In this way (3) simply expresses the conservation of probability.

Specifically, this function must obey the conservation of probability and the local conservation of energy (the particle cannot extract energy from the field indefinitely).

The text covers the elliptic equation and the Cauchy problem under certain conditions, one-dimensional theory, uniqueness results,

conservation of probability and maximum principles, properties of {T(t)} in spaces of continuous functions, uniform estimates for the derivatives of {T(t)}, point-wise estimates for the derivatives of {T(t)}, certain invariant measures in semigroups, the Ornstein-Uhlenbeck operator, a class of nonanalytic Markov semigroups, the Cauchy-Dirichlet problem, the Cauchy-Newman problem in the convex and nonconvex case, and a class of Markov semigroups associated with degenerate elliptic operators.