Construction of a Brownian sheet with values in a compact Riemannian manifold. (Q556577)

P. Malliavin constructed a Brownian sheet that takes its values in a compact Lie group. The purpose here is to give an alternative construction, valid for a compact Riemannian manifold. An embedding of the manifold \(M\) in \(\mathbb{R}^d\) and a \(\mathbb{R}^d\)-valued Brownian sheet being fixed, the construction proceeds roughly by conditioning the sheet to remain in an \(\varepsilon\)-neighbourhood of \(M\), along a discretization of the time parameters. In fact, to handle the second parameter, after the first one, tubes about a Lipschitzian curve are considered. The key step is a short-time asymptotics of the integral \(t^{-m/2}\int_M g(z)\exp({-|z-y|^2\over 2t})\lambda_M(dz)\). The proofs are sketched.