Averaging of random orthogonal transformations of domain of functions

We consider and study the notions of a random operator, random operator-valued function and a random semigroup defined on a Hilbert space as well as their averagings. We obtain conditions under which the averaging of a random strongly continuous function is also strongly continuous. In particular, we show that each random strongly continuous contractive operator-valued function possesses a strongly continuous contractive averaging.
We consider two particular random semigroups: a matrix semigroup of random orthogonal transformations of Euclidean space and a semigroup of operators defined on the Hilbert space of functions square integrable on the sphere in the Euclidean space such that these operators describe random orthogonal transformations of the domain these functions. The latter semigroup is called a random rotation semigroup; it can be interpreted as a random walk on the sphere. We prove the existence of the averaging for both random semigroups.
We study an operator-valued function obtained by replacing the time variable $t$ by $\sqrt t$ in averaging of the random rotation semigroup. By means of Chernoff theorem, under some conditions, we prove the convergence of the sequence of Feynman–Chernoff iterations of this function to a strongly continuous semigroup describing the diffusion on the sphere in the Euclidean space. In order to do this, we first find and study the derivative of this operator-valued function at zero being at the same time the generator of the limiting semigroup. We obtain a simple divergence form of this generator. By means of this form we obtain conditions ensuring that this generator is a second order elliptic operator; under these conditions we prove that it is essentially self-adjoint.