Control-Dependent Diffusion Coefficients

• Control-dependent diffusion coefficients are parameters whose values are explicitly modulated by system controls, impacting transport, fluctuations, and stability.
• They appear in diverse settings such as chaotic maps, stochastic differential equations, and fractional diffusion models, linking micro-dynamics with emergent behavior.
• Advanced numerical and control approaches, including policy gradient and MPC, address challenges in modeling nonlinear and scale-dependent control effects.

A control-dependent diffusion coefficient is a diffusion parameter whose value depends explicitly on a variable, mechanism, or parameter considered as a "control"—which may encompass system parameters (e.g., map nonlinearities), externally imposed fields, time-dependent factors, spatial structure, or direct control actions (e.g., engineered feedback, actuation, or decision variables in stochastic control problems). The mathematical and physical consequences of this dependence are diverse: such coefficients govern the macroscopic transport rate, the character of fluctuations, and the stability and controllability of the underlying system, with their variation often encoding intricate links between microscopic dynamics and emergent large-scale behavior.

1. Mathematical Formulations and Contexts

Control-dependent diffusion coefficients arise in multiple mathematical contexts, ranging from deterministic chaotic maps, stochastic processes with state- or control-dependent coefficients, fractional diffusion models, to partial differential equations (PDEs) and optimal control systems.

A prototypical example is the deterministic one-dimensional chaotic map, where a control parameter $h$ alters the map's dynamics and, consequently, the diffusion coefficient $D(h)$ characterizing the spreading of a probability density (Knight et al., 2010). In stochastic differential equations (SDEs), control-dependence can take the form

$dX_t = b(X_t, u_t)\,dt + \sigma(X_t, u_t)\,dW_t,$

with drift $b$ and diffusion coefficient $\sigma$ both controlled explicitly via $u_t$ (which may be a deterministic or feedback control: (Davey et al., 23 May 2025, Calixto et al., 6 Aug 2025)). In systems with spatial or parameter heterogeneity, the diffusion coefficient may have the structure $D(x, h)$, varying with space and a control variable $h$ or control function $u(x)$.

Fractional diffusion equations further extend this framework, where the time-dependent scalar function $f(t)$ in coefficients or source terms modulates the effective diffusion rate through the equation's temporal memory kernel (Fujishiro et al., 2015).

2. Deterministic Chaos and Fractal vs. Linear Control Dependence

In deterministic dynamical systems, especially families of chaotic maps, the relationship between the control parameter and the diffusion coefficient can be nontrivial and even fractal. For example, in the lifted Bernoulli shift and related maps, the diffusion coefficient $D_M(h)$ is obtained analytically as

$D_M(h) = \frac{h}{2} + T_M(h),$

where $T_M(h)$ is a generalized Takagi function capturing the fine-scale structure of $D$ as a function of $h$. The Taylor–Green–Kubo formula, evaluated via these functions, enables the extraction of exact analytical expressions (Knight et al., 2010).

Key findings include:

• In certain parameter ranges, $D(h)$ is fractal, with a dense distribution of local extrema tied to the nature of Markov partitions, periodicity, or pre-periodicity of critical orbits.
• Transitions to linear control dependence occur when non-ergodic regimes emerge (e.g., due to phase-space splitting or dominating branches), causing the diffusion coefficient to become robust against micro-dynamical changes despite continued microscopic complexity.
• Topological conjugacy and ergodic properties can yield maps with distinct microdynamics but identical asymptotic transport.

This dichotomy—fractal versus linear response—points to the topological and ergodic sensitivity (or insensitivity) to control, and is not unique to these specific maps but appears generically across minimal chaotic systems.

3. Stochastic Dynamics: State-Dependent, Scale-Dependent, and Control-Dependent Diffusion

In stochastic processes, state or control dependence of the diffusion coefficient introduces subtleties, especially in equilibrium and non-equilibrium statistical mechanics. Simply specifying $D(x)$ is insufficient: the equilibrium probability density $\pi(x)$ is not uniquely determined by $D(x)$ alone, leading to a paradox resolved by specifying both $D(x)$ and $\pi(x)$, with the drift then uniquely determined by the zero-flux (detailed balance) condition: $a(x) = \nabla D(x) + D(x) \nabla \log \pi(x).$ This formulation ensures that robust mesoscopic models and simulations, including those with externally controlled $D(x)$ or discontinuities, accurately yield the chosen steady-state distribution (Tupper et al., 2012).

