arXiv subjects

A. Truman

Publications and source records attributed to A. Truman.

5 recordsLinked to original sources

arXiv2015

Generalized Ito Formulae and Space-Time Lebesgue-Stieltjes Integrals of Local Times

Generalised Ito formulae are proved for time dependent functions of continuous real valued semi-martingales. The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded 2-dimensional variation. In particular a class of functions with discontinuous first derivative is included. An estimate of Krylov allows further weakening of these conditions when the semi-martingale is a diffusion.

math.PR↗

arXiv2010

The Gough-James Theory of Quantum Feedback Networks in the Belavkin Representation

The mathematical theory of quantum feedback networks has recently been developed by Gough and James \cite{QFN1} for general open quantum dynamical systems interacting with bosonic input fields. In this article we show, that their feedback reduction formula for the coefficients of the closed-loop quantum stochastic differential equation can be formulated in terms of Belavkin matrices. We show that the reduction formula leads to a non-commutative Mobius transformation based on Belavkin matrices, and establish a $\star$-unitary version of the Siegel identities.

math-ph↗

arXiv2010

A Burgers-KPZ Type Parabolic Equation \par\noindent for the Path-Independence of the Density of the Girsanov Transformation

Let $X_t$ solve the multidimensional Itô's stochastic differential equations on $\R^d$ $$dX_t=b(t,X_t)dt+σ(t,X_t)dB_t$$ where $b:[0,\infty)\times\R^d\to\R^d$ is smooth in its two arguments, $σ:[0,\infty)\times\R^d\to\R^d\otimes\R^d$ is smooth with $σ(t,x)$ being invertible for all $(t,x)\in[0,\infty)\times\R^d$, $B_t$ is $d$-dimensional Brownian motion. It is shown that, associated to a Girsanov transformation, the stochastic process $$\int^t_0\langle(σ^{-1}b)(s,X_s),dB_t\rangle+\frac{1}{2}\int^t_0|σ^{-1}b|^2(s,X_s)ds$$ is a function of the arguments $t$ and $X_t$ (i.e., path-independent) if and only if $b=σσ^\ast\nabla v$ for some scalar function $v:[0,\infty)\times\R^d\to\R$ satisfying the time-reversed KPZ type equation $$\frac{\partial}{\partial t}v(t,x)=-\frac{1}{2}\left[\left(Tr(σσ^\ast\nabla^2v)\right)(t,x) +|σ^\ast\nabla v|^2(t,x)\right].$$ The assertion also holds on a connected complete differential manifold.

math.PR↗

arXiv2007

A one dimensional analysis of turbulence and its intermittence for the d-dimensional stochastic Burgers equation

The inviscid limit of the stochastic Burgers equation is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one dimensional) action function. We demonstrate that the geometry of the caustic, level surfaces and Maxwell set can change infinitely rapidly causing turbulent behaviour which is stochastic in nature. The intermittence of this turbulence is demonstrated in terms of the recurrence of two processes.

math.PR↗

arXiv2007

A one dimensional analysis of singularities and turbulence for the stochastic Burgers equation in d-dimensions

The inviscid limit of the stochastic Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one dimensional) action function. We give an explicit expression for an algebraic surface containing the Maxwell set and caustic in the polynomial case. Those parts of the caustic and Maxwell set which are singular are characterised. We demonstrate how the geometry of the caustic, level surfaces and Maxwell set can change infinitely rapidly causing turbulent behaviour which is stochastic in nature, and we determine its intermittence in terms of the recurrent behaviour of two processes.

math.PR↗