AI 中文总结
研究由高斯噪声驱动的受控泛函微分方程的小噪声渐近性，通过布埃 - 迪皮伊变分表示等方法，证明小噪声对数价值函数的博弈论上下界，给出博弈有值条件，得到完整小噪声拉普拉斯原理。
AI 中文摘要
我们研究由加性高斯噪声驱动的受控泛函微分方程的小噪声渐近性。高斯噪声在抽象维纳空间上建模，涵盖经典布朗扰动和非马尔可夫扰动，如分数布朗运动。假设漂移系数是非预期的，在状态路径上是利普希茨连续的且具有线性增长。对于有界一致连续成本泛函，我们证明了小噪声对数价值函数的博弈论上下界，并在相关确定性零和博弈有值时确定其极限。在极限博弈中，一个玩家选择漂移控制，另一个选择高斯噪声的卡梅伦 - 马丁位移，并由相应的二次能量成本惩罚。我们进一步提供了博弈有值的充分的范型凸性和凹性条件，从而得到一个完整的小噪声拉普拉斯原理。证明结合了抽象维纳空间上的布埃 - 迪皮伊变分表示、受控解映射的路径稳定性以及卡梅伦 - 马丁位移的适配有限维逼近。
英文摘要
We study small-noise asymptotics for controlled functional differential equations driven by additive Gaussian noise. The Gaussian noise is modeled on an abstract Wiener space, covering both classical Brownian perturbations and non-Markovian perturbations such as fractional Brownian motion. The drift coefficient is assumed to be non-anticipative, Lipschitz continuous in the state path, and of linear growth. For bounded uniformly continuous cost functionals, we prove game-theoretic lower and upper bounds for the small-noise logarithmic value functions and identify their limit whenever the associated deterministic zero-sum game has a value. In the limiting game, one player chooses the drift control, while the other selects a Cameron--Martin shift of the Gaussian noise, penalized by the corresponding quadratic energy cost. We further provide sufficient Fan-type convexity and concavity conditions under which the game has a value, thereby obtaining a full small-noise Laplace principle. The proof combines the Boué--Dupuis variational representation on abstract Wiener spaces with pathwise stability of the controlled solution map and adapted finite-dimensional approximations of Cameron--Martin shifts.
