# Continuous-Time Autoregressive process and RKHS

Consider a stationary Continuous-time AutoRegressive (CAR) process on a bounded time-interval $$(a, \, b)$$. This article by Emmanuel Parzen describes the corresponding Reproducing Kernel Hilbert Space (RKHS) $$\mathcal{K}$$ and its inner product for the first and second-order CARs. For instance consider the first order
$$\frac{\text{d}}{\text{d}t} X(t) + \beta X(t) = \varepsilon(t)$$
where $$\varepsilon(t)$$ is a Gaussian white noise with unit variance and $$\beta >0$$ - a more rigorous formulation would be $$\text{d} X(t) + \beta X(t)\text{d}t = \text{d}W(t)$$ with $$W(t)$$ being a standard Wiener process. Then $$\mathcal{K}$$ contains all differentiable functions on $$(a, \,b)$$ and the inner product is given by formula (4.16) p. 477 \begin{aligned} \langle h, \, g \rangle_{\mathcal{K}} &= \frac{1}{2\beta} \,\int_{a}^b \left[h' + \beta h\right]\left[g' + \beta g\right] \text{d}t + h(a) g(a) \\ & = \frac{1}{2\beta} \,\int_{a}^b \left[h'g' + \beta^2 hg \right] \text{d}t + \frac{1}{2}\left\{ h(a) g(a) + h(b) g(b) \right\} \end{aligned}. The second form is more attractive because of its symmetry w.r.t. to the two end-points $$a$$ and $$b$$ which relates to the time-reversibilty of the process; it is easily derived from the non-symmetric form using integration by parts. Parzen gives the RKHS for the second-order CAR but only with a non-symmetric form for the inner product. The same is true for a $$\text{CAR}(p)$$ with order $$p$$ defined as the stationary solution (assumed to exist) of $$D^{p} X(t) + a_{1} D^{p-1} X(t) + \dots + a_p X(t) = \varepsilon(t)$$
where $$D := \text{d}/ d\text{t}$$ and again with a more rigorous form involving $$\text{d}W(t)$$. See formulas (4.7), (4.8) and (4.9) in the article.

For the $$\text{CAR}(p)$$ case, what is the expression of the RKHS inner product which is symmetric in the two end-points $$a$$ and $$b$$? I guess that this could involve the formal adjoint of the linear differential operator but did not find the answer with Google searches.