1 Fundamental concepts
Path space is the collection of all possible trajectories of a stochastic process, treated as a mathematical object in its own right. Rather than focusing only on values at isolated times, it records entire realizations over a time index set. This viewpoint is useful because many properties of a process are most naturally expressed in terms of complete paths.
1.1 Sample paths
A sample path is one realization of a stochastic process. For each outcome in the underlying probability model, the process produces a function that assigns a state to every time point in the index set. Sample paths may be continuous, piecewise constant, or irregular, depending on the process.
1.2 Time index sets
The time index set specifies the times at which the process is observed. It is often a subset of the real line, such as discrete time or continuous time, but more general index sets are also possible. The choice of index set affects the structure of the corresponding path space.
1.3 State spaces
The state space is the set of values that the process can take at each time. Common examples include the real numbers, Euclidean spaces, or more abstract metric spaces. A path is therefore a function from the time index set into the state space.
1.4 Trajectory interpretation
Under the trajectory interpretation, each point in path space represents a full evolution of the process. This allows one to study not only individual time marginals, but also the joint behavior across time. In many settings, path space is the natural domain for defining the law of a process.
2 Mathematical structure
Path space is typically modeled as a function space equipped with a topology and a sigma-algebra. These structures make it possible to discuss measurability, continuity, and convergence of stochastic processes. The precise choice depends on the regularity of the paths under consideration.
2.1 Function spaces
At its core, path space is a set of functions from time into state space. The choice of function class reflects the expected regularity of the trajectories. Some processes live naturally in spaces of continuous functions, while others require spaces that allow jumps.
2.1.1 Continuous path spaces
Continuous path spaces consist of functions whose trajectories vary continuously in time. They are used for processes such as Brownian motion and many diffusion models. These spaces often carry the uniform topology and a corresponding Borel sigma-algebra.
2.1.2 Cadlag path spaces
Cadlag path spaces contain functions that are right-continuous with left limits. This class is especially important for jump processes, including many Markov processes and counting processes. The space is commonly denoted by \(D\) and is equipped with the Skorokhod topology.
2.2 Sigma-algebras on path space
To treat paths probabilistically, one needs a sigma-algebra of measurable sets. This allows probability measures to be assigned to collections of trajectories. The most common sigma-algebras are generated either by coordinate projections or by the topology of the path space.
2.2.1 Product sigma-algebra
The product sigma-algebra is generated by cylinder sets determined by finitely many time coordinates. It is convenient because it is directly tied to finite-dimensional distributions. In many constructions, it is the minimal sigma-algebra needed to make all coordinate maps measurable.
2.2.2 Borel sigma-algebra
The Borel sigma-algebra is generated by the open sets of a chosen topology on path space. It is widely used when the space is equipped with a natural metric or topology. For well-behaved path spaces, the Borel and product viewpoints are closely related.
2.3 Topologies on path space
A topology on path space provides a notion of closeness between trajectories. This is essential for studying approximation and convergence of stochastic processes. Different topologies suit different classes of paths and different types of limiting behavior.
2.3.1 Uniform topology
The uniform topology measures the maximum deviation between two paths over the time interval. It is appropriate when paths are continuous and comparisons should be made pointwise in time. Convergence in this topology means that entire trajectories become uniformly close.
2.3.2 Skorokhod topology
The Skorokhod topology is designed for cadlag paths, where small time shifts may be allowed in comparisons. It captures convergence of jump processes more effectively than the uniform topology. This flexibility makes it central in functional limit theorems.
3 Probability measures on path space
A stochastic process can be viewed as a random element whose values are paths. Its distribution is then a probability measure on path space. This perspective turns questions about random evolution into questions about measures on function spaces.
3.1 Canonical process
The canonical process is the coordinate process on path space itself. At each time, it returns the value of the path at that time. When a measure is placed on path space, the canonical process becomes the stochastic process described by that measure.
3.2 Distribution of stochastic processes
The distribution of a stochastic process is the law induced on path space by the process. It summarizes all probabilistic information about trajectories. Two processes with the same distribution have the same pathwise behavior in law.
3.3 Finite-dimensional distributions
Finite-dimensional distributions describe the joint law of the process at finitely many time points. They are the basic building blocks for constructing a process on path space. Many results begin with these distributions before extending to the full trajectory law.
