论文标题
功能性较大的偏差，用于Stroock近似与一类高斯过程，并应用于小噪声扩散
Functional large deviations for Stroock's approximation to a class of Gaussian processes with application to small noise diffusions
论文作者
论文摘要
让〜$ n = \ left \ {n（t），t \ geq0 \ right \} $是标准的泊松过程，stroock〜 \ cite {strock-cite {strock-1981}构建了一个连续过程的家族。 $θ_ε（r）= \ frac {1}ε（-1）^{n（ε^{ - 2} r）} $，并证明它在连续函数拓扑下弱收敛到标准的布朗尼运动。我们建立了功能性较大偏差原理（LDP），用于由$θ_ε（t）$构建的一类高斯过程的近似值，并为费率函数找到明确的形式。 作为一个应用程序，我们考虑以下（非马克维亚）随机微分方程\ begin {qore*} \ begin {aLigned} x^ε（t） ＆= x_ {0}+\ int^{t} _ {0} b（x^ε（s））ds+λ（ε）\ int^{t} _ {0}σ（x^ε（x^ε（s））并将其Freidlin-Wentzell Type LDP建立为$ε\ rightarrow 0 $。速率函数表明相变现象为$λ（ε）$从一个区域移到另一个区域。
Letting~$N=\left\{N(t), t\geq0\right\}$ be a standard Poisson process, Stroock~ \cite{Stroock-1981} constructed a family of continuous processes by $$Θ_ε(t)=\int_0^tθ_ε(r)dr, \ \ \ \ \ 0 \le t \le 1,$$ where $θ_ε(r)=\frac{1}ε(-1)^{N(ε^{-2}r)}$, and proved that it weakly converges to a standard Brownian motion under the continuous function topology. We establish the functional large deviations principle (LDP) for the approximations of a class of Gaussian processes constructed by integrals over $Θ_ε(t)$, and find the explicit form for rate function. As an application, we consider the following (non-Markovian) stochastic differential equation \begin{equation*} \begin{aligned} X^ε(t) &=x_{0}+\int^{t}_{0}b(X^ε(s))ds+λ(ε)\int^{t}_{0}σ(X^ε(s))dΘ_ε(s), \end{aligned} \end{equation*} where $b$ and $σ$ are both Lipschitz functions, and establish its Freidlin-Wentzell type LDP as $ε\rightarrow 0$. The rate function indicates a phase transition phenomenon as $λ(ε)$ moves from one region to the other.
