Stochastic Processes & Markov Dynamics

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Abstractions about randomness evolving through time, spanning continuous-time Markov processes (transition-rate matrices, Kolmogorov's criterion, continuous-time Markov chains), diffusion and Brownian-motion constructions (geometric Brownian motion, reflected Brownian motion, Brownian snake), and applied stochastic models (ruin theory, stochastic Petri nets, tau-leaping).

29 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Asymmetric simple exclusion process — A continuous-time interacting-particle model on a lattice where biased nearest-neighbor jumps occur only into vacant sites.
• Borel right process — A strong Markov process on a suitable Borel state space with right-continuous paths and a transition semigroup satisfying the regularity needed for potential theory.
• Brownian meander — A Brownian-motion-derived stochastic process on a fixed interval conditioned to remain nonnegative after leaving zero.
• Brownian snake — A Markov process taking values in finite stopped paths whose lifetime evolves like reflected Brownian motion and whose path tips encode spatial genealogies such as superprocesses.
• Carré du champ operator — A symmetric bilinear operator derived from a Markov generator that measures its failure to satisfy the Leibniz derivation rule and plays the role of a squared gradient.
• Continuous-time Markov chain — A stochastic process with the Markov property on a discrete state space whose state changes occur in continuous time according to exponential holding rates and transition intensities.
• Demon algorithm — A microcanonical Monte Carlo method that augments a simulated system with a small energy reservoir whose exchanges enforce fixed total energy and reveal temperature.
• Diffusion-limited aggregation — A stochastic growth process in which randomly diffusing particles irreversibly attach upon first contact with a cluster, producing branched scale-dependent aggregates.
• Donsker classes — Classes of measurable functions for which the centered empirical process converges weakly in a uniform-function space to a tight Gaussian limit.
• Excursion probability — The probability that a stochastic process crosses a specified threshold somewhere over a declared index region.
• Gamma process — A nondecreasing Levy process with independent stationary increments distributed according to a gamma law, used to model cumulative random growth, wear, or activity.
• Generalized Wiener process — A continuous-time diffusion formed by adding state- or time-dependent drift and volatility to Brownian noise, commonly written as a stochastic differential equation.
• Geometric Brownian motion — A positive continuous-time stochastic process whose logarithm follows Brownian motion with drift, equivalently solving a multiplicative-noise stochastic differential equation.
• Itô isometry — The equality identifying the second moment of an Itô integral with the expected time integral of the squared adapted integrand.
• Itô's formula — The continuous Brownian Itô change-of-variables rule, which adds a quadratic-variation second-order term to a smooth function's transformed differential.
• Kolmogorov's criterion — A cycle-product condition characterizing when a Markov chain is reversible with respect to a positive stationary measure.
• Kramers–Moyal expansion — An expansion of a Markov process master equation into an infinite series of state derivatives weighted by conditional jump moments.
• Lindley equation — The reflected random-walk recursion W_(n+1)=max(0,W_n+X_n), canonically describing successive waiting times in a single-server queue.
• Reflected Brownian motion — A Brownian diffusion constrained to a domain by a regulating process that pushes sample paths inward whenever they reach the boundary.
• Reflection principle (Wiener process) — Reflect a Wiener path after its first hitting time of a level to obtain another process with the same law, converting barrier-crossing events into endpoint-distribution identities.
• Renewal theory — A probability framework for processes that restart after independent identically distributed waiting times, studying event counts, ages, residual lives and rewards over repeated cycles.
• Risk of ruin — The probability that accumulated losses drive capital or a bankroll below a specified absorbing or operational threshold before a declared horizon or target is reached.
• Ruin theory — An actuarial-probability framework modeling an insurer's surplus under premium inflow and random claims to quantify the probability and timing of insolvency.
• Runge–Kutta method (SDE) — A family of time-stepping schemes that approximates stochastic differential equations by combining drift and diffusion evaluations with simulated stochastic increments.
• Stationary sequence — A sequence of random variables whose finite-dimensional joint distributions are invariant under shifts of the index origin.
• Stochastic Petri net — A Petri net whose enabled transitions fire after random delays governed by assigned rates or distributions.
• Tau-leaping — An approximate stochastic-simulation method that advances a reaction or event system by a finite time step while sampling multiple event counts from Poisson distributions.
• Transition-rate matrix — The infinitesimal generator of a finite-state continuous-time Markov chain, with nonnegative off-diagonal jump rates and rows summing to zero.
• Weighting pattern — Represent a linear time-varying system's zero-state input–output action by a two-time kernel formed from output, state-transition, and input maps, reducing to an impulse-response convolution kernel in the time-invariant case.