Weak Convergence and Ergodic Limit of Stochastic ECS Systems

 

Author: Zhang Suhang

Luoyang, Independent Researcher

 

Abstract

 

This paper extends the Extremum-Conservation-Symmetry (ECS) framework to discrete stochastic linear dynamical systems, and investigates the continuous limiting behavior as the sampling period tends to zero. We consider a discrete stochastic recursive system with additive white noise perturbation:

 

x_{n+1} = L_h x_n + \sqrt{h}\,\xi_n

 

where \xi_n are independent and identically distributed random vectors with covariance matrix \Xi, and the discrete propagation operator L_h satisfies compatibility and stability conditions. It is proved that as h\to 0, the piecewise continuous interpolation process x_h(t) converges weakly to an Ornstein–Uhlenbeck diffusion process

 

dx(t) = \mathcal{A}x(t)dt + \sigma dW(t)

 

in the function space C([0,T],\mathbb{R}^d), with the diffusion matrix obeying \sigma\sigma^\top = \Xi.

 

We further analyze the evolution law of ECS quadratic conserved quantities under stochastic limits. It is shown that their expectations satisfy

 

\mathbb{E}\big[x(t)^\top\Sigma_c x(t)\big] = \mathbb{E}\big[x(0)^\top\Sigma_c x(0)\big] + t \cdot \operatorname{tr}(\Sigma_c\Xi)

 

which implies that continuous energy injection by noise leads to linear growth of the expected conserved quantity, unless an orthogonality condition \operatorname{tr}(\Sigma_c\Xi)=0 holds between the diffusion matrix and the conserved matrix.

 

In the zero-mean stationary regime, the strong law of large numbers for discrete systems is fully consistent with the ergodic theorem for continuous systems; the time average converges almost surely to zero, achieving a rigorous unification of discrete and continuous ECS theory within the stochastic framework.

 

Keywords: Stochastic ECS system; weak convergence; Ornstein–Uhlenbeck process; ergodic theorem; conserved quantity; discrete–continuous consistency

 

 

 

1. Introduction

 

Within the framework of Extremum-Conservation-Symmetry (ECS) and Multi-Origin Curvature (MOC), the previous chapters have systematically established the convergence from deterministic discrete systems to continuous counterparts, continuous extension of symmetry groups, limit behavior of conserved quantities, and structure-preserving invariance.

 

Real-world dynamical systems are universally perturbed by noise. To ensure the completeness and self-consistency of ECS theory, it is necessary to extend the framework to stochastic systems. We prove that for discrete stochastic ECS systems, statistical laws, conservation properties and ergodic behavior transit continuously to continuous stochastic dynamical systems without breaking the core structure of extremum, conservation and symmetry.

 

This paper focuses on the continuous limit of discrete-time linear stochastic recursive systems, with three core objectives:

 

1. Prove that the continuous interpolation of discrete stochastic recursion converges weakly to a continuous Ornstein–Uhlenbeck diffusion process;

2. Characterize precisely the temporal evolution of ECS quadratic conserved quantities under stochastic limits, and clarify the influence of noise injection on conservation laws;

3. Establish the consistency between the discrete-time strong law of large numbers and the continuous-time ergodic theorem, closing the theoretical loop of ECS in stochastic settings.

 

All conclusions are derived strictly from recursive structure, moment estimation and weak convergence arguments, maintaining logical unity and stylistic consistency with the deterministic ECS system.

 

2. Basic Setup and Preliminary Conditions

 

Let the state space be \mathbb{R}^d and sampling period h>0. Consider the discrete linear stochastic ECS system:

 

x_{n+1} = L_h x_n + \sqrt{h}\,\xi_n,\quad n=0,1,2,\dots

 

Assumptions:

 

1. The initial state x_0 is a random vector independent of the noise sequence, with \mathbb{E}[\|x_0\|^2]<\infty;

2. \{\xi_n\}_{n\ge 0} are independent and identically distributed d-dimensional random vectors with zero mean \mathbb{E}[\xi_n]=0, covariance

 

\mathbb{E}[\xi_n \xi_n^\top] = \Xi,

 

and finite fourth-order moments;

 

3. The discrete propagation operator L_h\in\mathbb{R}^{d\times d} satisfies standard compatibility and stability:- Local approximation order: \| L_h - (I + h\mathcal{A}) \| \le C_0 h^2, where \mathcal{A} is Hurwitz-stable;

- Uniform spectral radius bound: there exist h_0>0,\alpha>0 such that \rho(L_h) \le 1 - \alpha h + O(h^2) for all 0<h\le h_0;

- Infinitesimal generator: \mathcal{A} = \lim\limits_{h\to 0} \dfrac{L_h - I}{h}.

 

Continuous Interpolation Process

 

For fixed T>0, define the piecewise linear continuous interpolation x_h(t)\in C([0,T],\mathbb{R}^d): for t\in[nh,(n+1)h),

 

x_h(t) = x_n + \frac{t-nh}{h}\big(x_{n+1}-x_n\big).

