Large deviation principle for stochastic Cahn-Hilliard equation driven by fractional and colored noise(噪声驱动的随机Cahn-Hilliard方程的大偏差原理)

Abstract

This paper studies the existence and uniqueness of mild solution to stochastic Cahn-Hilliard equations, driven by fractional-colored noise, which is fractional in time and colored in space, with spatial kernel f. A local solution is found by truncating drift term and applying variable substitution to stochastic integral. We prove the tightness of truncated solution by estimating Green function. Finally, a weak convergence of local solution is explored to verify the existence and uniqueness for mild solution of original equation. Coefficient conditions related to Hurst exponent H is then revealed. Furthermore, regularity of the skeleton are checked by applying Cauchy-Schwarz, Burkholder's inequalities and estimating Green function. It makes use of Gronwall's lemma and Girsanov's theorem to reduce large deviation form. We obtain Freidlin-Wentzell inequality in a special space, in which extension of Garsia's lemma plays an important role. The large deviation principle with a small perturbation can then be established.(研究了时间分式空间有色噪声驱动的随机Cahn-Hilliard方程解的存在性和唯一性。利用截断函数处理漂移项，利用变量替换处理随机积分，得到了局部解；验证了局部解的弱收敛性，获得了原方程温和解与Hurst 指数之间的关系；验证了方程在特殊噪声下骨架函数的正则性，得到了Freidlin-Wentzell关系式。最后验证了大偏差原理。)

Keywords

• cahn-hilliard equation(cahn-hilliard方程)
• mild solution(温和解)
• fractional-colored noise(有色噪声)
• large deviation principle(大偏差原理)