Diffusion approximation of a birth-death process

For unit jumps with birth and death rates $b(x),d(x)$, the Fokker--Planck drift and diffusion in the conventionare $u=b-d$ and $D=(b+d)/2$.

Near a stable deterministic equilibrium, replace the drift by its linearization and the diffusion coefficient by its equilibrium value. The resulting Ornstein-Uhlenbeck approximation has a stationary normal distribution.

A reaction's stoichiometric vector outer product multiplied by its reaction propensity function is its contribution to the infinitesimal state-increment covariance. Summing over channels gives $B$. Its normalized form $D$ divides by the two mean copy numbers. The Fokker-Planck equation writes the associated second-derivative term with $B/2$; the covariance evolution and normalized stationary Lyapunov equation instead use $B$ and $D$ themselves. A simultaneous unit loss of two species contributes positive cross noise even when their long-time stationary covariance is negative.

Let $A$ be the drift Jacobian at a stable mean-field equilibrium, $S=\mathrm{diag}({\mu }_i)$, and $B={\sum }_r{a}_r{\nu }_r{\nu }_r^{\mathsf{T}}$ the reaction noise matrix. For normalized fluctuations, put $M=-S^{-1}AS$, $D=S^{-1}B{S}^{-1}$, and ${\eta }_{ij}=\mathrm{Cov}(X_i,X_j)/({\mu }_i{\mu }_j)$. Stationarity of the linear-noise covariance gives the displayed Continuous Lyapunov equation. It applies to driven reaction networks without requiring thermal equilibrium. A mode with zero restoring eigenvalue and nonzero noise has no stationary covariance.

// Target: mathematical-biology.bigb

For RNA birth/death rates ${\lambda }_1,{\beta }_1{x}_1$ and protein rates ${\lambda }_2{x}_{1}f(x_2),{\beta }_2{x}_2$, linearize at stationary means $m_1,m_2$. Put $H=1-\partial \log f/\partial \log x_2=1+h$. The relative-fluctuation drift is $\left(\begin{array}{cc}-{\beta }_1& 0\\ {\beta }_2& -{\beta }_{2}H\end{array}\right)$ and the diffusion matrix is $\mathrm{diag}(2{\beta }_1/m_1,2{\beta }_2/m_2)$. The stationary covariance equation gives the displayed normalized protein variance. Its extrinsic part factors as $H^{-2}[{\beta }_{2}H/({\beta }_1+{\beta }_{2}H)]m_1^{-1}$: static susceptibility squared times a time-averaging factor times environmental noise. Faster response alone weakens averaging; feedback reduces susceptibility and thereby lowers noise. The formula is exact for affine rates when $H=1$, and is a linear noise approximation for nonlinear feedback.

For a unit-jump population chain with one-step probabilities $\alpha k/n$ and $\beta k^2/n^2$, its equilibrium density is $u_0=\alpha /\beta$. Speed time by $n$ and center and scale the population as $(k-n{u}_0)/\sqrt{n}$. The accelerated conditional drift is $-\alpha x-\beta x^2/\sqrt{n}$, and the second moment is $2{\alpha }^2/\beta +3\alpha x/\sqrt{n}+\beta x^2/n$. Their compact-set limits give the displayed Ornstein-Uhlenbeck process. If $2{\alpha }^2<\beta$, the invalid-probability threshold stays a fixed positive density distance above equilibrium. Before that threshold, the generator of $x^2$ is at most one because its drift contribution is nonpositive. The stopped Lyapunov function bound proves compact containment and makes the probability of hitting that threshold on a fixed accelerated horizon tend to zero. This supplies the global hypotheses of the diffusion approximation theorem.

If boundary fluxes decay sufficiently fast, two integrations by parts givewhile the diffusion term contributes zero to the first moment.