@article{21894, abstract = {The Dean–Kawasaki equation—one of the most fundamental SPDEs of fluctuating hydrodynamics—has been proposed as a model for density fluctuations in weakly interacting particle systems. In its original form, it is highly singular and fails to be renormalizable, even by approaches such as regularity structures and paracontrolled distributions, hindering mathematical approaches to its rigorous justification. It has been understood recently that it is natural to introduce a suitable regularization, for example, by applying a formal spatial discretization or by truncating high-frequency noise: This yields well-posed equations that should still precisely approximate the law of the particle density fluctuations. In the present work, we prove that a regularization in the form of a formal discretization of the Dean–Kawasaki equation indeed accurately describes density fluctuations in systems of weakly interacting diffusing particles: We show that, in suitable weak metrics, the law of fluctuations as predicted by the discretized Dean–Kawasaki SPDE approximates the law of fluctuations of the original particle system, up to an error that is of arbitrarily high order in the inverse particle number and a discretization error. In particular, the Dean– Kawasaki equation provides a means for efficient and accurate simulations of density fluctuations in weakly interacting particle systems.}, author = {Cornalba, Federico and Fischer, Julian L and Ingmanns, Jonas and Raithel, Claudia}, issn = {2168-894X}, journal = {The Annals of Probability}, keywords = {Weakly interacting particle systems, fluctuating hydrodynamics, Dean-Kawasaki equation, stochastic PDEs, numerical approximation}, number = {1}, pages = {155--215}, publisher = {Institute of Mathematical Statistics}, title = {{Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation}}, doi = {10.1214/25-aop1763}, volume = {54}, year = {2026}, } @article{21379, abstract = {We study a (1 + 1)-dimensional semi-discrete random variational problem that can be interpreted as the geometrically linearized version of the critical 2-dimensional random field Ising model. The scaling of the correlation length of the latter was recently characterized in Probab. Duke Math. J. 172(9), 1781–1811 (2023) and arXiv:2011.08768v3, (2022); our analysis is reminiscent of the multi-scale approach of the latter work and of Combinatorica 9, 161–187 (1989) . We show that at every dyadic scale from the system size down to the lattice spacing the minimizer contains at most order-one Dirichlet energy per unit length. We also establish a quenched homogenization result in the sense that the leading order of the minimal energy becomes deterministic as the ratio system size / lattice spacing diverges. To this purpose we adapt arguments from arXiv:2401.06768, (2024) on the (d + 1)-dimensional version our the model, with a Brownian replacing the white noise potential, to obtain the initial large-scale bounds. Based on our estimate of the (p = 3)-Dirichlet energy, we give an informal justification of the geometric linearization. Our bounds, which are oblivious to the microscopic cut-off scale provided by the lattice spacing, yield tightness of the law of minimizers in the space of continuous functions as the lattice spacing is sent to zero.}, author = {Otto, Felix and Palmieri, Matteo and Wagner, Christian}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, publisher = {Springer Nature}, title = {{On minimizing curves in a Brownian potential}}, doi = {10.1007/s00440-026-01468-y}, year = {2026}, } @article{10011, abstract = {We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow-is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.}, author = {Hensel, Sebastian and Laux, Tim}, issn = {1945-743X}, journal = {Journal of Differential Geometry}, keywords = {Mean curvature flow, gradient flows, varifolds, weak solutions, weak-strong uniqueness, calibrated geometry, gradient-flow calibrations}, pages = {209--268}, publisher = {International Press}, title = {{A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness}}, doi = {10.4310/jdg/1747065796}, volume = {130}, year = {2025}, } @article{19505, abstract = {In this paper, we introduce and study the primitive equations with non-isothermal turbulent pressure and transport noise. They are derived from the Navier–Stokes equations by employing stochastic versions of the Boussinesq and the hydrostatic approximations. The temperature dependence of the turbulent pressure can be seen as a consequence of an additive noise acting on the small vertical dynamics. For such a model we prove global well-posedness in H^1 where the noise is considered in both the Itô and Stratonovich formulations. Compared to previous variants of the primitive equations, the one considered here presents a more intricate coupling between the velocity field and the temperature. The corresponding analysis is seriously more involved than in the deterministic setting. Finally, the continuous dependence on the initial data and the energy estimates proven here are new, even in the case of isothermal turbulent pressure.