We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation, ut−∆u = βu−u3, by noise. While a single multiplicative Itô noise of sufficient intensity will stabilise the origin, its Stratonovich counterpart leaves the dimension of the attractor essentially unchanged. We then show that a collection of multiplicative Stratonovich terms can make the origin exponent...

We extend the Itō-to-Stratonovich analysis or quantum stochastic differential equations, introduced by Gardiner and Collett for emission (creation), absorption (annihilation) processes, to include scattering (conservation) processes. Working within the framework of quantum stochastic calculus, we define Stratonovich calculus as an algebraic modification of the Itō one and give conditions for th...

The Langevin equation with a multiplicative Lévy white noise is solved. The noise amplitude and the drift coefficient have a power-law form. A validity of ordinary rules of the calculus for the Stratonovich interpretation is discussed. The solution has the algebraic asymptotic form and the variance may assume a finite value for the case of the Stratonovich interpretation. The problem of escapin...

In Burrage and Burrage (1996) it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In ...

We introduce the Stratonovich version of quantum stochastic calculus including integrals with respect to emission (creation), absorption (annihilation) and scattering (conservation) processes. The calculus allows us to consider the limit of regular open dynamical systems as a quantum Wong-Zakai approximation theorem. We introduce distinct definitions of Itô Dyson and Stratonovich Dyson time-ord...

o m=l 0 where o denotes generalized Stratonovich integration and the equality is a.s. (cf. lemma 4). We use the criteria we derive to provide some new relations between Stratonovich and Ogawa integrals which do not go through an intermediate chaos decomposition as in [1]. The results below are an outgrowth of some extensions of the Ito lemma pointed out in [2], especially lemma (4.2) there. We ...

Recently, a novel framework to handle stochastic processes has emerged from a series of studies in biology, showing situations beyond ‘Itô versus Stratonovich’. Its internal consistency can be demonstrated via the zero mass limit of a generalized Klein–Kramers equation. Moreover, the connection to other integrations becomes evident: the obtained Fokker–Planck equation defines a new type of stoc...

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...