Shot Noise Model (SNM) in Stochastic Analysis

• Shot Noise Model is a framework that quantifies fluctuations from discrete, stochastic events using Poisson or generalized renewal processes.
• Its methodology involves representing the observable as a sum of randomly timed impulses and deriving metrics such as power spectral density to understand noise characteristics.
• Applications span electronics, photonics, neuroscience, and finance, demonstrating the model's broad impact on analyzing systems with granular event arrivals.

A Shot Noise Model (SNM) provides a quantitative framework for fluctuations arising from the discrete, stochastic arrival of events in systems where the physical origin is the granularity of charge, photon, atom, or analogous entities. SNMs play a foundational role in fields as varied as electronic and photonic device physics, quantum transport, signal processing, stochastic systems theory, neuroscience, astrophysics, nonlinear optics, finance, and information systems. While the mathematical specifics vary with context, the defining principle is the explicit modeling of fluctuations imposed by the discrete nature of the fundamental carrier—represented as a Poisson or generalized renewal process, often with amplitude and temporal structure.

1. Fundamental Mathematical Structure

The canonical formulation of an SNM characterizes the observable as a sum of randomly timed, statistically independent pulses. For one-dimensional processes, the instantaneous signal—current, photon count, or analogue—is written as

$I(t) = \sum_{i} q_i\, \delta(t-t_i)$

where $q_i$ are the magnitudes ("quantum," not necessarily elementary charge) and $\{t_i\}$ are arrival times, typically modeled as points of a (possibly inhomogeneous) Poisson process or renewal process (Chen, 2018). The SNM postulates:

• The mean rate of arrivals is governed by $\lambda$.
• The distribution of $q_i$ reflects the physics of the process (e.g., charge $e$, photon energy $h\nu$).
• The observable is taken as the sum or convolution of impulses over some response function $g(t)$, introducing colored noise if $g$ is not delta.

In the time-integrated form, total quantities (charge, photon number, etc.) over interval $\Delta t$ are Poisson-distributed with mean and variance both equal to the expected number of arrivals.

The frequency-domain counterpart gives the power spectral density (PSD) for shot noise as

$q_i$0

where $q_i$1 is the average current, and the spectrum is white (flat) up to the frequency response roll-off of the measurement system (Chen, 2018).

Generalizations include:

• Non-Poissonian arrivals, time-dependent or correlated processes;
• Random parameter SNMs where shot amplitudes, decay times, or process rates are themselves stochastic, leading to modified cumulant structures and Volterra/delay equation representations (Chamayou, 2013).

2. Physical Domains and Instantiations

2.1 Electronics and Photon Detection

The SNM is foundational