}, author = {Agresti, Antonio and Hieber, Matthias and Hussein, Amru and Saal, Martin}, issn = {1050-5164}, journal = {Annals of Applied Probability}, number = {1}, pages = {635--700}, publisher = {Institute of Mathematical Statistics}, title = {{The stochastic primitive equations with nonisothermal turbulent pressure}}, doi = {10.1214/24-AAP2124}, volume = {35}, year = {2025}, } @article{19783, abstract = {We consider a local Cahn–Hilliard‐type model for tumor growth as well as a nonlocal model where, compared to the local system, the Laplacian in the equation for the chemical potential is replaced by a nonlocal operator. The latter is defined as a convolution integral with suitable kernels parametrized by a small parameter. For sufficiently smooth bounded domains in three dimensions, we prove convergence of weak solutions of the nonlocal model toward strong solutions of the local model together with convergence rates with respect to the small parameter. The proof is done via a Gronwall‐type argument and a convergence result with rates for the nonlocal integral operator toward the Laplacian due to Abels and Hurm.}, author = {Hurm, Christoph and Moser, Maximilian}, issn = {1522-2608}, journal = {GAMM-Mitteilungen}, number = {2}, publisher = {Wiley}, title = {{Nonlocal‐to‐local convergence for a Cahn–Hilliard tumor growth model}}, doi = {10.1002/gamm.70003}, volume = {48}, year = {2025}, } @article{19027, abstract = {Stochastic PDEs of fluctuating hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a multilevel Monte Carlo (MLMC) scheme for the Dean–Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant reduction in computational cost (with respect to a standard Monte Carlo method) in the simulation of the Dean–Kawasaki equation. Specifically, we link this reduction in cost to having a sufficiently large average particle density and show that sizeable cost reductions can be obtained even when we have solutions with regions of low density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean–Kawasaki equation being highly singular.}, author = {Cornalba, Federico and Fischer, Julian L}, issn = {1095-7170}, journal = {SIAM Journal on Numerical Analysis}, number = {1}, pages = {262--287}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics}}, doi = {10.1137/23M1617345}, volume = {63}, year = {2025}, } @article{18926, abstract = {We study weak solutions to mean curvature flow satisfying Young’s angle condition for general contact angles α ∈ (0, π). First, we construct BV solutions by using the Allen-Cahn approximation with boundary contact energy as proposed by Owen and Sternberg. Second, we prove the weak-strong uniqueness and stability for this solution concept. The main ingredient for both results is a relative energy, which can also be interpreted as a tilt excess. }, author = {Hensel, Sebastian and Laux, Tim}, issn = {0022-2518}, journal = {Indiana University Mathematics Journal}, number = {1}, pages = {111--148}, publisher = {Indiana University Mathematics Journal}, title = {{BV solutions for mean curvature flow with constant angle: Allen-Cahn approximation and weak-strong uniqueness}}, doi = {10.1512/iumj.2024.73.9701}, volume = {73}, year = {2024}, } @article{14797, abstract = {We study a random matching problem on closed compact 2 -dimensional Riemannian manifolds (with respect to the squared Riemannian distance), with samples of random points whose common law is absolutely continuous with respect to the volume measure with strictly positive and bounded density. We show that given two sequences of numbers n and m=m(n) of points, asymptotically equivalent as n goes to infinity, the optimal transport plan between the two empirical measures μn and νm is quantitatively well-approximated by (Id,exp(∇hn))#μn where hn solves a linear elliptic PDE obtained by a regularized first-order linearization of the Monge-Ampère equation. This is obtained in the case of samples of correlated random points for which a stretched exponential decay of the α -mixing coefficient holds and for a class of discrete-time Markov chains having a unique absolutely continuous invariant measure with respect to the volume measure.}, author = {Clozeau, Nicolas and Mattesini, Francesco}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, pages = {485--541}, publisher = {Springer Nature}, title = {{Annealed quantitative estimates for the quadratic 2D-discrete random matching problem}}, doi = {10.1007/s00440-023-01254-0}, volume = {190}, year = {2024}, } @article{14884, abstract = {We perform a stochastic homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii’s predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure.}, author = {Davoli, Elisa and D’Elia, Lorenza and Ingmanns, Jonas}, issn = {1432-1467}, journal = {Journal of Nonlinear Science}, number = {2}, publisher = {Springer Nature}, title = {{Stochastic homogenization of micromagnetic energies and emergence of magnetic skyrmions}}, doi = {10.1007/s00332-023-10005-3}, volume = {34}, year = {2024}, } @article{15098, abstract = {The paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier–Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere–ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.