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• Ren, Panpan; Tang, Hao & Wang, Feng-Yu (2024). Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations. Potential Analysis. ISSN 0926-2601. 61, p. 379–407. doi: 10.1007/s11118-023-10113-5. Full text in Research Archive Show summary

  By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.

• Tang, Hao & Wang, Zhi-An (2023). Strong solutions to a nonlinear stochastic aggregation-diffusion equation. Communications in Contemporary Mathematics. ISSN 0219-1997. doi: 10.1142/S0219199722500730. Full text in Research Archive
• Tang, Hao (2023). On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise. Journal of Functional Analysis. ISSN 0022-1236. 285(9). doi: 10.1016/j.jfa.2023.110075. Full text in Research Archive Show summary

  The stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise are considered in this work. We first establish two new cancellation properties on pseudo-differential operators, which considerably extend the previous results for transport type noise only involving gradient operator. Then, we obtain results on local solution, blow-up criterion, and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data is also studied for the multiplicative noise case.

• Miao, Yingting; Rohde, Christian & Tang, Hao (2023). Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities. Stochastics and Partial Differential Equations: Analysis and Computations. ISSN 2194-0401. doi: 10.1007/s40072-023-00291-z. Full text in Research Archive
• Tang, Hao & Yang, Anita (2023). Noise effects in some stochastic evolution equations: Global existence and dependence on initial data. Annales de l'I.H.P. Probabilites et statistiques. ISSN 0246-0203. 59(1), p. 378–410. doi: 10.1214/21-AIHP1241. Full text in Research Archive Show summary

  In this paper, we consider the noise effects on a class of stochastic evolution equations including the stochastic Camassa–Holm equations with or without rotation. We first obtain the existence, uniqueness and a blow-up criterion of pathwise solutions in Sobolev space Hs with s>3/2 . Then we prove that strong enough noise can prevent blow-up with probability 1, which justifies the regularization effect of strong nonlinear noise in preventing singularities. Besides, such strengths of noise are estimated in different examples. Finally, for the interplay between regularization effect induced by the noise and the dependence on initial conditions, we introduce and investigate the stability of the exiting time and construct an example to show that the multiplicative noise cannot improve both the stability of the exiting time and the continuity of the dependence on initial data simultaneously. Dans cet article, nous considérons les effets du bruit sur une classe d’équations d’évolution stochastiques y compris les équations stochastiques de Camassa–Holm avec ou sans rotation. Nous obtenons d’abord l’existence, l’unicité et un critère d’explosion de solutions pathwise dans l’espace de Sobolev Hs avec s>3/2. Ensuite, nous prouvons qu’un bruit suffisamment fort peut empêcher l’explosion avec probabilité 1, ce qui justifie l’effet régularisant du bruit non linéaire fort dans la prévention des singularités. De plus, de telles forces de bruit sont estimées dans les différents exemples. Enfin, pour l’interaction entre l’effet de régularisation induit par le bruit et la dépendance par rapport aux conditions initiales, nous introduisons et étudions la stabilité du temps de sortie et construisons un exemple pour montrer que le bruit multiplicatif ne peut pas améliorer simultanément la stabilité du temps de sortie et la continuité de la dépendance par rapport aux données initiales.

• Alonso-Orán, Diego; Miao, Yingting & Tang, Hao (2022). Global existence, blow-up and stability for a stochastic transport equation with non-local velocity. Journal of Differential Equations. ISSN 0022-0396. 335, p. 244–293. doi: 10.1016/j.jde.2022.06.025. Full text in Research Archive Show summary

  In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish a local-in-time theory, i.e., existence, uniqueness and blow-up criterion for pathwise solutions in Sobolev spaces Hs with s>3. Thereafter, we give a picture of the long time behavior of the solutions based on the type of noise we consider. On one hand, we identify a family of noises such that blow-up can be prevented with probability 1, guaranteeing the existence and uniqueness of global solutions almost surely. On the other hand, in the particular linear noise case, we show that singularities occur in finite time with positive probability, and we derive lower bounds of these probabilities. To conclude, we introduce the notion of stability of exiting times and show that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data.

• Alonso-Orán, Diego; Rohde, Christian & Tang, Hao (2021). A Local-in-Time Theory for Singular SDEs with Applications to Fluid Models with Transport Noise. Journal of nonlinear science. ISSN 0938-8974. 31(6), p. 1–55. doi: 10.1007/s00332-021-09755-9. Full text in Research Archive Show summary

  Abstract We establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in Hilbert spaces. The key requirement relies on an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. By establishing a cancellation estimate for certain differential operators of order one with suitable coefficients, we give the detailed constructions of such regular approximations for certain examples. In particular, we show novel local-in-time results for the stochastic two-component Camassa–Holm system and for the stochastic Córdoba–Córdoba–Fontelos model.

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