Well-posedness and uniform large deviation principle for stochastic generalized Burgers-Huxley equation perturbed by a multiplicative noise
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 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 large deviation principle (LDP) for the law of solutions to SGBH equation by using variational representation methods. Further, we derive the uniform large deviation principle (ULDP) for the law of solutions in two different topologies by using a weak convergence method. First, in the