Stochastic differential geometry of smooth surfaces of positive curvature

In this note, we derive a stochastic analogue of the Peterson-Codazzi equations for two-dimensional surfaces of positive curvature of the class $C^k$. To study these objects, methods of stochastic analysis are used, more precisely, the Ito formula and the properties of Brownian motion generated by the surface metric. An essential difference from the results of Backelman I. Ya. [3] is an application of the Ito formula and the second Ito derivative introduced in this paper. The technique of symmetric integrals (a deterministic analogue of Stratonovich's stochastic integrals) is also used.