Feynman integral and stochastic differential equations

O. O. Obrezkov

Zbigniew Haba

Anastasia Doikou, Simon J. A. Malham, Anke Wiese

We explore the connections between the theories of stochastic analysis and discrete quantum mechanical systems. Naturally these connections include the Feynman-Kac formula, and the Cameron-Martin-Girsanov theorem. More precisely, the notion of the quantum canonical transformation is employed for computing the time propagator, in the case of generic dynamical diffusion coefficients. Explicit computation of the path integral leads to a universal expression for the associated measure regardless of the form of the diffusion coefficient and the drift. This computation also reveals that the drift plays the role of a super potential in the usual super-symmetric quantum mechanics sense. Some simple illustrative examples such as the Ornstein-Uhlenbeck process and the multidimensional Black-Scholes model are also discussed. Basic examples of quantum integrable systems such as the quantum discrete non-linear hierarchy (DNLS) and the XXZ spin chain are presented providing specific connections between quantum (integrable) systems and stochastic differential equations (SDEs). The continuum limits of the SDEs for the first two members of the NLS hierarchy turn out to be the stochastic transport and the stochastic heat equations respectively. The quantum Darboux matrix for the discrete NLS is also computed as a defect matrix and the relevant SDEs are derived.

Alan E. E. Rogers

Various analytic results which combine fermionic Brownian motion with stochastic integration are described, and it is shown that a wide class of stochastic differential equations in superspace have solutions. Such solutions are then used to derive a Feynman-Kac formula for a supersymmetric system in terms of the supercharge whose square is the Hamiltonian of the system. This is achieved by introducing superpaths parametrized by a commuting and an anticommuting time variable.

Kazuo Takatsuka

S Leppard, A Rogers

Motivated by application to quantum physics, anticommuting analogues of Wiener measure and Brownian motion are constructed. The corresponding Ito integrals are defined and the existence and uniqueness of solutions to a class of stochastic differential equations is established. This machinery is used to provide a Feynman-Kac formula for a class of Hamiltonians. Several specific examples are considered.