Scale-dependence further enriches the scenario. For diffusion in liquids, the effective (renormalized) diffusion coefficient is not constant but varies with the observation scale. Stochastic advection-diffusion models incorporating thermal fluctuations demonstrate that the diffusion coefficient $\chi_\text{eff}$ acquires a scale dependence: $\chi_\text{eff} = \chi_0\,I + \chi(\delta),$ where $\chi(\delta)$ captures the contribution of unresolved advection at the coarse-graining scale $\delta$ (Donev et al., 2013). This effect highlights that even in classical fluids, measurement or control at different scales produces different effective transport coefficients, implying that the diffusion constant is not a purely material constant outside the Fickian regime.

4. Control-Dependent Diffusion in Fractional and Anomalous Diffusion Models

The control dependence of diffusion coefficients is relevant in fractional-order diffusion models, particularly in the presence of time-dependent source or coefficient functions. The inversion and stability theory for fractional diffusion often centers on the reconstruction of a time-dependent factor $f(t)$ from pointwise-in-time measurements. The methodology uses integral representations and fixed-point techniques, allowing for Lipschitz stable recovery: $$f(t) = \frac{D_t^\alpha u(x_0, t) + A u(x_0, t)}{R(x_0, t)},$$ when $R(x_0, t)$ is nonzero (Fujishiro et al., 2015). This ensures practical applicability in diagnostics and control, such as environmental monitoring, where the temporal evolution or control of anomalous diffusion rates is of interest.

5. Stochastic Control: HJB Equations and Control-Dependent Diffusion

In stochastic optimal control with control-dependent (and possibly degenerate) diffusion, the system behavior is governed by fully nonlinear degenerate elliptic Hamilton–Jacobi–Bellman (HJB) equations: $$\beta v^\beta(x) - \mathcal{H}\left(x, D_x v^\beta(x), D^2_x v^\beta(x)\right) = 0,$$ with the Hamiltonian

$$\mathcal{H}(x, g_x, H_x) = \inf_{u \in \mathcal{U}} \left\{ \langle \mu_x(x,u), g_x \rangle + \frac{1}{2} \operatorname{Tr} \left( H_x \sigma_x(x,u) \sigma_x(x,u)^\top \right) + L(x,u) \right\}.$$

State constraints are enforced through reflection and Neumann-type boundary conditions (Calixto et al., 6 Aug 2025).

The presence of control in the diffusion term introduces fully nonlinear effects in the HJB and challenges both theoretical and computational approaches. The value function is in general only the unique viscosity solution. For example, optimal controls may depend nonlinearly on gradients and exhibit piecewise or threshold structures.

6. Control Algorithms and Implementation for Control-Dependent Diffusion

The theoretical complexity of control-dependent diffusion translates into algorithmic challenges in policy gradient and model predictive control (MPC) contexts. For constrained stochastic control problems with control in both drift and diffusion, convergence of policy gradient methods is nontrivial. Under strong convexity, proximal policy gradient algorithms can be shown to converge linearly: $$u^{k+1}_t = \operatorname{prox}_U\left( u^k_t - \tau \partial_u H_t(X^k_t, u^k_t, Y^k_t, Z^k_t) \right).$$ The innovation is the adjoint representation of the policy gradient, bypassing the need for fine regularity of the backward SDE solution and accommodating fully nonlinear control dependence (Davey et al., 23 May 2025).

For nonlinear MPC, systems with state- and control-dependent coefficients are represented in pseudo-linear ("SCDC-form") or block-observable canonical forms, facilitating iterative quadratic programming (QP) approaches that update freeze the coefficients at each iteration and adjust the control accordingly (Kamaldar et al., 2023). Such iterative schemes handle complex effects, including nonlinear or saturating diffusion, by adapting QP constraints and the update trajectory in each iteration.

Deep neural network architectures and ODE-based methods further extend these approaches to high-dimensional settings, providing scalable and accurate control for systems with control-dependent diffusion coefficients.

7. Practical Implications and Physical Significance

Control-dependent diffusion coefficients have direct implications in physical, biological, and engineering systems:

• In chaotic transport, the controllable parameter may render diffusion robust or exquisitely sensitive, depending on ergodic properties.
• In cellular and soft-matter systems, specifying both the spatially varying diffusion $D(x)$ and equilibrium density $\pi(x)$ allows explicit control over occupancy and mobility under crowding and heterogeneity.
• Scale-dependent diffusion in fluids highlights the importance of matching modeling and measurement scales.
• In confined liquids, anisotropic and spatially varying diffusion (quantified via local lifetime statistics and corrections for drift (Höllring et al., 2022)) shapes macroscopic transport properties and device performance.
• In stochastic control, the influence of control-dependent dispersion must be addressed both analytically (via viscosity solutions to degenerate elliptic HJB PDEs) and numerically (using iterative optimization with appropriately defined adjoint gradients).

These advances emphasize that the interplay between control, structural heterogeneity, system scale, and the form of macroscopic transport coefficients is central to the predictive modeling, design, and control of complex systems.