Narrow Results

In this paper we have studied information dynamics of both internal and external noises driven harmonic oscillator. Based on the Fokker–Planck description of the stochastic processes and the Schwarz inequality principle we have calculated upper bound of rate of change of information entropy with time and its deviation from the time derivative of the entropy. The bound and the deviation monotonically decreases to zero during the relaxation of a given nonequilibrium state to a stationery state. The relaxation time increase with increase of noise correlation time of the internal colored thermal noise but it is independent on the correlation time of the external colored noise. However, an extremum behavior appears in the variation of the bound as a function of parameter of the system such as damping strength, noise correlation time of thermal and nonthermal noises etc.

Let be a Fock space and, for any non-negative integer k, let be the sum of all homogeneous chaos spaces of order at most k. For all non-negative integers m and n, the Wick product is a bounded bilinear operator from Γ (H_c)_m × Γ (H_c)_n into Γ (H_c)_{m +n} with norm greater than or equal to . In the monograph¹ S. Janson conjectured that this lower bound is exact. In this paper we prove this conjecture. In addition, we prove that a pair of nonzero vectors (ϕ, ψ)∈ Γ (H_c)_m × Γ (H_c)_n achieve this bound if and only if both vectors are multiples of the homogeneous products, i.e. ϕ =αu^⊗m, ψ =βu^⊗n, with u, v∈ H_c and α, β ∈ ℂ.

It is known that if X is a normally distributed random variable, and ♢ and E denote the Wick product and expectation, respectively, then for any non-negative integers m and n, and any polynomial functions f and g of degrees at most m and n, respectively, the following inequality holds:

We find the best constant c(m, n, r), such that the inequality:

An analogue of the Fourier transform will be introduced for all square integrable continuous martingale processes whose quadratic variation is deterministic. Using this transform we will formulate and prove a stochastic Heisenberg inequality.

It is known that for any normally distributed random variable X, any Borel measurable functions f and g, and any p and q positive numbers, such that 1/p + 1/q = 1, if and are both square integrable, then the Wick product f(X) ♢ g(X), of the random variables f(X) and g(X), is also square integrable, and the following inequality holds: