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Consider the Ornstein-Uhlenbeck process on $\mathbb R^n$ as the process $X$ that solves the SDE

$$dX_t = -BX_t dt + \sigma dW_t $$

where $B, \sigma$ are real square matrices and $W$ is a standard Brownian motion. Given this, the OU process follows a Gaussian distribution with mean $0$ and evolving covariance $$ S(t) = \int_0^t e^{-B(t-s)}\sigma \sigma^T \left (e^{-B(t-s)}\right)^T ds$$

I would like to know about the distribution of the OU process on path space. Say, given a path $X: [0,T]\to \mathbb R^N$, what is the probability of the OU process following this given path? In other words, if I look at the OU process as a path valued random variable, what is its distribution?

This is not straightforward as the distribution at each time is not sufficient to find the distribution on paths. Most likely one needs to start from the SDE and the Brownian motion will play a role in the sorts of paths that can be followed.

Any reference to a book/paper where they have worked out the law of the OU process on path space would be greatly appreciated. Thanks!

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First to be clear the path of OU is uniquely determined by its covariance $$\begin{align} \operatorname{cov}(x_s,x_t) & = = \frac{\sigma^2}{2\theta} \left( e^{-\theta|t-s|} - e^{-\theta(t+s)} \right) \end{align} $$ because a Gaussian process is uniquely determined by its covariance.

If you are asking for support theorems eg. Brownian motion staying close to a continuous function eg. see references in The Wiener measure of an open set, then you can take a look at support theorems for SDEs. A main result there is the Stroock–Varadhan support theorem.

Stroock–Varadhan support theorem Assume that $V = (V_1,..., V_d)$ is a collection of $Lip_2$-vector fields on $\mathbb{R}$, and $V_0$ is a $Lip_1$-vector field on $\mathbb{R}$ Let B be a d-dimensional Brownian motion and consider the unique (up to indistinguishability) Stratonovich SDE solution $Y$ on [0, T] to

$$dY=V_0(Y)dt+\sum V_k(Y)\circ dB_k$$

and let us write $y^h = \pi(V, V_0) (0, y_0; (h, t))$ for the ODE solution to

$$dY=V_0(Y)dt+\sum V_k(Y)dh_k$$

started at $y_0 \in \mathbb{R}$ where $h$ is a Cameron–Martin path, i.e. $h\in W^{1,2}([0,T],\mathbb{R}^{d})$. Then, for any $\alpha \in [0, 1/2)$ and any $\delta > 0$,

$$\lim_{\epsilon\to 0}P[|Y-y^{h}|_{Hol-\alpha}<\delta||B-h|_{\infty,[0,T]}<\epsilon]=1,$$

(where the Euclidean norm is used for conditioning $|B − h|_{\infty,[0,T]}<\epsilon$).

So here the Y solution stays close to a deterministic path if the "noises" are close. The setting is general enough to contain multivariate Ornstein-Uhlenbeck also.

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  • $\begingroup$ Thank you. For future readers I will also add that the OU process is a Gaussian process and defines a Gaussian measure on the space of continuous paths, where the covariance is the covariance in your post, and the mean is the solution to the ODE as one sets $\sigma=0$. $\endgroup$
    – Lance
    Commented Oct 30, 2023 at 9:08

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