On the Geometry of the Hamilton-jacobi-bellman Equation

We show how a minimal deformation of the geometry of the classical Hamilton-Jacobi equation provides a probabilistic theory whose cornerstone is the Hamilton-Jacobi-Bellman equation. This is the basis for a novel dynamical system approach to Stochastic Analysis. 1. Stochastic deformation of classical dynamical systems. The geometrical study of the Hamilton-Jacobi theory lies at the heart of Analytical Mechanics. Hamilton’s original motivation to study light rays in terms of wave fronts, as related to Fermat’s variational principle, became almost irrelevant (at least in Jacobi’s perspective) when it was realized that the dynamical equations of motion of Analytical Mechanics have the same structure as those characterizing the critical orbits of Fermat’s principle. What we call today classical Hamilton-Jacobi theory was created by Jacobi during the first half of the 19th century. He introduced, in particular, the notion of complete solution of the Hamilton-Jacobi (HJ) equation and was thereby able to construct all solutions of Hamilton’s canonical equations in phase space. This original and seemingly indirect integration method along with the new difficult problems it allowed to solve are at the origin of the theory of integrable systems, whose modern ramifications such as Algebraic Geometry are impressive. Along the way, Jacobi developed the geometric theory of transformations (diffeomorphisms) of phase space that leave invariant the form of Hamilton’s canonical equations. He proved, in fact, that a complete solution of the HJ equation generates such a transformation. In this review article we wish to explain what a Stochastic Deformation of Jacobi’s program should look like. Of course, there are many ways to introduce a random noise into Analytical Mechanics. A recent interesting possibility is partly inspired by works of J.-M. Bismut (cf. [4, 16] and references therein). We shall, therefore, try to show why the choice we advocate is natural. We claim our way is natural because our deformation is essentially the same as that which makes Quantum Mechanics look as a deformation of Analytical Mechanics. 2000 Mathematics Subject Classification. Primary: 37J05; Secondary: 60H10, 60H30, 60M07.