Geodesic
Ito equation as a geodesic flow on $\widehat {\text{Diff}\sp {s}(S\sp 1) \bigodot C\sp {\infty }(S\sp 1)}$
Archivum mathematicum, Tome 36 (2000) no. 4, pp. 305-312.

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The Ito equation is shown to be a geodesic flow of $L^2$ metric on the semidirect product space ${\widehat{{\it Diff}^s(S^1) \bigodot C^{\infty }(S^1)}}$, where ${\it Diff}^s(S^1)$ is the group of orientation preserving Sobolev $H^s$ diffeomorphisms of the circle. We also study a geodesic flow of a $H^1$ metric.
Classification : 35Q53, 37K10, 37K65, 58D05
Keywords: Bott-Virasoro Group; Ito equation 
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     author = {Guha, Partha},
     title = {Ito equation as a geodesic flow on $\widehat {\text{Diff}\sp {s}(S\sp 1) \bigodot C\sp {\infty }(S\sp 1)}$},
     journal = {Archivum mathematicum},
     pages = {305--312},
     publisher = {mathdoc},
     volume = {36},
     number = {4},
     year = {2000},
     mrnumber = {1811175},
     zbl = {1049.37045},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2000__36_4_a7/}
}
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Guha, Partha. Ito equation as a geodesic flow on $\widehat {\text{Diff}\sp {s}(S\sp 1) \bigodot C\sp {\infty }(S\sp 1)}$. Archivum mathematicum, Tome 36 (2000) no. 4, pp. 305-312. http://geodesic.mathdoc.fr/item/ARM_2000__36_4_a7/