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Navier-Stokes Equations with Fractional Dissipation and Associated Doubly Stochastic Yule Cascades
Radu Dascaliuc, Tuan N. Pham, Enrique Thomann, Edward C. Waymire
https://arxiv.org/abs/2509.10806

Navier-Stokes Equations with Fractional Dissipation and Associated Doubly Stochastic Yule Cascades
Parametric regions are identified in terms of the spatial dimension $d$ and the power $γ$ of the Laplacian that separate explosive from non-explosive regimes for the self-similar doubly stochastic Yule cascades (DSY) naturally associated with the deterministic fractional Navier-Stokes equations (FNSE) on $\mathbb{R}^d$ in the scaling-supercritical setting. Explosion and/or geometric properties of the DSY, are then used to establish non-uniqueness, local existence, and finite-time blow-up resul…

Strong convergence rates of stochastic theta methods for index 1 stochastic differential algebraic equations under non-globally Lipschitz conditions
Lin Chen, Ziheng Chen, Jing Zhao
https://arxiv.org/abs/2509.11618

Strong convergence rates of stochastic theta methods for index 1 stochastic differential algebraic equations under non-globally Lipschitz conditions
This work investigates numerical approximations of index 1 stochastic differential algebraic equations (SDAEs) with non-constant singular matrices under non-global Lipschitz conditions. Analyzing the strong convergence rates of numerical solutions in this setting is highly nontrivial, due to both the singularity of the constraint matrix and the superlinear growth of the coefficients. To address these challenges, we develop an approach for establishing mean square convergence rates of numerical …

Trust Region Algorithm for Stochastic Minimax Problems with Decision-Dependent Distributions
Yan Gao, Yongchao Liu, Zili Luo
https://arxiv.org/abs/2509.12868 https://

Trust Region Algorithm for Stochastic Minimax Problems with Decision-Dependent Distributions
Stochastic minimax optimization has drawn much attention over the past decade due to its broad applications in machine learning, signal processing and game theory. In some applications, the probability distribution of uncertainty depends on decision variables, as the environment may respond to decisions. In this paper, we propose a trust region algorithm for finding the stationary points of stochastic minimax problems with decision-dependent distributions. At each iteration, the algorithm local…

Geometric Phase of Stochastic Oscillators
Yangyang Du
https://arxiv.org/abs/2509.10900 https://arxiv.org/pdf/2509.10900

Geometric Phase of Stochastic Oscillators
Several definitions of phase have been proposed for stochastic oscillators, among which the mean-return-time phase and the stochastic asymptotic phase have drawn particular attention. Quantitative comparisons between these two definitions have been done in previous studies, but physical interpretations of such a relation are still missing. In this work, we illustrate this relation using the geometric phase, which is an essential concept in both classical and quantum mechanics. We use properties…

Random attractor and SRB measure for stochastic Hopf bifurcation under discretization
Chuchu Chen, Jialin Hong, Yibo Wang
https://arxiv.org/abs/2509.12865 https://

Random attractor and SRB measure for stochastic Hopf bifurcation under discretization
Chaotic phases in stochastic differential equations are characterized by two essential long-time dynamical features: a random attractor capturing asymptotic geometry and a Sinai-Ruelle-Bowen (SRB) measure describing statistical information. This paper investigates whether the stochastic Hopf bifurcation under discretization could inherit both features. We establish that the stochastic Hopf bifurcation under discretization induces a discrete random dynamical system. Further, we prove that this d…

Localised solution of stochastic Schr\"odinger equation describing simultaneous continuous measurement of position and momentum
Daniel J. Bedingham
https://arxiv.org/abs/2509.11832

Localised solution of stochastic Schrödinger equation describing simultaneous continuous measurement of position and momentum
A stochastic Schrödinger equation is presented to describe simultaneous continuous measurement of the position and momentum of a non-relativistic particle. The equation is solved to yield a state localised in position and momentum contingent on the uncertainty principle. The state is understood as that to which the particle tends after a period of continuous measurement. The solution takes the form of a wave packet of fixed Gaussian shape whose centre undergoes random motion in phase space. It…

Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning
Jia-Qi Yang, Lei Shi
https://arxiv.org/abs/2509.11070 https://arxiv.org/pdf…

Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning
We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which admit discrete spectral decompositions, and (ii) diagonal kernels of the form $K(x,x')=k(x,x')T$, where $k$ is a scalar-valued kernel and $T$ is a positive operator on the output space. This broad setting induces expressive vector-valued reproducing kernel Hilbe…

Model Predictive Control with High-Probability Safety Guarantee for Nonlinear Stochastic Systems
Zishun Liu, Liqian Ma, Yongxin Chen
https://arxiv.org/abs/2509.11584 https://

Model Predictive Control with High-Probability Safety Guarantee for Nonlinear Stochastic Systems
We present a model predictive control (MPC) framework for nonlinear stochastic systems that ensures safety guarantee with high probability. Unlike most existing stochastic MPC schemes, our method adopts a set-erosion that converts the probabilistic safety constraint into a tractable deterministic safety constraint on a smaller safe set over deterministic dynamics. As a result, our method is compatible with any off-the-shelf deterministic MPC algorithm. The key to the effectiveness of our method…

Existence and uniqueness on $L^1$ solutions of multidimensional BSDEs with generators of stochastic one-sided Osgood type
Yuru Lai, Xinying Li, Shengjun Fan
https://arxiv.org/abs/2509.11927

Existence and uniqueness on $L^1$ solutions of multidimensional BSDEs with generators of stochastic one-sided Osgood type
By imposing an additional integrability condition on the first component of the solution, this paper establishes an existence and uniqueness result for $L^1$ solutions of multidimensional backward stochastic differential equations (BSDEs) with a general terminal time when the generator $g$ satisfies a stochastic one-sided Osgood condition along with a general growth condition in the state variable $y$, and a stochastic Lipschitz condition in the state variable $z$, extending and strengthening T…