[Math] Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

brownian motionhilbert-spacesprobability theorystochastic-integralsstochastic-processes

Consider the standard Brownian motion on $[0,1]$:

$$
dB_t, \; B_0 = 0,
$$

defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times [0,1]$. The RKHS with reproducing kernel $K$ is the Sobolev space

$$
\mathcal{H}_K = \{f \; {\tt absolutely \; continuous}, \; f(0) = 0, f'\in L^2[0,1] \}
$$

with inner product

$$
\langle f, g \rangle_{\mathcal{H}_K} = \int f'g'.
$$

This can be seen by noting that $t \mapsto \min \{ s,t\}$ has weak derivative $1_{[0, s]}$.

Questions

  1. $\mathcal{H}_K$ is isomorphic to the Hilbert space generated by $\{ B_t\}_{t \in [0,1]}$, with isomorphism $K_t(s) = K(t,s) \mapsto B_t \in L^2(\Omega, P)$. So it looks like one can define a stochastic integral against $dB_t$ with deterministic integrands. Is there a name for this integral? It looks a bit strange when compared to the Ito integral. For example, the increment $B_{s_2} – B_{s_1}$ is identified with

$$
\min(s_2, t) – \min(s_1, t).
$$

  1. I've encountered the claim that "(the differential operator) $-\frac{d^2}{dx^2}$ is the reproducing kernel of $B_t$". How is $-\frac{d^2}{dx^2}$ related to $K$? Subject to boundary conditions, integrating by parts can recover the inner product on $\mathcal{H}_K$ but I don't see an identification with the Sobolev space $\mathcal{H}_K$:

$$
– \int f''g = \int f'g'.
$$

  1. Related to 2.: the infinitesmal generator of $B_t$ as a Markov process happens to be the Laplacian $\frac{d^2}{dx^2}$. Is this a related to the above?

  2. The Cameron-Martin space of $B_t$ also just happens to be $\mathcal{H}_K$.

Same objects, all related to the Brownian motion $B_t$, keep coming up via (apparently) different constructions…what's happening here? Is there a way they fit together?

Best Answer

For #1: Let's call your isomorphism $T$, the isometry from $\mathcal{H}_K$ to $L^2(P)$ which maps $K_t$ to $B_t$. Yes, $T$ really is a stochastic integral: $Tf$ is the Itô integral, not of $f$, but of $f'$. So $$Tf = \int_0^1 f'(t)\,dB_t$$ or in other words, for deterministic $g \in L^2([0,1])$, $$\int_0^1 g(t)\,dB_t = T\left(\int_0^\cdot g(s)\,ds\right).$$ So this recovers the Itô integral for deterministic $L^2$ integrands. This special case of the Itô integral is sometimes called the Wiener integral.

Another way of thinking about this is that the map $f \mapsto f'$ is an isometric isomorphism from $\mathcal{H}_K$ to $L^2([0,1])$. So if you identify $\mathcal{H}_K$ with $L^2([0,1])$ under this map, then $T$ really is the stochastic integral.

I will think about your other questions some more.

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