Stochastic analogue of fundamental theorem of surface theory for surfaces of bounded distortion and positive curvature

Abstract

In this paper, we prove a stochastic analogue of Gauss–Peterson–Codazzi equations and provide a stochastic analogue of the fundamental theorem in the theory of surfaces for surfaces of a bounded distortion and a positive curvature. In 1956, I.Ya. Bakelman derived the Gauss–Peterson–Codazzi equations for surfaces of bounded distortion, that is, for the surfaces defined by functions with continuous first derivatives and square summable second generalized derivatives in the sense of Sobolev. In 1988, Yu.E. Borovskii proved that the Gauss–Peterson–Codazzi equations (derived by I.Ya. Bakelman) uniquely determined the surface of a bounded curvature.The aim of this paper is to present the results of I.Ya. Bakelman and Yu.E. Borovskii in terms of the theory of random processes in the case of a surface of a positive bounded distortion and a positive curvature.By means of two fundamental forms of the surface, we construct two random processes and derive a system of equations relating the characteristics (transition functions) of these processes. The resulting system is a stochastic analogue of the system of Gauss–Peterson–Codazzi equations and is a criterion determining uniquely the surface up to a motion. The generators of random processes are second order operators generated by the fundamental forms of the surface. For instance, if the surface metrics is given by the expression $ I = ds^2 = g_{ij} dx^i dx^j$, then the generator of the corresponding process is $ A = g^{ij} \partial_i \partial_j $. We establish a relationship between the transition functions of the random process and the generator coefficients. The obtained expressions are substituted into the generalized Gauss–Peterson–Codazzi equations, which leads us to the desired result.