proving the brownian motion on the sphere equation the

stratonovich form differential equation $$\partial X=n(X)\times \partial B$$ the equation in ito's form becomes $$dX=n(X)\times dB+H(X)n(X)dt$$ where mean curvature is given by: $$H(X)=-\frac{1}{2}(div \ n)(x)$$ where $X$ is a brownian motion process in $\mathbb{R^3}$.$n(X)$ is a unit normal to the sphere .

for a unit sphere parameterized by $\theta ,\phi$ $$x=\cos \theta \sin\phi,y=\sin\theta\sin\phi,z=\cos \phi$$ the normal to the surface $$n(X)=\cos\theta\sin\phi ,\sin\theta\sin\phi,\cos \phi$$ but i am doubtful about how the representation of the vector be

$$d(\cos\theta\sin\phi ,\sin\theta\sin\phi,\cos \phi)=(\cos\theta\sin\phi ,\sin\theta\sin\phi,\cos \phi)\times dB +(\cos\theta\sin\phi ,\sin\theta\sin\phi,\cos \phi)dt $$


mean curvature $$H(X)=1 $$ for a sphere .

the result in the book are given

as where $B,W $ are the independent motion

$$d\phi = dB +\frac{1}{2}\cot\phi d\phi$$

$$\theta _t =W\big{(}\int^{t}_{0}(\sin \phi_u )^{-2} du\big{)}$$

any help will be thank full , i am mainly confused with the vector part representation.


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