Lagrangian description of Brownian motion? I'm interested in the existence of a Lagrangian field theory description of Bronwnian motion, does such a thing exist? Given a particle of some spin $\sigma$, which has a  Lagrangian associated with it $\mathcal{L}_{\sigma}$ (which, using the Euler-Lagrange equations, produces Klein-Gordon for $\sigma$ = 0 etc) is there a way I can allow for Brownian type freedoms in this description? Hopefully such a stochastic freedom is allowed in the Lagrangian description.
 A: The Brownian motion $x(t)$ is non-differentiable, so a particular trajectory $x(t)$ can't extremize an action $S$ which would be a functional of $x(t)$ and its derivative, $\dot x(t)$, because the derivative isn't even well-defined and any expression of the type $\int [\dot x(t)]^2 dt$, the usual kinetic term in the action, diverges. (See e.g. middle of page 2 of this paper to see the statement that there is no Lagrangian, too. The paper does its best to construct something that is "as close as possible" to the normal Lagrangian formulation.)
However, when you mention field theory, it's interesting to point out that the typical trajectories $x(t)$ that contribute to Feynman's path integral computation of ordinary quantum mechanics do resemble the Brownian trajectories very closely. But the amount of zigzag motion is determined by the uncertainty principle and Planck's constant, not by adjustable collisions with the molecules of a liquid etc. There are many other differences in the physical interpretation, too.
