On Uniqueness of"SDE Decomposition"in A-type Stochastic Integration

An innovative theoretical framework for stochastic dynamics based on a decomposition of a stochastic differential equation (SDE) has been developed with an evident advantage in connecting deterministic and stochastic dynamics, as well as useful applications in physics, engineering, chemistry and biology. It introduces the A-type stochastic integration for SDE beyond traditional Ito’s or Stratonovich’s interpretation. Serious question on its uniqueness was recently raised. We provide here both mathematical and physical demonstrations that the uniqueness is guaranteed. Such discussion leads to a better understanding on the robustness of the novel framework. We also discuss the limitation of a related approach of obtaining potential function from steady state distribution. 1 ar X iv :1 60 3. 07 92 7v 1 [ co nd -m at .s ta tm ec h] 2 5 M ar 2 01 6 Recently Zhou and Li (ZL) [1] extensively discussed connections among three known ways of finding potential landscapes in generic nonequilibrium processes in biology, chemistry, engineering and physics described by stochastic differential equations. Some good questions were formulated, along with a few insightful results. They speculated that a set of differential equations proposed 10 years ago, referred to as “SDE decomposition” by ZL, would have generally no unique solution. In this comment we show that such speculation is not supported by either mathematical or physical reasoning. A few more points raised by ZL are further clarified. We start by review the original definition [2, 3] of the “SDE decomposition” into three components: a potential function φ(q) (a scalar function), a friction matrix S(q) (symmetric and semi-positive definite) and a Lorentz-like force represented by a transverse matrix A(q) (antisymmetric). Matrices S(q) and A(q) are determined by the potential condition Eq. (1a) and the generalized Einstein relation Eq. (1b): ∇× {[S(q) + A(q)] f(q)} = 0 , (1a) [S(q) + A(q)]D(q)[S(q)− A(q)] = S(q) . (1b) where f(q) is the deterministic drift velocity and D(q) the diffusion matrix given by the SDE. The ∇ × x = ∂ixj − ∂jxi for arbitrary n-dimensional vector x. In principle, the n unknowns in [S(q) +A(q)] can be determined by solving the n(n− 1)/2 partial differential equations in Eq. (1a) under proper boundary conditions for matrices S(q) and A(q), together with the n(n+ 1)/2 equations given by Eq. (1b) (n unknowns and n equations). Eq. (1a) and Eq. (1b) are the same as Eq. (28a) and (28b) in [1]. Equation (1) may be transformed into a more standard but equivalent form. Notice that Eq. (1b) implies a relation that [S + A] = [D +Q]−1, where Q is an antisymmetric matrix. We therefore rewrite Eq. (1a) and Eq. (1b) as ∇× { [D(q) +Q(q)] f(q) } = 0 (2) where n(n − 1)/2 partial differential equations determine the n(n − 1)/2 unknowns in the antisymmetric matrix Q with necessary boundary conditions for Q. One class of boundary conditions has been employed is that near fixed points every component of Q is a smooth