| Abstract: |
| The LIL is a fundamental probabilistic limit theorem that characterizes the almost sure maximum possible fluctuation of a stochastic process over the long term. However, whether a numerical approximation can preserve this asymptotic pathwise behavior remains an open problem. In this talk, we first establish the LIL of symplectic methods for linear stochastic Hamiltonian systems in Hilbert space, providing a new perspective to reveal the superiority of symplectic methods in capturing the utmost fluctuation of the underlying solution process. For general time-homogeneous Markov processes with a unique invariant measure, we derive the functional LIL for the numerical approximation under verifiable assumptions, where the underlying process is discretized by a decreasing time-step strategy. Finally, we present illustrative examples, including SODEs and SPDEs, demonstrating that our results can be flexibly applied to a broad class of stochastic systems. |
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