Canonical Path Space for Diffusion Laws

From a Random Process to a Measure on Paths

A weak solution is built on a probability space that may contain randomness beyond the solution itself. Its path distribution can nevertheless be represented on one common space: the space of continuous functions. On that space, each outcome is a path, and the process is obtained by reading the path at the chosen time. This representation separates the law of the solution from the particular probability space used to construct it.

Fix a finite horizon \(T>0\) and a state dimension \(d\geq1\). Let \(C_T=C([0,T],\mathbb{R}^d)\) be the set of continuous paths, equipped with the uniform metric \(d_T(\omega,\eta)=\sup_{0\leq t\leq T}|\omega(t)-\eta(t)|\) and its Borel sigma-field \(\mathcal{B}(C_T)\). For an infinite time horizon, the corresponding space \(C([0,\infty),\mathbb{R}^d)\) is given the topology of uniform convergence on every bounded time interval. We first work on \(C_T\).

The raw filtration records exactly what the path has revealed by time \(t\). One may complete it by adding subsets of \(P_X\)-null sets and make it right-continuous; these operations depend on the chosen measure. Thus the raw canonical space and filtration are defined before a particular path law is selected.

For context, a scalar weak solution of \(dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\) on \([0,T]\) consists of a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})\), a Brownian motion \(B_t\) relative to that filtration, and a continuous adapted process \(X_t\) satisfying the integral equation almost surely. The drift integral must be defined, for example by \(\int_0^T|\mu(t,X_t)|\,dt<\infty\) almost surely. For an Itô integral in the class used here, its integrand \(H_t=\sigma(t,X_t)\) is progressively measurable and belongs to \(\mathcal{L}^2\): \(\mathbb{E}\int_0^T H_t^2\,dt<\infty\). The Brownian motion and solution need not be defined on a prescribed probability space. The law of the solution path is the probability measure \(P_X=\mathbb{P}\circ X^{-1}\) on \(C_T\).

Why the Path Map Is Measurable

To define \(P_X\), the random path \(X:\Omega\to C_T\) must be measurable. This follows from continuity of its paths and measurability of its time evaluations. The underlying fact is that the Borel sigma-field on continuous path space is generated by evaluation maps.

Proof. Each evaluation \(\omega\mapsto\omega(t)\) is continuous for the uniform metric, so \(\sigma(X_t:0\leq t\leq T)\subseteq\mathcal{B}(C_T)\). For the reverse inclusion, let \(D\) be a countable dense subset of \([0,T]\) containing the endpoints. For any fixed \(\eta\in C_T\), continuity of \(\omega-\eta\) gives \(d_T(\omega,\eta)=\sup_{q\in D}|\omega(q)-\eta(q)|\). Thus every open ball with center \(\eta\) and radius \(r>0\) belongs to \(\sigma(X_q:q\in D)\): the condition \(d_T(\omega,\eta)<r\) can be written as the union, over rational \(a<r\), of the events \(\{|\omega(q)-\eta(q)|\leq a\text{ for every }q\in D\}\). The space \(C_T\) is separable, so its open sets are countable unions of balls from a countable base. Therefore every open set belongs to \(\sigma(X_q:q\in D)\), proving the reverse inclusion. For a process with continuous paths, the inverse image of any such open set is consequently measurable; changing the path map on a measurable null set preserves measurability. \(\square\)

The measure \(P_X\) is a pushforward: for \(A\in\mathcal{B}(C_T)\), \(P_X(A)=\mathbb{P}(X\in A)\). In particular, for times \(t_1,\ldots,t_n\), the joint distribution of the canonical coordinates \((X_{t_1},\ldots,X_{t_n})\) under \(P_X\) is the joint distribution of the original solution \((X_{t_1},\ldots,X_{t_n})\) under \(\mathbb{P}\). This is the precise sense in which the solution’s path law is represented canonically.

Path Laws and Finite-Dimensional Distributions

A finite-dimensional distribution is the joint law of the process at any finite list of times. These distributions are often convenient to calculate, but the canonical path law records more: it assigns probabilities to measurable sets of entire paths. On continuous path space, however, the finite-dimensional distributions determine that law.

Proof. Take a countable dense set \(D\subseteq[0,T]\) containing \(0\) and \(T\). The proposition shows that \(\mathcal{B}(C_T)=\sigma(X_q:q\in D)\). Consider events specified by finitely many coordinates, such as \(\{\omega:(\omega(q_1),\ldots,\omega(q_n))\in A\}\), where \(q_i\in D\) and \(A\) is Borel in \((\mathbb{R}^d)^n\). These events form a class closed under finite intersections and generate \(\mathcal{B}(C_T)\). The assumed agreement of finite-dimensional distributions gives \(P=Q\) on this class. The standard pi-lambda theorem for probability measures then gives \(P=Q\) on the sigma-field it generates, namely \(\mathcal{B}(C_T)\). \(\square\)

This theorem explains how equality in law can be checked using finite-time observations. In the language of weak solutions, uniqueness in law means that any two weak solutions with the same specified initial law and coefficients induce the same probability measure on path space. It does not assert that the solutions are equal on a common probability space; that stronger type of comparison is pathwise uniqueness.

What a Canonical Law Does—and Does Not—Record

Replacing the original probability space by \((C_T,\mathcal{B}(C_T),P_X)\) preserves the distribution of every measurable path observation. If \(F:C_T\to\mathbb{R}\) is measurable, then \(\mathbb{E}^{P_X}[F(X)]=\mathbb{E}[F(X)]\), whenever either expectation is defined. For example, this applies to evaluations at fixed times and to events describing the maximum of a continuous path.

The replacement does not automatically carry along the original Brownian motion \(B_t\). The canonical coordinate \(X_t\) is the solution path; it is not, by definition, the noise that drove the solution. A path law alone therefore does not generally specify a joint law of solution and driver, nor does it directly provide a Brownian motion relative to the canonical filtration. Conclusions about a particular driving noise require additional structure or a separate argument.

A Common Misinterpretation

A canonical path space is not a claim that all weak solutions can be driven by the same Brownian motion on that space. For example, after constructing \(P_X\), it is valid to say that the coordinate process has the same path distribution as the original \(X\). It is not valid, without further proof, to say that \(X_t\) on canonical space satisfies the original SDE with a Brownian driver measurable in the canonical filtration. The pushforward construction transports the solution’s distribution, not every random object on the original probability space.

Keep the roles distinct: the canonical space \(C_T\) supplies possible paths; the coordinate process reads those paths; the measure \(P_X\) assigns their probabilities; and the natural filtration records their observed history. A weak solution law is precisely this probability measure on continuous paths. Equality of such measures is equality in law, whereas statements about a common noise or sample-by-sample equality require more than equality of path distributions.