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• Pang, Ho Cheung (2026). The viscous variational wave equation with transport noise. Stochastics and Partial Differential Equations: Analysis and Computations. ISSN 2194-0401. doi: 10.1007/s40072-025-00409-5. Fulltekst i vitenarkiv Vis sammendrag

  This article considers the variational wave equation with viscosity and transport noise as a system of three coupled nonlinear stochastic partial differential equations. We prove pathwise global existence, uniqueness, and temporal continuity of solutions to this system in L2x. Martingale solutions are extracted from a two-level Galerkin approximation via the Skorokhod–Jakubowski theorem. We use the apparatus of Dudley maps to streamline this stochastic compactness method, bypassing the usual martingale identification argument. Pathwise uniqueness for the system is established through a renormalisation procedure that involves double commutator estimates and a delicate handling of noise and nonlinear terms. New model-specific commutator estimates are proven.

• Fjordholm, Ulrik Skre; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2025). Convergent Finite Difference Schemes for Stochastic Transport Equations. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 63(1), s. 149–192. doi: 10.1137/23M159946X. Fulltekst i vitenarkiv Vis sammendrag

  We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish 𝐿2 stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The 𝐿2 estimate, crucial for the analysis, is obtained through a discrete duality argument and a comprehensive examination of a class of backward parabolic difference schemes.

• Galimberti, Luca; Holden, Helge; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2024). Global existence of dissipative solutions to the Camassa–Holm equation with transport noise. Journal of Differential Equations. ISSN 0022-0396. 387, s. 1–103. doi: 10.1016/j.jde.2023.12.021. Fulltekst i vitenarkiv Vis sammendrag

  We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term.

• Holden, Helge; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2022). Strong solutions of a stochastic differential equation with irregular random drift. Stochastic Processes and their Applications. ISSN 0304-4149. 150, s. 655–677. doi: 10.1016/j.spa.2022.05.006. Fulltekst i vitenarkiv Vis sammendrag

  We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form where the drift coefficient is random and irregular, with a weak derivative satisfying for some , . The random and regular noise coefficient may vanish. The main contribution is a pathwise uniqueness result under the assumptions that as , and satisfies the one-sided gradient bound , where the process exhibits an exponential moment bound of the form for small times , for some . This study is motivated by ongoing work on the well-posedness of the stochastic Hunter–Saxton equation, a stochastic perturbation of a nonlinear transport equation that arises in the modelling of the director field of a nematic liquid crystal. In this context, the one-sided bound acts as a selection principle for dissipative weak solutions of the stochastic partial differential equation.

• Holden, Helge; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2022). Global well-posedness of the viscous Camassa–Holm equation with gradient noise. Discrete and Continuous Dynamical Systems. Series A. ISSN 1078-0947. 43(2), s. 568–618. doi: 10.3934/dcds.2022163. Fulltekst i vitenarkiv Vis sammendrag

  We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa–Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in () using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod–Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-Itô correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional Itô formula and a DiPerna–Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators.

• Chen, Gui-Qiang G. & Pang, Ho Cheung (2021). Nonlinear anisotropic degenerate parabolic-hyperbolic equations with stochastic forcing. Journal of Functional Analysis. ISSN 0022-1236. 281(12), s. 1–48. doi: 10.1016/j.jfa.2021.109222. Fulltekst i vitenarkiv
• Holden, Helge; Karlsen, Kenneth Hvistendahl & Pang, Ho Cheung (2021). The Hunter-Saxton equation with noise. Journal of Differential Equations. ISSN 0022-0396. 270, s. 725–786. doi: 10.1016/j.jde.2020.07.031. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter–Saxton equation (1.1), and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking and stochastic effects unobserved in the deterministic problem. Notably, we derive an explicit law for the random wave-breaking time.

Se alle arbeider i NVA

• Pang, Ho Cheung (2023). Convergence of stochastic integrals with weakly convergent integrands: applications to SPDEs. Fulltekst i vitenarkiv
• Pang, Ho Cheung (2023). The stochastic compactness method in SPDEs. Fulltekst i vitenarkiv
• Pang, Ho Cheung (2022). Second order commutator estimates in renormalisation theory for SPDEs with gradient-type noise. Fulltekst i vitenarkiv
• Pang, Ho Cheung (2020). The Hunter-Saxton equation with noise. Fulltekst i vitenarkiv

Se alle arbeider i NVA