Results and publications

Stochastic Reaction Networks (SRNs) are a broad class of continuous-time Markov chains used to model interacting particle systems in biochemistry, epidemiology, ecology, and population dynamics. The dynamics of an SRN are encoded by a finite directed graph — the reaction graph — and one of the central challenges in the field is to relate structural properties of this graph to the long-term probabilistic behaviour of the system, independently of specific rate parameters.

ConStRAINeD addressed this challenge along two complementary axes. Objective 1 targeted the stationary behaviour of SRNs in the classical well-mixed setting; Objective 2 extended the framework to spatially heterogeneous systems via random interaction graphs and reaction-diffusion dynamics. These two axes correspond to the Limit behavior and Spatial heterogeneity threads described on the project page; the precise mathematical setting is on Mathematical details. The paragraphs below summarise the main outcomes; numbers in square brackets refer to the full publication list further down the page.

Objective 1 — Stationary behaviour of SRNs

(See also project page → Limit behavior.)

The headline result of the project is a complete proof of the long-standing chemical recurrence conjecture in dimension two [16]: every weakly reversible SRN with two species is positive recurrent, regardless of rate constants. The argument combines a Wentzell–Freidlin-style coarse-graining near the axes of the state space — where Poisson product-form stationary measures rule out outward drift — with a chemical Lyapunov function patched together from local, weakly-endotactic Lyapunov functions in the bulk. This is the most substantial progress on the problem in decades; the full conjecture in arbitrary dimension remains open. A preview was presented at the ConStRAINeD Workshop in June 2025.

A new general tool for convergence rates of SRNs is the path method for exponential ergodicity of CTMCs on ℤ^d [1]: it provides sufficient conditions for the positivity of the spectral gap without requiring time-reversibility, and yields, as an immediate corollary, that every open complex-balanced network is exponentially ergodic. Stability under random Markovian switching between network configurations was studied in [11], which establishes matrix conditions for both exponential ergodicity and transience in the high- and low-switching regimes — and shows that stability can change, potentially many times, as the switching rate varies.

Several further contributions enlarge the toolkit for stationary analysis. A systematic theory of stochastic ordering for CTMCs and mass-action SRNs [15] identifies a finite set of explicit linear conditions — checkable by linear programming — that guarantee the existence of a pathwise coupling preserving a specified component-wise ordering for all time, with applications to Michaelis–Menten kinetics, SIS epidemiology, signalling cascades, and Lotka–Volterra-type competition models, and a companion open-source algorithm. An explicit example of noise-induced stabilisation [12] exhibits a network that blows up in finite time under deterministic mass-action kinetics yet is positive recurrent stochastically, with a stabilisation mechanism independent of boundary effects. Ongoing work on stationary circuit balance [17] generalises detailed balance to a broader class of non-reversible CTMCs, allowing the explicit computation of stationary measures for SRNs that fall outside the classical reversible setting. On the applications side, a stochastic model for nanoparticle growth in an unusual parameter regime, where the growth rate of the first newly formed particle is of the same order as the nucleation rate, was analysed in [2]; the paper provides strong evidence for the emergence of a deterministic limit for the final particle size distribution.

Objective 2 — Spatial extensions of SRNs

(See also project page → Spatial heterogeneity.)

Spatial heterogeneity was attacked from two directions: random interaction graphs and stochastic PDEs. A statistical-mechanics approach to spatial Marcus–Lushnikov coagulation processes [4] derives an explicit Gibbsian representation for the empirical particle configuration, from which a large-deviation principle, a law of large numbers, criteria for gelation, and a connection to the Smoluchowski equation all follow. Complementary results on gelation in a general cluster coagulation framework appear in [5], and a study of large deviations for Brownian particles with coagulation is in preparation [18]. A further paper [3] establishes a large-deviation principle for the covariance process in fully connected Gaussian neural networks, transporting interacting-particle techniques into modern machine-learning architectures.

On the reaction-diffusion side, an Edwards–Wilkinson limit [13] shows that, under diffusive rescaling, the fluctuations of the quenched density of a diffusion in a Gaussian random environment converge to an additive stochastic heat equation, characterising the first-order correction to the quenched CLT in the divergence-free case. A homogenisation result [14] for a passive scalar transported by a locally supported white noise velocity field establishes convergence to an effective diffusion equation; work on particle pair separation in 2D Gaussian incompressible velocity fields [19] is in preparation. Finally, an SRN-inspired diffusion model was applied to the data-driven study of solar photovoltaic adoption in Italy [6].

Other contributions

Beyond the two main objectives, project members contributed to neighbouring fields: random splitting schemes for point vortex flows [8]; fair currency incentives in repeated weighted congestion games [9]; scalable Bayesian inference for generalised linear mixed models [10]; and a Markovian model for the householder solar-panel consumer [7].

Full list of publications

Names of project members are highlighted.