences |language=en |doi=10.1007/978-3-319-76895-3 |issn=0066-5452}}</ref><ref>{{Cite journal |last=Schuss |first=Zeev |date=2010 |title=Theory and Applications of Stochastic Processes |url=https://link.springer.com/book/10.1007/978-1-4419-1605-1 |journal=Applied Mathematical Sciences |language=en |doi=10.1007/978-1-4419-1605-1 |issn=0066-5452}}</ref>

In systems with limit cycle attractors, such as those seen in oscillatory biological systems, the exit time distribution deviates from the classical Poisson law, the survival probability <math>P(t)</math> of the process decaying from the limit cycle is modulated in time:

:<math> P(t) = \mathbb{P}(\tau > t) \not\sim e^{-\lambda t}, </math>

Instead, the survival probability contains oscillatory terms reflecting the periodic nature of the attractor. The conditional exit time density <math>f(t)</math> is no longer exponential and is better described through an expansion in Hermite polynomials:

:<math> Pr(\tau >t) =\sum_{n=1}^\infty c_n \exp (- \lambda_n t) \cos (\omega n t) </math>,

where the coefficients <math>c_n</math> reflect how strongly the exit probability deviates from the exponential form due to phase preference along the limit cycle. This expansion reveals that escape occurs with higher probability at particular phases of the cycle, breaking the memoryless (Markovian) nature of classical escape theory.<ref>{{Cite journal |last=Dao Duc |first=K. |last2=Schuss |first2=Z. |last3=Holcman |first3=D. |date=January 2016 |title=Oscillatory Survival Probability: Analytical and Numerical Study of a Non-Poissonian Exit Time |url=https://epubs.siam.org/doi/abs/10.1137/151004100 |journal=Multiscale Modeling & Simulation |volume=14 |issue=2 |pages=772–798 |doi=10.1137/151004100 |issn=1540-3459}}</ref> The rate of escape depends not only on the noise intensity but also on geometric and dynamical anisotropies along the attractor. This phenomenon is particularly relevant for modeling neuronal excitability, biological clocks, and cardiac rhythms, where timing and variability of transitions are tightly regulated but also susceptible to random perturbations.

===Asymptotic MFPT in Gradient Systems===

In systems with small noise and a drift given by the gradient of a potential, <math>b(x) = -\nabla\phi(x),</math> the stochastic process

:<math> dX_t = -\nabla \phi(X_t)\,dt + \sqrt{2\varepsilon}\,dW_t </math>

models the overdamped Langevin dynamics of a particle in a potential landscape <math>\phi(x)</math>. The associated mean first passage time <math>u(x),</math> which satisfies the backward Kolmogorov equation:

:<math> \varepsilon \Delta u(x) - \nabla \phi(x) \cdot \nabla u(x) = -1, </math>

subject to <math>u=0</math> on the exit boundary <math>\partial\Omega_a \subset \partial\Omega,</math>  has the following asymptotic solution in the limit <math>\varepsilon\to 0,</math> when <math>x</math> is near a local minimum <math>x_0</math> of <math>\phi</math> and escape occurs over a saddle point <math>x_s</math> of the potential:

:<math> \mathbb{E}[\tau] \sim \frac{2\pi}{\sqrt{|\det H(x_s)|}} \cdot \frac{e^{[\phi(x_s) - \phi(x_0)]/\varepsilon}}{\sqrt{\det H(x_0)}}, </math>

where: 

* <math>H(x_0)</math> is the Hessian matrix of <math>\phi</math> at the stable point <math>x_0</math>,
* <math>H(x_s)</math> is the Hessian at the saddle point <math>x_s,</math> with one negative eigenvalue,
* <math>\phi(x_s) - \phi(x_0)</math> is the energy barrier or quasi-potential difference the system must cross.

This formula generalizes Kramers' escape time to n-dimensional gradient systems and shows the exponential sensitivity of MFPT to potential barriers, with prefactors determined by second-order variations (local curvatures) of the potential at critical points. This result connects with large deviation theory and WKB asymptotics, where the action functional (or quasi-potential) governs the probability of rare events. It underpins modern approaches to metastability in physics, chemistry, and biology—such as chemical reaction rates, ion channel gating, or noise-induced switching in gene networks.