In this paper, we introduce a generalized continuous-time quantum walk (CTQW) on a hypercube, which can be interpreted as a CTQW with a “coin”. We examine its properties from a perspective of probability distribution and spectral theory, and obtain formulae for calculating its probability distributions and spectra, respectively. In particular, we prove that, for a wide range of initial states, the walk produces the uniform measure on the hypercube as its a stationary measure. Finally, we show that the time-reversal symmetry of the walk can be described directly by a unitary operator. Some other results are also proven.
In this paper, Lie symmetry analysis method is applied to a nonlinear fractional Burgers-type equation with time-dependent diffusion coefficient, which contains the mixed derivative of the Riemann-Liouville time-fractional derivative and the first order x-derivative. We obtained all the Lie symmetries admitted by the equation and used them to reduce the governing fractional partial differential equation to some fractional ordinary differential equations with Erd & eacute;lyi-Kober fractional derivative or Riemann-Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.
As an application of a general theory of quasi-invariant states developed in Ref. [1], we prove a classical de Finetti type structure theorem for quasi-invariant states and, for this class of states, we also extend the conditional version of this theorem, first proved in Ref. [2].
In this paper, we consider the following McKean-Vlasov SDE driven by fractional Brownian motion with small noise X-t,X-epsilon = X0 +integral(t)(0)b(s,X s,epsilon,& Laplacetrf;X(s,epsilon))ds + epsilon(H)integral(t)(0)sigma(s,& Laplacetrf; X-s,X-epsilon)dB(s)(H,)t is an element of [0,T], where the initial condition X-0 is a real number, & Laplacetrf;X(t,epsilon )denotes the law of X-t,X-epsilon,X- {B-t(H),t >= 0} is a fractional Brownian motion with Hurst index H is an element of (1 /2, 1) and epsilon is a small parameter. We study the rate of Fisher information convergence in the central limit theorem for the solution of small noise SDE, then we show that the convergence rate epsilon(2H) is of optimal order.
