Abstract: We generalize the systems of Hamiltonian diffusions, which were introduced and studied by Bismut, to accommodate arbitrary Jacobi manifolds as phase spaces and general continuous semimartingales as forcing noises. As is well-known, Jacobi structures are the natural generalization of Poisson structure and in particular of symplectic, cosymplectic and Lie–Poisson structures. However, very interesting manifolds like contact manifolds and locally conformal symplectic manifolds are also Jacobi but not Poisson. In this talk, we focus on the stochastic Hamiltonian systems whose phase flows preserve characteristic structures, and develop a Hamilton–Jacobi theory which is regarded as an alternative method for formulating the dynamics. A particular example is the case of thermodynamic dynamics in which we apply our methods on a manifold with its canonical contact form.