The talk is dedicated to self-adjoint differential operators in the case of infinite- dimensional coordinate spaces. In particular, it is considered the space of real- valued sequences. It is well-known that in the case of finite-dimensional coordinate spaces the Laplace operator can be defined by the following scheme. 1. The first step is to consider semigroups of shift operators on the space of square-integrable functions (with respect to the Lebesgue measure) and prove that the semigroups are continuous with respect to the strong operator topology. 2. The second step is to average semigroups of shifts with respect to the standard Gaussian vector. The result is a strongly continuous operator semi- group. 3. The last step is to calculate the generator of the averaged semigroup. The generator is equal to the Laplace operator. The aim of this talk is to generalize the scheme to the case of infinite- dimensional coordinate spaces. The infinite-dimensional case is more compli- cated due to the following facts. 1. The Lebesgue measure on an infinite-dimensional topological vector space does not exist. 2. Semigroups of shifts could be discontinuous. It will be discussed how to overcome these obstacles and apply the scheme to the space of real-valued sequences.