A Guided Path Through the Large Deviations Series

-
  This post serves as a short guide to the four-part series on large deviations and their applications to stochastic processes, biology, and weak-noise dynamical systems. Each article can be read independently, but together they form a coherent narrative that moves from foundational principles to modern applications. 1. Sanov’s Theorem and the Geometry of Rare Events The series begins with an intuitive introduction to Sanov’s theorem , highlighting how empirical distributions deviate from their expected behavior and how the Kullback-Leibler divergence emerges as the natural rate functional. This post lays the conceptual groundwork for understanding rare events in high-dimensional systems. Read the post → 2. Sanov’s Theorem in Living Systems The second article explores how Sanov’s theorem applies to biological and neural systems . Empirical measures, population variability, and rare transitions in gene expression or neural activity are framed through ...
Post a Comment

Freidlin–Wentzell Theory and Rare Transitions in Diffusion Processes

-

 

After introducing Sanov’s theorem, Girsanov transformations, and their applications to diffusion processes, we now turn to the broader framework provided by Freidlin–Wentzell theory. This theory describes how stochastic systems with weak noise escape from basins of attraction, offering a geometric and variational perspective on rare transitions, metastability, and escape times.

Notation

For clarity, we summarize the variables and symbols used throughout this post:

  • \(X_t^\varepsilon\): diffusion process with small noise intensity \(\varepsilon\).
  • \(\varepsilon\): noise amplitude, assumed to satisfy \(0 < \varepsilon \ll 1\).
  • \(b(x)\): deterministic drift field defining the noiseless dynamics \(\dot{x} = b(x)\).
  • \(\sigma\): diffusion coefficient (scalar or matrix), assumed non-degenerate.
  • \(W_t\): standard Brownian motion.
  • \(\phi(t)\): candidate path used in the variational formulation of large deviations.
  • \(\dot{\phi}(t)\): time derivative of the path \(\phi(t)\).
  • \(I_{0,T}(\phi)\): Freidlin–Wentzell action functional over the interval \([0,T]\).
  • \(\|\cdot\|_{\sigma^{-1}}\): norm induced by the inverse diffusion matrix, \(\|v\|_{\sigma^{-1}}^2 = v^\top (\sigma\sigma^\top)^{-1} v\).
  • \(V(x,y)\): quasi-potential, i.e., minimal action required to move the system from \(x\) to \(y\).
  • \(\tau_{\text{escape}}\): first exit time from a basin of attraction.

1. Small-noise diffusions

Consider a diffusion process with small noise:

\[ dX_t^\varepsilon = b(X_t^\varepsilon)\,dt + \sqrt{\varepsilon}\,\sigma\,dW_t, \]

where \(0 < \varepsilon \ll 1\). When the noise is weak, the process typically follows the deterministic flow \(\dot{x} = b(x)\), but occasionally performs rare transitions between attractors.

2. The Freidlin–Wentzell action functional

The probability of observing an atypical trajectory \(\phi\) on \([0,T]\) is governed by the action functional:

\[ I_{0,T}(\phi) = \frac{1}{2} \int_0^T \|\dot{\phi}(t) - b(\phi(t))\|^2_{\sigma^{-1}}\,dt. \]

This functional is the continuous analogue of the rate functional in Sanov’s theorem: it quantifies the “energetic cost” required to force the system to follow a non-typical path.

3. Quasi-potential and metastability

The quasi-potential between two points \(x\) and \(y\) is defined as:

\[ V(x,y) = \inf_{\phi(0)=x,\;\phi(T)=y} I_{0,T}(\phi). \]

It represents the minimal cost needed to push the system from \(x\) to \(y\). When \(x\) and \(y\) belong to different basins of attraction, the quasi-potential determines the escape time:

\[ \mathbb{E}[\tau_{\text{escape}}] \asymp \exp\!\left(\frac{V(x, \partial A)}{\varepsilon}\right). \]

This exponential scaling is the hallmark of metastability.

4. Rare transitions and optimal paths

Rare transitions occur along optimal paths, i.e., trajectories that minimize the Freidlin–Wentzell action. These paths are the continuous analogue of the optimal empirical measures in Sanov’s theorem and the drift-adjusted dynamics obtained via Girsanov transformations.

In many systems, the optimal path satisfies the reverse-time equation:

\[ \dot{\phi}(t) = b(\phi(t)) - \sigma^2 \nabla V(\phi(t)). \]

This describes the most probable route by which the system crosses an energy barrier.

5. Connections to Schrödinger bridges and optimal transport

The Freidlin–Wentzell framework is deeply connected to several modern mathematical structures:

  • Schrödinger bridges: stochastic optimal transport with entropic regularization;
  • Optimal transport: small-noise limits of quadratic transport costs;
  • Girsanov transformations: the change of measure that realizes the optimal path;
  • Importance sampling: constructing tilted measures for efficient rare-event simulation.

These connections make Freidlin–Wentzell theory a natural bridge between stochastic analysis, statistical physics, and modern machine learning.

6. Concluding remarks

Freidlin–Wentzell theory provides a geometric and variational description of rare transitions in weak-noise dynamical systems. Together with Sanov’s theorem and Girsanov transformations, it forms a unified framework for understanding empirical deviations, pathwise deviations, and optimal changes of measure in diffusion processes.

Comments

Post a Comment

Popular posts from this blog

Understanding Anaerobic Threshold (VT2) and VO2 Max in Endurance Training

-
Image
  Introduction: The Science Behind Ventilatory Thresholds VT1 vs. VT2: Defining Energy Transitions p (Aerobic Threshold) : The transition from lipid metawwwpp (fat-burning) to a mixed metabolism, where both carbohydrates and fats fuel muscle activity. V0T2 (Anaerobic Threshold) : The point where the  shifts to a predominantly carbohydrate-based wmetabolism , leading to rapid lactate waccumpulation and increased reliance on anaerobic energy pathways. www w In p VT1 and VT2 Wwww ewAgep & Sex Younger athletes generally exhibit higher VT2 values , while aging naturally reduces aerobicw capacity. Men tend to have a higher VO2 Max due to greater lung capacity and muscle mass, but women can achieve similar endurance levels through optimized training. Body Composition & Health Status High body fat percentage may reduce VT2 efficiency, as excess weight increases ...
Read more

Owen's Function: A Simple Solution to Complex Problems

-
Definition   If you are interested in statistics or probability, you may have encountered the Owen's function , a special function that arises in various applications involving bivariate normal distributions. Let’s explain what Owen's function is, how it is defined and notated, and why it is useful.  The Owen's function is defined as follows: for any real numbers h and a, Owen's function T(h,a) is given by T ( h , a ) = 1 2 π · ∫ 0 a exp ( - h 2 2 · ( 1 ...
Read more

Cell Count Analysis with cycleTrendR

-
Image
  1. Introduction Long‑term monitoring of adherent cell cultures (e.g., Vero, MDCK) is central to vaccine development, viral propagation, and QC workflows in CRO environments. Standard analyses typically focus on short‑term growth metrics such as doubling time, confluence progression, and endpoint viability. While informative, these metrics overlook slow drifts, cyclic environmental influences, and structural transitions that emerge over multi‑day or multi‑week monitoring. Operational routines (e.g., weekly medium changes), environmental oscillations (e.g., daily temperature cycles), and metabolic rhythms can introduce periodic components into cell‑density trajectories. These patterns are rarely analyzed explicitly, despite their potential impact on viral yield, infection kinetics, and batch‑to‑batch reproducibility. This post demonstrates how cycleTrendR can extract long‑term trends, dominant cycles, and structural transitions from a realistically simulated adherent cell‑co...
Read more