本文研究了随机广义Burgers-Huxley（SGBH）方程在时间上受小乘性白噪声扰动和空间上受有色噪声扰动下解的全局可解性和一致大偏差. SGBH方程具有多项式阶的非线性，本文所考虑的噪声是无穷维的，其系数具有线性增长。首先，我们利用截断变元和压缩映射原理证明了Walsh到SGBH方程意义下的\textsl{unique local mild solution}的存在性。然后利用局部温和解的一致界、停时参数、紧性和Skorokhod表示定理，得到了该模型的全局可解性结果.利用一致Laplace原理，利用变分表示方法，得到了SGBH方程解的规律的\textsl{large deviation principle}（LDP）.进一步，我们利用弱收敛方法导出了两种不同拓扑下解的规律的\textsl{uniform large deviation principle}（ULDP）。首先，在$\mathrm{C}([0, T ];\mathrm{L}^p([0,1])) $拓扑中，一致性是在初始条件的$\mathrm{L}^p([0,1])$有界集上，其次，在$\mathrm{C} ([0, T ] \times[0,1])$拓扑中，一致性是在$\mathrm{C}([0,1])$范数中的有界子集上。最后，我们考虑了噪声系数有界的时空白噪声扰动下的SGBH方程，建立了了解的ULDP定律。在这项工作中得到的结果适用于随机Burgers'以及Burgers-Huxley方程。
英文摘要：
In this work, we focus on the global solvability and uniform large deviations for the solutions of stochastic generalized Burgers-Huxley (SGBH) equation perturbed by a small multiplicative white in time and colored in space noise. The SGBH equation has the nonlinearity of polynomial order and noise considered in this work is infinite dimensional with a coefficient having linear growth. First, we prove the existence of a \textsl{unique local mild solution} in the sense of Walsh to SGBH equation with the help of a truncation argument and contraction mapping principle. Then the global solvability results are established by using uniform bounds of the local mild solution, stopping time arguments, tightness properties and Skorokhod's representation theorem.
By using the uniform Laplace principle, we obtain the \textsl{large deviation principle} (LDP) for the law of solutions to SGBH equation by using variational representation methods. Further, we derive the \textsl{uniform large deviation principle} (ULDP) for the law of solutions in two different topologies by using a weak convergence method. First, in the $\mathrm{C}([0, T ];\mathrm{L}^p([0,1])) $ topology where the uniformity is over $\mathrm{L}^p([0,1])$-bounded sets of initial conditions, and secondly in the $\mathrm{C} ([0, T ] \times[0,1])$ topology with uniformity being over bounded subsets in the $\mathrm{C}([0,1])$-norm.
Finally, we consider SGBH equation perturbed by a space-time white noise with bounded noise coefficient and establish the ULDP for the laws of solutions. The results obtained in this work hold true for stochastic Burgers' as well as Burgers-Huxley equations.
Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients
LMAM, School of Mathematical Sciences, Peking University（北京大学数学科学学院）；School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