3.4 Consistency conditions
Finite-dimensional distributions must satisfy compatibility conditions in order to define a probability measure on path space. These conditions ensure that the marginal laws agree when coordinates are omitted. In classical constructions, such conditions are a key step toward existence theorems.
4 Common examples
Different stochastic processes lead to different path spaces, depending on their regularity and state structure. Standard examples illustrate how the abstract framework is used in practice. The same path-space language applies to both continuous and jump-type models.
4.1 Brownian motion path space
Brownian motion is naturally studied on the space of continuous functions. Its sample paths are almost surely continuous but highly irregular. This makes continuous path space the standard setting for its analysis.
4.2 Random walk path space
A random walk often lives on a discrete-time path space. Its trajectories are sequences of states indexed by integers. When scaled appropriately, these paths can converge to continuous limits such as Brownian motion.
4.3 Markov process path space
Markov processes are commonly represented on path spaces that allow either continuous or jump behavior. Their future evolution depends only on the present state, a property that can be expressed in terms of the canonical process and its filtration. Path space provides the setting in which transition mechanisms are studied.
4.4 Diffusion process path space
Diffusion processes typically have continuous trajectories and are modeled on continuous function spaces. They are used to describe random motion influenced by drift and noise. Their path laws are central in both theory and applications.
5 Canonical constructions
Canonical constructions identify the sample space with a path space itself. This removes the need to specify an external probability space in detail. Instead, the trajectories and their coordinate maps carry the main structure.
5.1 Canonical sample space
The canonical sample space is the set of all admissible paths for the process. It serves as the underlying space on which the probability measure is defined. In this framework, each outcome corresponds directly to a trajectory.
5.2 Coordinate mappings
Coordinate mappings extract the value of a path at a chosen time. They are basic measurable functions on path space and generate much of its probabilistic structure. Through them, one recovers finite-dimensional distributions from the full path law.
5.3 Evaluation maps
Evaluation maps are another name for the mappings that send a path to its value at a specified time. They are fundamental in defining measurability and in linking trajectories to random variables. Their joint behavior across times determines many properties of the process.
5.4 Filtrations generated by paths
A filtration generated by paths is the increasing family of sigma-algebras formed from information observed up to each time. It formalizes the idea of available information as the path unfolds. This structure is crucial in martingale theory, stopping times, and process dynamics.
6 Convergence and approximation
Path space is especially important when studying limits of stochastic processes. Many sequences of processes are analyzed by examining whether their path laws converge to a limit measure. Approximation methods often rely on simpler discrete models.
6.1 Weak convergence of measures
Weak convergence describes the convergence of probability measures on path space. It means that expectations of suitable test functions converge. This notion is central to understanding limiting behavior of stochastic processes.
6.2 Tightness criteria
Tightness criteria provide conditions ensuring that a family of measures does not spread out too widely. They are often needed to prove the existence of convergent subsequences. In path spaces, tightness usually depends on controlling oscillations of trajectories.
6.3 Functional limit theorems
Functional limit theorems identify the limit of a sequence of stochastic processes as a process-valued random element. These results extend classical limit theorems from numbers to whole trajectories. They are a major reason path space is so widely used.
6.4 Approximation by discrete paths
Many continuous processes are approximated by discrete paths such as lattice walks or time-discretized schemes. Such approximations are useful in simulation and in theoretical proofs. Path-space convergence provides the framework for justifying these approximations.
7 Applications
Path space appears throughout modern probability theory because it allows processes to be studied as objects with entire time evolution. It unifies existence, regularity, approximation, and limiting arguments. The same framework also supports more specialized analytical techniques.
7.1 Stochastic process theory
In stochastic process theory, path space provides the standard setting for defining and comparing process laws. It supports the study of continuity, jumps, dependence, and temporal structure. Many foundational results are formulated most cleanly in this language.
7.2 Large deviations
Large deviation theory examines rare events by assigning asymptotic probabilities to unlikely trajectories. Path space is the natural arena for such results because the objects of interest are entire sample paths. Rate functionals often measure the cost of deviations from typical behavior.
7.3 Optimal stopping
Optimal stopping problems involve choosing a time to act based on observed path information. The filtration generated by the path is central to formulating the decision rule. Path-space methods help describe admissible stopping times and evaluate expected rewards.
7.4 Pathwise analysis
Pathwise analysis studies individual trajectories rather than only averaged quantities. It is useful for understanding regularity, variation, and sample-path properties. This approach often reveals features that are hidden in marginal or distributional descriptions.