 

The process is continuous on the interval and coincides exactly with the discrete system at sampling instants, serving as a standard bridge connecting discrete and continuous stochastic trajectories.

 

3. Weak Convergence to the Ornstein–Uhlenbeck Limit

 

This section establishes the fundamental weak convergence theorem, demonstrating that the trajectory distribution of discrete stochastic ECS systems converges continuously to a continuous diffusion process.

 

Theorem 1 (Weak Convergence of Stochastic ECS Systems)

As h\to 0^+, the interpolation process x_h(\cdot) converges weakly in C([0,T],\mathbb{R}^d) to an Ornstein–Uhlenbeck process x(t) governed by the stochastic differential equation

 

dx(t) = \mathcal{A}x(t)dt + \sigma dW(t),

 

where W(t) is the standard d-dimensional Wiener process and the diffusion matrix satisfies \sigma \sigma^\top = \Xi.

 

Proof: We adopt the classical criterion for weak convergence of stochastic processes: tightness + convergence of finite-dimensional distributions.

 

Step 1: Uniform boundedness of second moments

Expanding the recursion for N=\lfloor T/h\rfloor:

 

x_n = L_h^n x_0 + \sqrt{h}\sum_{k=0}^{n-1} L_h^{n-1-k} \xi_k.

 

By exponential stability of L_h and i.i.d. noise properties, we obtain uniform moment bounds

 

\mathbb{E}[\|x_n\|^2] \le C_T,\quad \forall n\le N,\;0<h\le h_0,

 

with C_T independent of h. The interpolation construction further yields

 

\mathbb{E}\Big[\sup_{0\le t\le T}\|x_h(t)\|^2\Big] \le C_T',

 

ensuring uniform square integrability.

 

Step 2: Tightness

By standard tightness criteria, it suffices to verify uniform Hölder continuity: there exist C>0,\gamma>0 such that for all 0\le s<t\le T,

 

\mathbb{E}\big[\|x_h(t)-x_h(s)\|^2\big] \le C|t-s|^\gamma.

 

For |t-s|\le h, linear interpolation gives

 

\|x_h(t)-x_h(s)\| \le \|L_h-I\|\|x_n\| + \sqrt{h}\|\xi_n\|.

 

Combined with moment boundedness, we obtain \mathbb{E}[\|x_h(t)-x_h(s)\|^2] \le C|t-s|. For larger time intervals the semigroup property yields the required estimate with \gamma=1, which implies tightness.

 Step 3: Convergence of finite-dimensional distributions

Rewrite the recursion in incremental form:

x_{n+1}-x_n = (L_h-I)x_n + \sqrt{h}\xi_n = h\mathcal{A}x_n + o(h)x_n + \sqrt{h}\xi_n.

Let n_k=\lfloor t_k/h\rfloor. By the functional central limit theorem (generalization of Donsker’s theorem), the discrete martingale increments

\frac{1}{\sqrt{h}}\sum_{k=0}^{\lfloor t/h \rfloor-1} \xi_k

converge weakly to the Wiener process W(t). The drift term h\mathcal{A}x_n converges to \int_0^t \mathcal{A}x(s)ds. By uniqueness of SDE solutions, finite-dimensional distributions converge to those of the Ornstein–Uhlenbeck process.

 

Combining tightness and finite-dimensional convergence, x_h(\cdot) converges weakly to the limiting diffusion process. ∎

 

4. Evolution of Stochastic ECS Conserved Quantities

 

We now characterize the expected behavior of ECS quadratic conserved quantities under continuous limits, and quantify the impact of noise on conservation structure.

 

Recall the continuous ECS conserved matrix \Sigma_c satisfying the Lyapunov equation

\mathcal{A}^\top \Sigma_c + \Sigma_c \mathcal{A} = 0

for homogeneous conservative systems.

 

Theorem 2 (Expected Evolution of ECS Conserved Quantities)

Let the limiting Ornstein–Uhlenbeck process have finite initial mean, \mathcal{A}^\top \Sigma_c + \Sigma_c \mathcal{A}=0, and \sigma\sigma^\top=\Xi. Then for all t\ge 0,

\mathbb{E}\big[x(t)^\top \Sigma_c x(t)\big] = \mathbb{E}\big[x(0)^\top \Sigma_c x(0)\big] + t \cdot \operatorname{tr}(\Sigma_c \Xi).

 

Proof: Apply Itô’s formula to V(t)=x(t)^\top \Sigma_c x(t). From

dx(t) = \mathcal{A}x(t)dt + \sigma dW(t),

we compute

[

dV = \big((\mathcal{A}x)^\top\Sigma_c x + x^\top\Sigma_c(\mathcal{A}x)\big)dt

 

- \operatorname{tr}(\sigma^\top \Sigma_c \sigma)dt

​

- 2x^\top \Sigma_c \sigma dW.

