Fluid limit

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.