Numerically asymptotical preservation of the large deviations principles for invariant measures of Langevin equations

Abstract

In this paper, we focus on two kinds of large deviations principles (LDPs) of the invariant measures of Langevin equations and their numerical methods, as the noise intensity $ε\to 0$ and the dissipation intensity $ν\to\infty$ respectively. First, by proving the weak LDP and the exponential tightness, we conclude that the invariant measure $\{μ_{ν,ε}\}$ of the exact solution satisfies the LDPs as $ε\to0$ and $ν\to\infty$ respectively. Then, we study whether there exist numerical methods asymptotically preserving these two LDPs of $\{μ_{ν,ε}\}$ in the sense that the rate functions of invariant measures of numerical methods converge pointwise to the rate function of $\{μ_{ν,ε}\}$ as the step-size tends to zero. The answer is positive for the linear Langevin equation. For the small noise case, we show that a large class of numerical methods can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ε>0}$ as $ε\to0$. For the strong dissipation case, we study the stochastic $θ$-method ($θ\in[1/2,1]$) and show that only the midpoint scheme ($θ=1/2$) can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ν>0}$ as $ν\to\infty$. These results indicate that in the linear case, the LDP as $ε\to0$ and the LDP as $ν\to\infty$ for the invariant measures of numerical methods have intrinsic differences: the common numerical methods can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ε>0}$ as $ε\to0$ while the asymptotical preservation of numerical methods for the LDP of $\{μ_{ν,ε}\}_{ν>0}$ as $ν\to\infty$ depends on the choice of numerical methods. To the best of our knowledge, this is the first result of investigating the relationship between the LDPs of invariant measures of stochastic differential equations and those of their numerical methods.