We analyze the long-time behavior of numerical schemes for a class of monotone stochastic partial differential equations(SPDEs)driven by multiplicative noise.By deriving several time-independent a priori estimates for...We analyze the long-time behavior of numerical schemes for a class of monotone stochastic partial differential equations(SPDEs)driven by multiplicative noise.By deriving several time-independent a priori estimates for the numerical solutions,combined with the ergodic theory of Markov processes,we establish the exponential ergodicity of these schemes with a unique invariant measure,respectively.Applying these results to the stochastic Allen-Cahn equation indicates that these schemes always have at least one invariant measure,respectively,and converge strongly to the exact solution with sharp time-independent rates.We also show that these numerical invariant measures are exponentially ergodic and thus give an affirmative answer to a question proposed in[J.Cui et al.,Stochastic Process.Appl.,134(2021)],provided that the interface thickness is not too small.展开更多
We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise.The main ingredient of our method d...We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise.The main ingredient of our method depends on the satisfaction of a Lyapunov condition followed by a uniform moments'estimate,combined with the regularity property for the full discretization.We transform the original stochastic equation into an equivalent random equation where the discrete stochastic convolutions are uniformly controlled to derive the desired uniform moments'estimate.Applying the main result to the stochastic Allen-Cahn equation driven by multiplicative white noise indicates that this full discretization is uniquely ergodic for any interface thickness.Numerical experiments validate our theoretical results.展开更多
基金supported by the National Natural Science Foundation of China(No.12101296)by the Basic and Applied Basic Research Foundation of Guangdong Province(No.2024A1515012348)by the Shenzhen Basic Research Special Project(Natural Science Foundation)Basic Research(General Project)(Nos.JCYJ20220530112814033,JCYJ20240813094919026).
文摘We analyze the long-time behavior of numerical schemes for a class of monotone stochastic partial differential equations(SPDEs)driven by multiplicative noise.By deriving several time-independent a priori estimates for the numerical solutions,combined with the ergodic theory of Markov processes,we establish the exponential ergodicity of these schemes with a unique invariant measure,respectively.Applying these results to the stochastic Allen-Cahn equation indicates that these schemes always have at least one invariant measure,respectively,and converge strongly to the exact solution with sharp time-independent rates.We also show that these numerical invariant measures are exponentially ergodic and thus give an affirmative answer to a question proposed in[J.Cui et al.,Stochastic Process.Appl.,134(2021)],provided that the interface thickness is not too small.
基金supported by the National Natural Science Foundation of China(No.12101296)by the Basic and Applied Basic Research Foundation of Guang dong Province(No.2024A1515012348)by the Shenzhen Basic Research Special Project(Natural Science Foundation)Basic Research(General Project)(Nos.JCYJ20220530112814033,JCYJ20240813094919026).
文摘We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise.The main ingredient of our method depends on the satisfaction of a Lyapunov condition followed by a uniform moments'estimate,combined with the regularity property for the full discretization.We transform the original stochastic equation into an equivalent random equation where the discrete stochastic convolutions are uniformly controlled to derive the desired uniform moments'estimate.Applying the main result to the stochastic Allen-Cahn equation driven by multiplicative white noise indicates that this full discretization is uniquely ergodic for any interface thickness.Numerical experiments validate our theoretical results.
