The relationship of the Ito-Stratonovich stochastic calculus to studies of weakly colored noise is explained. A functional calculus approach is used to obtain an effective Fokker-Planck equation for the weakly colored noise regime. In a smooth limit, this representation produces the Stratonovich version of the ItoStratonovich calculus for white noise. It also provides an approach to steady stat...

We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itôtype equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of inva...

We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process (not necessarily a semi-martingale). No adaptedness of initial point or vector fields is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. This condition is s...

We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Str...

The many-body dynamics of a quantum computer can be reduced to the time evolution of non-interacting quantum bits in auxiliary fields by use of the Hubbard-Stratonovich representation of two-bit quantum gates in terms of one-bit gates. This makes it possible to perform the stochastic simulation of a quantum algorithm, based on the Monte Carlo evaluation of an integral of dimension polynomial in...

A scheme is presented to decompose the exponential of a two-body operator in a discrete sum over exponentials of one-body operators. This discrete decomposition can be used instead of the Hubbard-Stratonovich transformation in auxiliary-field quantum Monte-Carlo methods. As an illustration, the decomposition is applied to the Hubbard model, where it is equivalent to the discrete Hubbard-Straton...

Recently, two different approaches were put forward to extend the supersymmetry method in random matrix theory from Gaussian ensembles to general rotation invariant ensembles. These approaches are the generalized Hubbard– Stratonovich transformation and the superbosonization formula. Here, we prove the equivalence of both approaches. To this end, we reduce integrals over functions of supersymme...

We consider the Langevin equation with multiplicative noise term which depends on time and space. The corresponding Fokker-Planck equation in Stratonovich approach is investigated. Its formal solution is obtained for an arbitrary multiplicative noise term given by g(x, t) = D(x)T (t), and the behaviors of probability distributions, for some specific functions of D(x), are analyzed. In particula...