Stochastic Gravity: A Primer with Applications
  Stochastic semiclassical gravity of the 90's is a theory naturally evolved from semiclassical gravity of the 70's and 80's. It improves on the semiclassical Einstein equation with source given by the expectation value of the stress-energy tensor of quantum matter fields in curved spacetimes by incorporating an additional source due to their fluctuations. In stochastic semiclassical gravity the main object of interest is the noise kernel, the vacuum expectation value of the (operator-valued) stress-energy bi-tensor, and the centerpiece is the (stochastic) Einstein-Langevin equation. We describe this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity. The functional approach uses the Feynman-Vernon influence functional and the Schwinger-Keldysh close-time-path effective action methods which are convenient for computations. It also brings out the open systems concepts and the statistical and stochastic contents of the theory such as dissipation, fluctuations, noise and decoherence. We then describe the application of stochastic gravity to the backreaction problems in cosmology and black hole physics. Intended as a first introduction to this subject, this article places more emphasis on pedagogy than completeness. 
B. L. Hu and E. Verdaguer