We consider the continuous parabolic Anderson model with Gaussian fields under measure-valued initial conditions, covariances of which are homogeneous or nonhomogeneous in time and fractional rough space. mainly study spatial behaviors for Feynman–Kac formulas Stratonovich sense. Benefited from application formula based on Brownian bridge, precise asymptotics can be obtained more general condit...

In this article, we address the generalized BBM equation with white noise dispersion which reads du− duxx + ux ◦ dW + uuxdt = 0, in Stratonovich formulation, where W (t) is a standard real valued Brownian motion. It is equivalent to, in Ito’s formulation, du + (1−∆)uxdW − 1 2 (1−∆)uxxdt + (1−∆)uuxdt = 0. The global well posedeness for the initial value problem is shown for any p ≥ 1. Furthermor...

Generalized Langevin equations (GLE) with multiplicative white Poisson noise pose the usual prescription dilemma leading to different evolution equations (master equations) for the probability distribution. Contrary to the case of multiplicative Gaussian white noise, the Stratonovich prescription does not correspond to the well-known midpoint (or any other intermediate) prescription. By introdu...

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingal...

From the notion of stochastic Hamiltonians and the flows that they generate, we present an account of the theory of stochastic derivations over both classical and quantum algebras and demonstrate the natural way to add stochastic derivations. Our discussion on quantum stochastic processes emphasizes the origin of the Itô correction to the Leibniz rule in terms of normal ordering of white noise ...

We study the convergence of a semi-discretized version of a numerical scheme for a stochastic nonlinear Schrödinger equation. The nonlinear term is a power law and the noise is multiplicative with a Stratonovich product. Our scheme is implicit in the deterministic part of the equation as is usual for conservative equations. We also use an implicit discretization of the noise which is better sui...

where ww(/) = Wt(cS). Usually, this approach is not followed because of the technical difficulties that appear when one tries to solve (det) for arbitrary continuous w. One is therefore forced to use other lines of attack, and to study (stoch) directly. This gives rise to at least two nonequivalent theories, namely, the one due to Ito, and that of Fisk and Stratonovich. The purpose of this note...