}, author = {Agresti, Antonio and Luongo, Eliseo}, issn = {1432-1807}, journal = {Mathematische Annalen}, pages = {2727--2766}, publisher = {Springer Nature}, title = {{Global well-posedness and interior regularity of 2D Navier-Stokes equations with stochastic boundary conditions}}, doi = {10.1007/s00208-024-02812-0}, volume = {390}, year = {2024}, } @article{15119, abstract = {In this paper we consider an SPDE where the leading term is a second order operator with periodic boundary conditions, coefficients which are measurable in (t,ω) , and Hölder continuous in space. Assuming stochastic parabolicity conditions, we prove Lp((0,T)×Ω,tκdt;Hσ,q(Td)) -estimates. The main novelty is that we do not require p=q . Moreover, we allow arbitrary σ∈R and weights in time. Such mixed regularity estimates play a crucial role in applications to nonlinear SPDEs which is clear from our previous work. To prove our main results we develop a general perturbation theory for SPDEs. Moreover, we prove a new result on pointwise multiplication in spaces with fractional smoothness.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {0246-0203}, journal = {Annales de l'institut Henri Poincare Probability and Statistics}, number = {1}, pages = {413--430}, publisher = {Institute of Mathematical Statistics}, title = {{Stochastic maximal Lp(Lq)-regularity for second order systems with periodic boundary conditions}}, doi = {10.1214/22-AIHP1333}, volume = {60}, year = {2024}, } @article{15334, abstract = {We consider the sharp interface limit of a Navier-Stokes/Allen Cahn equation in a bounded smooth domain in two space dimensions, in the case of vanishing mobility mε=ε√, where the small parameter ε>0 related to the thickness of the diffuse interface is sent to zero. For well-prepared initial data and sufficiently small times, we rigorously prove convergence to the classical two-phase Navier-Stokes system with surface tension. The idea of the proof is to use asymptotic expansions to construct an approximate solution and to estimate the difference of the exact and approximate solutions with a spectral estimate for the (at the approximate solution) linearized Allen-Cahn operator. In the calculations we use a fractional order ansatz and new ansatz terms in higher orders leading to a suitable ε-scaled and coupled model problem. Moreover, we apply the novel idea of introducing ε-dependent coordinates.}, author = {Abels, Helmut and Fei, Mingwen and Moser, Maximilian}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {4}, publisher = {Springer Nature}, title = {{Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility}}, doi = {10.1007/s00526-024-02715-7}, volume = {63}, year = {2024}, } @article{12178, abstract = {In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal L² regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms. The adaptation to Stratonovich type noise and, more generally, to variable viscosity and/or conductivity are discussed as well.}, author = {Agresti, Antonio and Hieber, Matthias and Hussein, Amru and Saal, Martin}, issn = {2194-041X}, journal = {Stochastics and Partial Differential Equations: Analysis and Computations}, keywords = {Applied Mathematics, Modeling and Simulation, Statistics and Probability}, pages = {53--133}, publisher = {Springer Nature}, title = {{The stochastic primitive equations with transport noise and turbulent pressure}}, doi = {10.1007/s40072-022-00277-3}, volume = {12}, year = {2024}, } @article{12485, abstract = {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.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, pages = {957--1015}, publisher = {Springer Nature}, title = {{The critical variational setting for stochastic evolution equations}}, doi = {10.1007/s00440-023-01249-x}, volume = {188}, year = {2024}, } @article{12486, abstract = {This paper is concerned with the problem of regularization by noise of systems of reaction–diffusion equations with mass control. It is known that strong solutions to such systems of PDEs may blow-up in finite time. Moreover, for many systems of practical interest, establishing whether the blow-up occurs or not is an open question. Here we prove that a suitable multiplicative noise of transport type has a regularizing effect. More precisely, for both a sufficiently noise intensity and a high spectrum, the blow-up of strong solutions is delayed up to an arbitrary large time. Global existence is shown for the case of exponentially decreasing mass. The proofs combine and extend recent developments in regularization by noise and in the Lp(Lq)-approach to stochastic PDEs, highlighting new connections between the two areas.}, author = {Agresti, Antonio}, issn = {2194-041X}, journal = {Stochastics and Partial Differential Equations: Analysis and Computations}, pages = {1907--1981}, publisher = {Springer Nature}, title = {{Delayed blow-up and enhanced diffusion by transport noise for systems of reaction-diffusion equations}}, doi = {10.1007/s40072-023-00319-4}, volume = {12}, year = {2024}, } @article{13129, abstract = {We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior ahom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨⋅⟩ and the corresponding linear elliptic operator −∇⋅a∇. In line with the theory of homogenization, the method proceeds by computing d=3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨⋅⟩. We make this point by investigating the bias (or systematic error), i.e., the difference between ahom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O(L−1). In case of a suitable periodization of ⟨⋅⟩ , we rigorously show that it is O(L−d). In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨⋅⟩ of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function.