Some extensions of the Pr\'ekopa-Leindler inequality using Borell's stochastic approach

We present an abstract form of the Pr'ekopa-Leindler inequality that includes several known -and a few new- related functional inequalities on Euclidean spaces. The method of proof and also the formulation of the new inequalities are based on Christer Borell’s stochastic approach to Brunn-Minkowski type inequalities.