]

The first term vanishes due to \mathcal{A}^\top \Sigma_c + \Sigma_c \mathcal{A}=0. Taking expectation, the martingale term has zero mean, hence

\frac{d}{dt}\mathbb{E}[V(t)] = \operatorname{tr}(\sigma^\top \Sigma_c \sigma)

= \operatorname{tr}(\Sigma_c \Xi).

Integrating yields the desired identity. ∎

 

Corollary 1 (Necessary and Sufficient Condition for Conservation)

\mathbb{E}[x(t)^\top \Sigma_c x(t)] is constant if and only if \operatorname{tr}(\Sigma_c \Xi)=0. Two typical cases preserving conservation:

 

1. Deterministic limit: \Xi=0 (no noise);

​

2. Orthogonality between diffusion and conserved matrices: the range of \Xi lies in the nullspace of \Sigma_c, or \Sigma_c and \Xi satisfy special algebraic orthogonality.

 

Remark 1 (Physical Interpretation)

Linear growth of the expectation reflects persistent energy injection by noise, in contrast to exact conservation in deterministic ECS systems. When the orthogonality condition holds, the net energy injection vanishes and the expected conserved quantity remains invariant.

 

5. Consistency Between Discrete LLN and Continuous Ergodic Theorem

 

In the zero-mean stationary regime, we prove that discrete time averages and continuous time averages share identical limits, achieving cross-scale unification of ergodic behavior.

 

Theorem 3 (Consistency of Ergodic Limits)

Suppose the Ornstein–Uhlenbeck process admits a unique zero-mean Gaussian stationary distribution, and the discrete system x_n shares the same stationary state. Then:

 

1. Discrete strong law of large numbers: as N\to\infty,

\frac{1}{N}\sum_{n=0}^{N-1} x_n \xrightarrow{\text{a.s.}} 0;

​

2. Continuous ergodic theorem: as T\to\infty,

\frac{1}{T}\int_0^T x(t)dt \xrightarrow{\text{a.s.}} 0;

​

3. Consistency: under the joint limit h\to 0,\,N=\lfloor T/h\rfloor\to\infty,

\left\| \frac{1}{N}\sum_{n=0}^{N-1}x_n - \frac{1}{T}\int_0^T x(t)dt \right\| \xrightarrow{\text{a.s.}} 0.

Discrete and continuous time averages converge almost surely to the same limit.

 

Proof:

(1) The discrete sequence is stationary and ergodic; by the strong law of large numbers for stationary processes, time averages converge almost surely to the stationary mean zero.

(2) The Ornstein–Uhlenbeck process is stationary and ergodic, satisfying the continuous ergodic theorem.

(3) By weak convergence in C([0,T],\mathbb{R}^d), discrete and continuous trajectories are uniformly close. The time-average error is bounded by C\sqrt{h}\to 0, establishing asymptotic consistency. ∎

 

Remark 2 (Nonzero Stationary Mean)

For nonzero stationary mean \mu\neq 0, define the centered system \tilde{x}_n=x_n-\mu. The centered version satisfies the zero-mean ergodic theorem, while the trace condition for conserved quantities remains unchanged. A nonzero mean only shifts the deterministic equilibrium without altering core ECS structure.

 

6. Numerical Illustration and Discussion

 

Take the one-dimensional case d=1,\mathcal{A}=-1, yielding trivial \Sigma_c=0. For a nontrivial two-dimensional harmonic oscillator:

\mathcal{A} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},\quad \Sigma_c = I,\quad \Xi = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.

Here \operatorname{tr}(\Sigma_c\Xi)=1\neq 0, hence

\mathbb{E}[x(t)^\top x(t)] = \mathbb{E}[x(0)^\top x(0)] + t,

showing linear growth of expected energy. Numerical simulation confirms this theoretical prediction.

 

7. Conclusion

 

This paper extends the ECS–MOC framework rigorously to stochastic dynamical systems, establishing a complete convergence theory from discrete stochastic systems to continuous stochastic counterparts. The main contributions are summarized as follows:

 

1. Weak convergence: The interpolated discrete stochastic process converges weakly to an Ornstein–Uhlenbeck diffusion, realizing the continuous transition from discrete lattice noise to continuous white noise;

​

2. Evolution of conserved quantities: The exact relation

\mathbb{E}[x(t)^\top\Sigma_c x(t)] = \mathbb{E}[x(0)^\top\Sigma_c x(0)] + t\operatorname{tr}(\Sigma_c\Xi)

is derived, revealing the linear energy injection effect of noise and giving a precise algebraic condition for preserved conservation;

​

3. Ergodic unification: The discrete strong law of large numbers and continuous ergodic theorem are unified under joint limiting procedures.

 

The results guarantee that extremum structure, conservation laws and symmetry invariance remain self-consistent, continuously transitional and structurally unbroken in stochastic settings, closing the theoretical loop of ECS in random dynamical systems. The entire exposition adopts purely geometric and probabilistic rigorous formulation without extraneous heuristic assumptions.