For a stochastic energy exchange model derived numerically from billiard dynamics, the paper proves a law of large numbers to a discrete heat equation, a central limit theorem, and an O(M^(-1)) approximation by a mesoscopic stochastic differential equation.
Using coupling methods to estimate sample quality for stochastic differential equations
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abstract
A probabilistic approach for estimating sample qualities for stochastic differential equations is introduced in this paper. The aim is to provide a quantitative upper bound of the distance between the invariant probability measure of a stochastic differential equation and that of its numerical approximation. In order to extend estimates of finite time truncation error to infinite time, it is crucial to know the rate of contraction of the transition kernel of the SDE. We find that suitable numerical coupling methods can effectively estimate such rate of contraction, which gives the distance between two invariant probability measures. Our algorithms are tested with several low and high dimensional numerical examples.
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From deterministic dynamics to thermodynamic laws II: Fourier's law and mesoscopic limit equation
For a stochastic energy exchange model derived numerically from billiard dynamics, the paper proves a law of large numbers to a discrete heat equation, a central limit theorem, and an O(M^(-1)) approximation by a mesoscopic stochastic differential equation.