Fluid model

A fluid model replaces discrete counts by continuous quantities whose evolution follows averaged large-scale drifts. A heuristic drift calculation motivates such a model; proving a fluid limit requires a separate convergence argument.

For a two-coordinate fluid model with $\dot{s}=g(n/s)$, $\dot{n}=f(n/s)$ and $s>0$, set $\kappa =n/s$ and $du/dt=1/s$. Then $d\kappa /du=f(\kappa )-\kappa g(\kappa )$ and $d\log s/du=g(\kappa )$. A bounded ratio eventually confined where $g\le -\eta <0$ makes $s$ decay exponentially in the new time. Since $dt/du=s$, the original fluid time has a finite terminal value and both coordinates tend to zero. This is a useful finite-time draining of a fluid model argument; the singular origin is treated by a stopped trajectory.

A nonnegative fluid model drains in finite time if all its coordinates approach zero at a finite terminal time. A ratio-based ordinary differential equation can become undefined at that endpoint; an absorbing extension is an additional modeling convention.

A fluid limit is a scaling limit of a stochastic process obtained by scaling space and time so that random fluctuations vanish and a deterministic trajectory remains. It can justify a fluid model when the required convergence conditions hold.

For a density-dependent Markov jump process with finitely many jump vectors, bounded nonnegative rates and Lipschitz drift $F(x)={\sum }_{\ell }\ell {\beta }_{\ell }(x)$, an initial state converging in probability to $x_0$ gives uniform convergence in probability on every finite horizon to the ODE $\dot{x}=F(x)$, $x(0)=x_0$. The compensated martingale has expected maximal squared size $O(N^{-1})$ by the Doob L2 maximal inequality; Gronwall's inequality transfers that estimate to the solution error.

// Target: probability-and-statistics.bigb

If the rate functions are bounded and Lipschitz on an invariant compact state space, the Poisson time-change representation of a Markov chain has drift $F={\sum }_j{v}_j{\beta }_j$. Each centered Poisson term is uniformly small after division by $n$. The Gronwall inequality then bounds the distance to $\dot{z}=F(z)$ by the initial error and the centered counting error, multiplied by $e^{LT}$. A Poisson maximal concentration bound gives exponentially decreasing error probabilities for a fixed horizon and fixed error tolerance.