噪声驱动的随机Cahn-Hilliard方程的大偏差原理

Large deviation principle for stochastic Cahn-Hilliard equation driven by fractional and colored noise

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。

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.