IL NUOVO CIMENTO Aprile 1998 VOL. 113 B, N. 4 Asymptotic globally homogeneous domain distribution in stochastic inflation M. BELLINI (*), P. D. SISTERNA (**) and R. R. DEZA (***) Departamento de Física, Facultad de Ciencias Exactas y Naturales Universidad Nacional de Mar del Plata - Deán Funes 3350, 7600 Mar del Plata, Argentina (ricevuto il 17 Aprile 1997; approvato il 25 Giugno 1997) Summary. — Within the stochastic approach to inflation we investigate under what conditions can the solution to Starobinsky’s equation be mapped onto an Ornstein-Uhlenbeck process. These conditions restrict the Lagrangian of the inflaton field. New solutions for physically appealing potentials are obtained, which show asymptotically global homogeneity in F-space. PACS 98.80.Hw – Mathematical and relativistic aspects of cosmology; quantum cosmology. PACS 04.60.Gw – Covariant and sum-over-histories quantization. 1. – Introduction The possibility of self-reproduction of inflationary domains [1] has dramatically changed our point of view on the history and on the fate of our universe, as well as on its global structure. It has become one of the most important issues in the current inflationary cosmology, since it would allow to account for observational data without any reference to a “Big Bang”. In the stochastic inflation scenario one assumes “our” universe to be a member of a collection of inflationary domains, each of which would grow exponentially with a variety of symmetry breakings and space-time compactifications. The stochastic inflationary scenario has steadily ripen during the last years [2-7], Starobinsky’s Fokker-Planck equation being one of the corner- stones in the development of its formalism [4]. The global structure of “the” universe is suitably described by the probability distribution P c ( F , t ) of finding a given value of the field F at a given time t, in a given domain. Several stationary solutions of the (*) E-mail: mbelliniHmdp.edu.ar (**) E-mail: sisternaHargenet.com.ar (***) E-mail: dezaHmdp.edu.ar G Società Italiana di Fisica 471