}, author = {Clozeau, Nicolas and Josien, Marc and Otto, Felix and Xu, Qiang}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics}, pages = {1305--1387}, publisher = {Springer Nature}, title = {{Bias in the representative volume element method: Periodize the ensemble instead of its realizations}}, doi = {10.1007/s10208-023-09613-y}, volume = {24}, year = {2024}, } @article{14451, abstract = {We investigate the potential of Multi-Objective, Deep Reinforcement Learning for stock and cryptocurrency single-asset trading: in particular, we consider a Multi-Objective algorithm which generalizes the reward functions and discount factor (i.e., these components are not specified a priori, but incorporated in the learning process). Firstly, using several important assets (BTCUSD, ETHUSDT, XRPUSDT, AAPL, SPY, NIFTY50), we verify the reward generalization property of the proposed Multi-Objective algorithm, and provide preliminary statistical evidence showing increased predictive stability over the corresponding Single-Objective strategy. Secondly, we show that the Multi-Objective algorithm has a clear edge over the corresponding Single-Objective strategy when the reward mechanism is sparse (i.e., when non-null feedback is infrequent over time). Finally, we discuss the generalization properties with respect to the discount factor. The entirety of our code is provided in open-source format.}, author = {Cornalba, Federico and Disselkamp, Constantin and Scassola, Davide and Helf, Christopher}, issn = {1433-3058}, journal = {Neural Computing and Applications}, number = {2}, pages = {617--637}, publisher = {Springer Nature}, title = {{Multi-objective reward generalization: Improving performance of Deep Reinforcement Learning for applications in single-asset trading}}, doi = {10.1007/s00521-023-09033-7}, volume = {36}, year = {2024}, } @article{17372, abstract = {In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the d-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g. the Allen-Cahn equation), and dissipative systems (e.g. equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka-Volterra equations for d∈{1,2,3,4} and the Brusselator for d∈{1,2,3}. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on Itô's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new Lζ -coercivity/dissipativity conditions and the development of an Lp(Lq) -framework for systems of reaction-diffusion equations, which are needed when treating dimensions d∈{2,3} in the case of cubic or higher order nonlinearities.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1095-7154}, journal = {SIAM Journal on Mathematical Analysis}, number = {4}, pages = {4870--4927}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems}}, doi = {10.1137/23M1562482}, volume = {56}, year = {2024}, } @article{17462, abstract = {We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale l in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter L >>l around the support of the charge. We show that the algorithm in [J. Lu, F. Otto, and L. Wang, Optimal Artificial Boundary Conditions Based on Second-Order Correctors for Three Dimensional Random Ellilptic Media, preprint, arXiv:2109.01616, 2021], suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion [P. Bella, A. Giunti, and F. Otto, Comm. Partial Differential Equations, 45 (2020), pp. 561–640], still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of l, L, and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multiscale logarithmic Sobolev inequality, where our main tool is an extension of the semigroup estimates in [N. Clozeau, Stoch. Partial Differ. Equ. Anal. Comput., 11 (2023), pp. 1254–1378]. As part of our strategy, we construct sublinear second-order correctors in this correlated setting, which is of independent interest.}, author = {Clozeau, Nicolas and Wang, Lihan}, issn = {1540-3467}, journal = {Multiscale Modeling and Simulation}, number = {3}, pages = {973--1029}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Artificial boundary conditions for random elliptic systems with correlated coefficient field}}, doi = {10.1137/23M1603819}, volume = {22}, year = {2024}, } @article{17481, abstract = {Phase-field models such as the Allen–Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen–Cahn equation with a potential with N≥3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow. In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: as long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen–Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter ε↘0. We even establish the rate of convergence O(ε 1/2 ). Our approach is based on the gradient-flow structure of the Allen–Cahn equation and its limiting motion: building on the recent concept of “gradient-flow calibrations” for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen–Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the Allen–Cahn operator or additional convergence hypotheses for the energy at positive times.}, author = {Fischer, Julian L and Marveggio, Alice}, issn = {1873-1430}, journal = {Annales de l'Institut Henri Poincare C}, number = {5}, pages = {1117--1178}, publisher = {EMS Press}, title = {{Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow}}, doi = {10.4171/AIHPC/109}, volume = {41}, year = {2024}, }