  Let $M$ be a compact oriented $d$-dimensional smooth manifold. Chas and
Sullivan have defined a structure of Batalin-Vilkovisky algebra on
$\mathbb{H}_*(LM)$. Extending work of Cohen, Jones and Yan, we compute this
Batalin-Vilkovisky algebra structure when $M$ is a sphere $S^d$, $d\geq 1$. In
particular, we show that $\mathbb{H}_*(LS^2;\mathbb{F}_2)$ and the Hochschild
cohomology $HH^{*}(H^*(S^2);H^*(S^2))$ are surprisingly not isomorphic as
Batalin-Vilkovisky algebras, although we prove that, as expected, the
underlying Gerstenhaber algebras are isomorphic. The proof requires the
knowledge of the Batalin-Vilkovisky algebra $H_*(\Omega^2 S^3;\mathbb{F}_2)$
that we compute in the Appendix.
