article published yes Antonio Agresti author 673cd0cc-9b9a-11eb-b144-88f30e1fbb720000-0002-9573-2962 Mark Veraar author JuFi department Bridging Scales in Random Materials project In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. Moreover, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn-Hilliard equations, tamed Navier-Stokes equations, and Allen-Cahn equation. https://research-explorer.ista.ac.at/download/12485/17296/2024_ProbTheory_Agresti.pdf application/pdfno Springer Nature2024 eng 0178-8051 1432-2064 2206.00230 00115422650000110.1007/s00440-023-01249-x 188957-1015 A. Agresti, M. Veraar, Probability Theory and Related Fields 188 (2024) 957–1015. Agresti, Antonio, and Mark Veraar. “The Critical Variational Setting for Stochastic Evolution Equations.” <i>Probability Theory and Related Fields</i>, vol. 188, Springer Nature, 2024, pp. 957–1015, doi:<a href="https://doi.org/10.1007/s00440-023-01249-x">10.1007/s00440-023-01249-x</a>. A. Agresti and M. Veraar, “The critical variational setting for stochastic evolution equations,” <i>Probability Theory and Related Fields</i>, vol. 188. Springer Nature, pp. 957–1015, 2024. Agresti, A., &#38; Veraar, M. (2024). The critical variational setting for stochastic evolution equations. <i>Probability Theory and Related Fields</i>. Springer Nature. <a href="https://doi.org/10.1007/s00440-023-01249-x">https://doi.org/10.1007/s00440-023-01249-x</a> Agresti A, Veraar M. 2024. The critical variational setting for stochastic evolution equations. Probability Theory and Related Fields. 188, 957–1015. Agresti, Antonio, and Mark Veraar. “The Critical Variational Setting for Stochastic Evolution Equations.” <i>Probability Theory and Related Fields</i>. Springer Nature, 2024. <a href="https://doi.org/10.1007/s00440-023-01249-x">https://doi.org/10.1007/s00440-023-01249-x</a>. Agresti A, Veraar M. The critical variational setting for stochastic evolution equations. <i>Probability Theory and Related Fields</i>. 2024;188:957-1015. doi:<a href="https://doi.org/10.1007/s00440-023-01249-x">10.1007/s00440-023-01249-x</a> 124852023-02-02T10:45:15Z2025-09-04T11:27:46Z