This paper investigates in depth how stochastic perturbations affect the integrable structure of Hamiltonian systems and develops a Kolmogorov-Arnold-M & ouml;ser (KAM) theory for stochastic Hamiltonian dynamics, in the sense of the most probable path. We first derive the Onsager-Machlup functional for stochastic Hamiltonian systems driven by time-dependent noise coefficients and identify the most probable path of the system trajectories. Building on this, we establish a large deviation principle and obtain an explicit rate function that quantitatively characterizes trajectory deviations, in particular for rare events. The main contribution of this work is to prove that, under stochastic noise, the original quasi-periodic invariant tori persist in the sense of the most probable path, thereby demonstrating the stability of KAM structures in random environments. Moreover, we show that the Onsager-Machlup functional coincides exactly with the large deviation rate function, thereby providing a quantitative characterization of both the structural persistence of quasi-periodic motions and the geometry of fluctuations in stochastic Hamiltonian systems. Overall, our results extend the classical KAM framework to stochastic settings and offer new insights into the behavior of complex dynamical systems under noise.
Computer experiments have extensive applications in various fields, such as industry and manufacturing. Maximin distance designs are commonly utilized in computer experiments due to their excellent space-filling properties and asymptotic optimality. Numerous construction methods have been proposed for generating maximin distance designs, and several upper bounds on the maximin Lp-distance have been provided for evaluating the distance properties of the generated designs. However, existing upper bounds are too loose to be attained in most cases. Consequently, many well-performing designs may be considered mediocre-performing. To address this issue, we integrate the characteristics of U-type designs with the sphere packing concept. More appropriate upper bounds on the maximin Lp-distance for U-type designs are established. Furthermore, we demonstrate the effectiveness of the new upper bounds by constructing U-type designs that achieve or approach these bounds and comparing the new upper bounds with existing upper bounds in the scenario of Latin hypercube designs.
In this paper, we propose deep fitted value iteration (DFVI), an iterative algorithm designed to estimate the optimal state-value function, where deep neural networks with ReLU activations are used as the hypothesis class. By exploiting the contraction property of the optimal Bellman operator, DFVI constructs an iterative procedure that refines state-value function estimates through a sequence of least-squares regression updates. Unlike action-value-based approaches, DFVI approximates a function on the state space alone and therefore avoids neural-network approximation over the joint state-action space. For finite action spaces, we establish a rigorous non-asymptotic error bound between the state-value function induced by the estimated greedy policy and the optimal state-value function. Through an error-propagation analysis, we show that the overall error is primarily governed by the estimation error accumulated across iterations. Moreover, we decompose this estimation error into statistical and approximation components, establishing explicit bounds for each. These results yield principled insights into the required network expressivity and sample complexity for achieving reliable value estimation in practice.
In this paper, we study the free boundary problem for an incompressible viscoelastic fluid system. Under a natural force balance condition on the free boundary in the absence of surface tension, we establish the short-time vanishing viscosity limit. To this end, we derive a uniform a priori estimate with respect to the viscosity. By reformulating the problem in Lagrangian coordinates, we observe that the elastic term exerts a smoothing effect, which enables us to obtain normal derivative estimates via an ordinary differential equation (ODE) structure. Notably, our framework for justifying the vanishing viscosity limit is built upon standard Sobolev spaces, rather than conormal Sobolev spaces.
Structural breaks frequently arise in economic policy evaluation, biomedical monitoring and other dynamic systems, posing substantial challenges for partially linear models (PLMs) that contain nonparametric nuisance components. In this paper, we develop a unified tail-adaptive cumulative sum (CUSUM) framework for structural break testing and estimation in PLMs. The proposed methodology integrates cross-fitted semiparametric composite quantile regression with a weighted CUSUM process, allowing robust inference under heterogeneous and heavy-tailed error distributions. A key theoretical contribution is the establishment of a uniform oracle equivalence result, showing that the feasible plug-in CUSUM process converges uniformly to its oracle linear-model counterpart over the time index. This equivalence provides the theoretical foundation for valid tail-adaptive testing and multiplier bootstrap inference in high-dimensional settings. We further establish non-asymptotic localization bounds, showing that the proposed estimator attains the optimal rate up to logarithmic factors. The procedure is implemented within a seeded binary segmentation scheme to accommodate multiple change-points. Simulation studies and an empirical application to energy policy data demonstrate the effectiveness and robustness of the proposed method.
