Lagrangian Formulation of a Solution to the Cosmological Constant Problem

A covariant Lagrangian formulation of a solution to the cosmological constant problem, based on vizualising the fluctuations of the vacuum energy as a non-equilibrium process with stochastic behaviour, is presented. The variational principle yields equations of motion for the cosmological ``constant" $\mathrm{\Lambda }$, treated as a dynamical field, together with an equation for a Lagrange multiplier field $\varphi$, and the standard Einstein field equations with a variable cosmological constant term. A stochastic model of $\mathrm{\Lambda }$ yields a natural explanation for the smallness or zero value of the constant in the present epoch and its large value in an era of inflation in the early universe.