Correlated Process Noise and Componentwise Positivity

Why Correlation Is Not a Positivity Condition

Correlated process noise can represent shared biological or environmental variation across compartments. But correlation describes the dependence between random increments; it does not, by itself, keep any component of the state from crossing zero. Componentwise nonnegativity is a boundary property: at the face where one compartment is zero, its own instantaneous noise and drift must be compatible with remaining in the nonnegative orthant.

Let \(X_t=(X_{1,t},\ldots,X_{d,t})^\mathsf{T}\) be the vector of compartment states. Write a diffusion driven by a standard \(m\)-dimensional Brownian motion \(W_t\) as

where \(b:\mathbb{R}^d\to\mathbb{R}^d\) is the drift and \(S:\mathbb{R}^d\to\mathbb{R}^{d\times m}\) is the effective diffusion coefficient. Its instantaneous covariance matrix is \(a(x)=S(x)S(x)^\mathsf{T}\). The nonnegative orthant is \(\mathbb{R}_+^d=\{x:x_i\geq 0\text{ for every }i\}\). Invariance means that a solution started anywhere in this orthant stays there at all later times, almost surely.

The criterion concerns the effective diffusion coefficient, not a particular bookkeeping choice of Brownian drivers. A model may be written with correlated Brownian motions, or transformed to independent ones; the boundary test uses the resulting covariance. Under standard existence and regularity conditions, it is the variance of the coordinate at the face—not the sign or size of correlations with other coordinates—that determines whether noise can push that coordinate across zero.

Correlated Drivers and Boundary Conditions

Suppose instead that the model is written

where \(R\) is a fixed positive semidefinite correlation matrix. Factoring \(R=R^{1/2}(R^{1/2})^\mathsf{T}\) gives the effective coefficient \(S(x)=G(x)R^{1/2}\), so

Here \(G_i(x)\) denotes row \(i\) of \(G(x)\). Thus the diffusion boundary condition is \(G_i(x)R G_i(x)^\mathsf{T}=0\) whenever \(x_i=0\). If \(R\) is positive definite, this is equivalent to \(G_i(x)=0\). If \(R\) is singular, a nonzero row can still have zero effective variance when it lies in a null direction of \(R\). Correlation can therefore matter to the calculation of variance, but it does not generally cancel a coordinate’s noise.

When the boundary variance is zero, all instantaneous cross-covariances involving that coordinate also vanish. Indeed, positive semidefiniteness of \(a\) implies \(|a_{ij}|^2\leq a_{ii}a_{jj}\), so \(a_{ii}=0\) forces \(a_{ij}=0\) for every \(j\). This is a consequence of zero variance, not a separate way to achieve positivity.

A Boundary Theorem and Its Justification

The following version gives a precise setting in which the boundary test is necessary and sufficient. It assumes regular coefficients so that solutions exist uniquely and the usual moment calculations are valid. More singular models require their own existence and boundary analysis.

The equivalence between \(S_i(x)=0\) and \(a_{ii}(x)=0\) follows from \(a_{ii}(x)=\sum_k S_{ik}(x)^2\). To see why the conditions are necessary, start the process at a point \(x\) on face \(x_i=0\). Global Lipschitz regularity gives \(E|X_s-x|^2=O(s)\). Consequently, \(E[b_i(X_s)]=b_i(x)+O(\sqrt{s})\), and integrating the drift yields

If the process stays nonnegative, then \(E[X_{i,t}]\geq0\); the expansion rules out \(b_i(x)<0\). For the variance condition, continuity of \(S\) and the same small-time moment bound imply

The drift divided by \(\sqrt{t}\) tends to zero, and the diffusion increment has the displayed normal limit. If \(a_{ii}(x)>0\), that limit assigns probability \(1/2\) to negative values, contradicting invariance from \(X_{i,0}=0\). Hence \(a_{ii}(x)=0\), or equivalently \(S_i(x)=0\).

For sufficiency, let \(X^-_i=\max(-X_i,0)\), \(X^-=(X^-_1,\ldots,X^-_d)\), and \(V(X)=\sum_i(X^-_i)^2\). Let \(X^+\) be the componentwise projection onto the orthant. When \(X_i<0\), the point \(X^+\) is on the face \(x_i=0\), so \(b_i(X^+)\geq0\) and \(S_i(X^+)=0\). Lipschitz continuity, with constant \(L\), then gives

Apply Itô’s formula to \(V\), whose first derivative is continuous and whose second derivative exists except at zero (the corresponding generalized Itô formula applies). For each negative coordinate, the drift contribution is \(-2X_i^-b_i(X)\), bounded above by \(2L X_i^-|X^-|\); its diffusion contribution is \(|S_i(X)|^2\), bounded above by \(L^2|X^-|^2\). Summing and using \(\sum_i X_i^-\leq\sqrt{d}|X^-|\) gives a constant \(C\) such that the drift of \(V\) is at most \(CV\). After stopping to ensure integrability, taking expectations and applying Grönwall’s inequality yields

when \(X_0\in\mathbb{R}_+^d\). Since \(V\geq0\), it follows that \(V(X_t)=0\) almost surely. Removing the stopping by localization proves invariance. \(\square\)

Worked Compartment Models

Diagnostics, Alternatives, and Limitations

For a proposed compartment model, check each face separately. Evaluate the drift component \(b_i\) at \(x_i=0\), then calculate \(a_{ii}\) there from the full covariance structure. Checking only that \(R\) is a valid correlation matrix, or that the off-diagonal covariance has the expected sign, does not answer either boundary question. A numerical simulation that happens to stay positive over a finite grid is also not a proof of continuous-time invariance.

These are conditions for invariance when the process may start anywhere in the closed orthant, including on its boundary. A model initialized strictly in the interior may have a boundary that it never reaches; that is a different claim and needs an accessibility analysis. Also, the theorem above assumes globally Lipschitz coefficients. Square-root coefficients are not Lipschitz at zero, so passing the boundary test alone does not establish existence, uniqueness, or invariance for such a model. The earlier material on Square-Root Diffusions and the Feller Condition addresses the additional scalar boundary issues.

A common mistake is to argue that strongly positively correlated compartment increments “move together,” so neither component can cross zero alone. The additive-noise example disproves that reasoning: each coordinate’s marginal increment still has its own nonzero variance at its boundary. The correction is to test the marginal instantaneous variance \(a_{ii}\) and the corresponding drift at every face. Correlation affects joint behavior, but it cannot replace these componentwise boundary conditions.