Stochastic Processes & Markov Dynamics

← Back to Domain-Specific Families

Abstractions about random systems evolving through time, including Markov chains, martingales, Brownian and Poisson processes, filtration, renewal, stopping, and stochastic differential methods. They formalize transition rates, observability, stationarity, excursions, and probabilistic simulation.

38 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.
• Compound Poisson process — A jump process formed by summing independent random jump sizes at event times of a Poisson counting process.
• 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.
• Continuous-time stochastic process — A collection of random variables indexed by a continuous parameter set, usually a real time interval, without implying that its sample paths are continuous.
• Cox process — A point process that is conditionally Poisson given a random intensity measure, thereby representing clustered or environment-driven event rates.
• Discrete-time Markov chain — A stochastic sequence whose next-state distribution depends on the current state and transition step but not on the earlier path once the present is known.
• Ergodic process — A stochastic process for which specified long-run time averages along almost every realization equal the corresponding ensemble expectations, allowing one sufficiently long trajectory to represent the regime.
• Excursion probability — The probability that a stochastic process crosses a specified threshold somewhere over a declared index region.
• Filtering problem (stochastic processes) — The sequential inference problem of estimating a hidden stochastic state from noisy partial observations available up to the present time.
• 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.
• 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.
• Leimkuhler–Matthews method — A discretization of overdamped Langevin dynamics using correlated noise to improve configurational sampling accuracy.
• 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.
• Local martingale — A stochastic process that becomes a martingale when stopped along an increasing sequence of stopping times tending to the time horizon.
• Markov kernel — A measurable assignment sending each source point to a probability measure on a target space, generalizing a stochastic transition matrix to arbitrary measurable spaces.
• Martingale (probability theory) — Model an adapted integrable stochastic process whose conditional expected future value, given present information, equals its current value.
• Palm calculus — The probability calculus relating a stationary point process as seen from a typical event to its ordinary time- or space-average law.
• Partially observable Markov decision process — A sequential decision model with Markovian hidden states, stochastic observations and actions chosen from observation histories or belief-state probability distributions.
• Progressively measurable process — A stochastic process whose restriction through every time t is jointly measurable with respect to Borel time and the information available by t.
• Projection filters — Nonlinear state-estimation algorithms that approximate an evolving conditional probability density by projecting infinite-dimensional filtering dynamics onto a finite-dimensional statistical manifold.
• 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.
• 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 process — A stochastic process whose probabilistic law is invariant under shifts of its time index, with weaker forms preserving selected moments instead.
• 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.
• Stopping time — A random time whose occurrence can be determined from information available up to that time, without access to future states of the stochastic process.
• 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.
• Σ-Algebra of τ-past — The stopped sigma-algebra Fτ containing exactly those events whose truth is knowable by a stopping time τ, defined by compatibility of each event with {τ≤t} and the filtration Ft.