String brownian motion

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I construct a string brownian motion, replacing the paths in the manifold by surfaces. On the real line, the string can be developed by Fourier series, so that the string brownian motion can be decomposed in the Fourier coefficient as :

$$x(t) (\theta ) =\sum_n a_n (t) e^{2in \theta}$$

$$P(a_n (t)-a_n (s) \in H_n)=(1/\sum_n r_n) \sum_n r_n \frac{1}{\sqrt{2\pi (t-s)}}\int_{H_n} e^{-\frac{x^2}{2(t-s)}}dx$$

where $P$ is the probability. We can also take infinitely number of independent brownian motions.

Can we have bifurcations in the motion for surfaces of different genre type?

Can you give definitions of $H_n$, $r_n$?
$H_n$ is any Lebesgue mesurable set and we could take $r_n=1/n^2$, or $r_n>0$ and the series $\sum_n r_n$ convergent.
$H_n$ is any Lebesgue mesurable set and we could take $r_n=1/n^2$, or $r_n>0$ and the series $\sum_n r_n$ convergent.
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