# Set Invariance with Probability One for Controlled Diffusion: Score-based Approach

arXiv: https://arxiv.org/abs/2507.22385 · HTML: https://mlsanity.com/paper/2507.22385/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 69 references that are in the map.
- First posted on arXiv 30 Jul 2025.

## Builds on (most cited first, 16 of 16)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations](https://mlsanity.com/paper/2202.02514/) (2022)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Safety Certification for Stochastic Systems via Neural Barrier Functions](https://mlsanity.com/paper/2206.01463/) (2022)
- [A Barrier Function Approach to Finite-Time Stochastic System Verification and Control](https://mlsanity.com/paper/1909.05109/) (2019)
- [Control Barrier Functions for Stochastic Systems](https://mlsanity.com/paper/2003.03498/) (2020)
- [Stochastic reachability of a target tube: Theory and computation](https://mlsanity.com/paper/1810.05217/) (2018)
- [Control Barrier Functions for Stochastic Systems and Safety-critical Control Designs](https://mlsanity.com/paper/2209.08728/) (2022)
- [Reach-Avoid Analysis for Polynomial Stochastic Differential Equations](https://mlsanity.com/paper/2208.10752/) (2022)
- [Safety-Critical Control of Stochastic Systems using Stochastic Control Barrier Functions](https://mlsanity.com/paper/2104.02585/) (2021)
- [On a notion of stochastic zeroing barrier function](https://mlsanity.com/paper/2004.01882/) (2020)
- [Mean viability theorems and second-order Hamilton-Jacobi equations](https://mlsanity.com/paper/2208.13276/) (2022)
- [Safety Under Uncertainty: Tight Bounds with Risk-Aware Control Barrier Functions](https://mlsanity.com/paper/2304.01040/) (2023)

## Abstract (arXiv)

Given a controlled diffusion and a connected, bounded, Lipschitz set, when is it possible to guarantee controlled set invariance with probability one? In this work, we answer this question by deriving the necessary and sufficient conditions for the same in terms of gradients of certain log-likelihoods -- a.k.a. score vector fields -- for two cases: given finite time horizon and infinite time horizon. The deduced conditions comprise a score-based test that provably certifies or falsifies the existence of Markovian controllers for given controlled set invariance problem data. Our results are constructive in the sense when the problem data passes the proposed test, we characterize all controllers guaranteeing the desired set invariance. When the problem data fails the proposed test, there does not exist a controller that can accomplish the desired set invariance with probability one. The computation in the proposed tests involve solving certain Dirichlet boundary value problems, and in the finite horizon case, can also account for additional constraint of hitting a target subset at the terminal time. We illustrate the results using several semi-analytical and numerical examples.