State-Dependent Diffusion in the Forward Equation

Why the Diffusion Coefficient Appears Inside a Derivative

When random fluctuations depend on the current concentration, the diffusion contribution to the density’s evolution is not generally a constant multiple of its second spatial derivative. The state-dependent variance and the density occur together inside a second derivative. This placement matters: differentiating the product creates additional terms involving derivatives of the variance. An equivalent probability-flux form makes those terms visible in a different way and is useful for checking boundary conditions and probability conservation.

We consider a scalar Itô diffusion on an interval \(I\), with state \(X_t\), drift \(b(x,t)\), diffusion coefficient \(\sigma(x,t)\), and variance coefficient \(a(x,t)=\sigma(x,t)^2\). The density of \(X_t\), when it exists, is denoted by \(p(x,t)\). All formulas below refer to the Itô drift. If a model is specified in the Stratonovich convention, its drift must first be converted to the Itô convention.

The Scalar Forward Operator

The assumptions for the classical density equation can be stated locally. Suppose \(b\) and \(\sigma\) are sufficiently regular for the SDE to have a nonexplosive solution on the time interval of interest, and suppose its law has a density \(p\). To assert the equation pointwise, assume in addition that \(p\) is continuously differentiable in time and twice continuously differentiable in space, and that the coefficient products below have the derivatives used in the calculation. These smoothness conditions are sufficient conditions for the classical form; the equation can also be interpreted weakly under weaker assumptions.

Here subscripts denote partial derivatives with respect to \(x\). The theorem’s weak justification follows from Itô’s formula. For a smooth test function \(\varphi\) with compact support in the interior of \(I\), Itô’s formula gives the generator \(b\varphi_x+\tfrac12a\varphi_{xx}\). Taking expectations and using the density yields $$ \frac{d}{dt}\int_I \varphi(x)p(x,t)\,dx =\int_I\left(b(x,t)\varphi_x(x)+\frac12a(x,t)\varphi_{xx}(x)\right)p(x,t)\,dx. $$ Compact support removes boundary terms in two integrations by parts. Thus the right side equals $$ \int_I\varphi(x)\left[-\partial_x(bp)+\frac12\partial_{xx}(ap)\right]dx. $$ Since this identity holds for every such test function, it is the forward equation in the weak sense. If \(p\) and the displayed derivatives are continuous, the weak identity gives the stated pointwise equation. Expanding \(\partial_x(bp)=b_xp+bp_x\), and then \(\tfrac12\partial_{xx}(ap)=\tfrac12(a_{xx}p+2a_xp_x+ap_{xx})\), proves the expanded version. Finally, $$ -\partial_xJ=-\partial_x(bp)+\frac12\partial_{xx}(ap), $$ which proves the flux form with the specified convention. \(\square\)

The factor \(a=\sigma^2\), rather than \(\sigma\), is the variance rate in the forward operator. The coefficient derivatives in the expanded version are not optional corrections: they follow from differentiating the product \(ap\). For constant \(a\), both derivatives \(a_x\) and \(a_{xx}\) vanish, and the diffusion term reduces to \(\tfrac12a p_{xx}\). That familiar simplification does not hold for state-dependent diffusion.

Worked Forms for Pharmacokinetic Models

Flux, Conservation, and the Difference Between Forms

The flux form is a conservation law: the density at a location changes according to the net probability flowing into or out of that location. For a finite subinterval \([u,v]\subset I\), integrate the equation to obtain $$ \frac{d}{dt}\int_u^v p(x,t)\,dx=-J(v,t)+J(u,t). $$ Thus probability in the interval decreases when outward flux at \(v\) exceeds inward flux at \(u\). On a bounded state space, zero-flux boundaries \(J=0\) at both endpoints imply conservation of total probability. On the real line, the corresponding conclusion requires suitable decay so the flux vanishes at both infinities. If probability can leave the state space, total mass need not be conserved within that space.

A frequent source of error is replacing \(\partial_{xx}(ap)\) with \(\partial_x(a p_x)\). These are generally different: $$ \partial_{xx}(ap)=a_{xx}p+2a_xp_x+a p_{xx}, \qquad \partial_x(a p_x)=a_xp_x+a p_{xx}. $$ The second expression omits \(a_{xx}p+a_xp_x\). The flux can also be written as $$ J=\left(b-\frac12a_x\right)p-\frac12a p_x. $$ This identity follows by expanding \(\partial_x(ap)=a_xp+ap_x\). The adjusted coefficient in this flux expression is not a change to the Itô drift in the SDE; it is simply an algebraic way of writing the same probability current.

Diagnostics, Alternatives, and Limitations

The conservative form \(-\partial_xJ\) and the second-derivative form \(-\partial_x(bp)+\tfrac12\partial_{xx}(ap)\) are equivalent, not competing models. The conservative form is often clearest for interpreting probability flow and imposing boundary conditions. The expanded form is useful for inspecting local coefficients or substituting a particular model, but it is more vulnerable to product-rule errors. A practical check is to derive the flux from \(J=bp-\tfrac12\partial_x(ap)\), differentiate it, and compare term by term with the expanded operator.

• Check the variance rate. If the SDE has noise coefficient \(\sigma\), the coefficient inside the forward operator is \(a=\sigma^2\), not \(\sigma\).
• Check every product derivative. For variable \(a\), confirm that the expanded diffusion term includes \(\tfrac12a_{xx}p+a_xp_x+\tfrac12ap_{xx}\).
• Check the flux sign. With the convention used here, \(\partial_t p=-\partial_xJ\), not \(+\partial_xJ\).
• Check probability accounting. Integrating over the state space should produce boundary flux terms; unexplained loss of mass may indicate a missing boundary condition or an algebraic error.

The classical pointwise equation assumes enough smoothness for the derivatives to exist. Some diffusion models have degenerate coefficients, nonsmooth densities, or boundaries that require weak formulations and careful boundary analysis. The equation alone also does not guarantee that a proposed model respects a desired state space. Those questions require additional analysis of the SDE and its boundaries, not just expansion of its forward operator.