In this paper, the one-dimensional α-fractional diffusion equation is revisited. This equation is a particular case of the timeand space-fractional diffusion equation with the quotient of the orders of the timeand space-fractional derivatives equal to one-half. First, some integral representations of its fundamental solution including the Mellin-Barnes integral representation are derived. Then ...

The paper contains a systematic presentation of how the so-called “W-transform” can be used to study stability of stochastic functional differential equations. The W-transform is an integral transform which typically is generated by a simpler differential equation (“reference equation”) via the Cauchy representation of its solutions (“variation-of-constant formula”). This other equation is supp...

We derive bridges from general multidimensional linear non time-homogeneous processes using only the transition densities of the original process giving their integral representations (in terms of a standard Wiener process) and so-called anticipative representations. We derive a stochastic differential equation satisfied by the integral representation and we prove a usual conditioning property ...

We perform the stochastic quantization of Yang–Mills theory in configuration space and derive the Faddeev-Popov path integral density. Based on a generalization of the stochastic gauge fixing scheme and its geometrical interpretation this result is obtained as the exact equilibrium solution of the associated Fokker–Planck equation. Included in our discussion is the precise range of validity of ...

We investigate the problem of using a Riemannian sum with random subintervals to approximate the iterated Itô integral ∫ w dw—or, equivalently, solving the corresponding stochastic differential equation by Euler’s method with variable step sizes. In the past this task has been used as a counterexample to illustrate that variable step sizes must be used with extreme caution in stochastic numeric...