Brownian motion on the n-sphere

Question

From course notes on SDE's. We consider a Stratonovich equation.

$dX_t=\left(I-\frac{1}{|X_t|^2}X_tX_t^T \right)\circ dB_t$

With $X_t\in \mathbb{R}^n$ and $\{B_t\}$ being n-dimensional brownian motion. We wish to show that $|X_t|^2$ is constant along trajectories, so that there cannot exist a unique stationary trajectory.

Now, does the forward Kolmogorov equation work in the same way for a Stratonovich equation as it does the Ito case? If that is the case, how would I look at trajectories in general? Ergodic theory is not a part of the notes.

Answer 1

That $|X_t|^2$ is constant can be seen as follows:

\begin{align} d(|X_t|^2) & = 2X_t^T\circ dX_t\\ & = 2\left(X_t^T-\frac{1}{|X_t|^2}X_t^TX_tX_t^T\right)\circ dB_t\\ & = 0. \end{align}

(I used $P_t\circ (Q_t\circ dB_t)=(P_tQ_t)\circ d B_t$.)