The $T\overline T$ deformation of quantum field theory as a stochastic process.

We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant $\det T$ of the stress tensor, commonly referred to as $T\overline T$. Infinitesimally this is equivalent to a random coordinate transformation, with a local action which is, however, a total derivative and therefore gives a contribution only from boundaries or nontrivial topology. We discuss in detail the examples of a torus, a finite cylinder, a disk, and a truncated cone. In all cases the partition function evolves according to a linear diffusion-type equation, and the deformation may be viewed as a kind of random walk in moduli space in which domains shrink and become more symmetrical. The truncated cone allows access to information about the behavior of correlations, and of entanglement entropies, under the deformation. We also suggest a generalization of the whole formalism to higher dimensions.