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  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 \(b(x)\) : 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...

  In the context of large deviation theory, it is valuable to explore how  Sanov’s theorem and the Girsanov transformation interact in the context of two fundamental diffusion models: Brownian motion and the Ornstein–Uhlenbeck process. 1. Introduction Sanov’s theorem describes the exponential decay of probabilities associated with atypical empirical distributions of i.i.d. random variables. Girsanov’s theorem, on the other hand, provides a way to modify the drift of a stochastic process by changing the underlying probability measure. Together, they form a natural bridge between empirical deviations and pathwise deviations in diffusion processes. Notation For clarity, we summarize the variables and symbols used throughout this post: \(X_t\) : generic diffusion process. \(B_t\) : Brownian motion with volatility \(\sigma\). \(\sigma\) : diffusion coefficient (scalar or matrix). \(W_t\) : standard Brownian motion under the reference measure...

Biological and neural systems operate under substantial intrinsic stochasticity. Despite this, they maintain stable distributions of molecular, cellular, and population-level states. Occasionally, however, these systems exhibit rare, high-impact deviations. Sanov’s theorem provides a unifying framework for quantifying the improbability of such events through the geometry of empirical distributions. 1. Introduction Rare events in living systems often correspond not to isolated fluctuations but to atypical empirical distributions emerging from many interacting components. Sanov’s theorem characterizes the exponential decay of the probability of such deviations using the Kullback–Leibler divergence. This post surveys applications across molecular biology, cell dynamics, and neuroscience. Notation The following symbols and variables are used throughout this post: \(X_1, \dots, X_n\) : observations or molecular states sampled from a biological system. \(P\) : bas...

1. Introduction Evaluating the difference between two probability distributions is an important task in many fields of statistics, machine learning, and data science. For example, we may want to compare the performance of different models, test hypotheses about the underlying data-generating process, or measure the divergence or similarity between two populations or groups. There are different ways to evaluate the difference between two probability distributions, depending on the type and nature of the data, the assumptions we make about the distributions, and the goal of the analysis. Some common methods are: ·          The Kolmogorov-Smirnov test : This is a non-parametric test that compares the empirical cumulative distribution functions (CDFs) of two samples and measures the maximum vertical distance between them. The test can be used to check if two samples come from the same distribution or if one distribution is stochastically larger th...