We study the long-time behaviour of solutions to some classes of fourth-order nonlinear PDEs with non-monotone nonlinearities, which include the Landau–Lifshitz–Baryakhtar (LLBar) equation (with all relevant fields and spin torques) and the convective Cahn–Hilliard/Allen–Cahn (CH-AC) equation with a proliferation term, in dimensions d=1,2,3. Firstly, we show the global well-posedness, as well as the existence of global and exponential attractors with finite fractal dimensions for these problems. In the case of the exchange-dominated LLBar equation and the CH-AC equation without convection, an estimate for the rate of convergence of the solution to the corresponding stationary state is given. Finally, we show the existence of a robust family of exponential attractors when d≤ 2. As a corollary, exponential attractor of the LLBar equation is shown to converge to that of the Landau–Lifshitz–Bloch equation in the limit of vanishing exchange damping, while exponential attractor of the convective CH-AC equation is shown to converge to that of the convective Allen–Cahn equation in the limit of vanishing diffusion coefficient.

We develop a fully discrete, semi-implicit mixed finite element method for approximating solutions to a class of fourth-order stochastic partial differential equations (SPDEs) with non-globally Lipschitz and non-monotone nonlinearities, perturbed by spatially smooth multiplicative Gaussian noise. The proposed scheme is applicable to a range of physically relevant nonlinear models, including the stochastic Landau-Lifshitz-Baryakhtar (sLLBar) equation, the stochastic convective Cahn-Hilliard equation with mass source, and the stochastic regularised Landau-Lifshitz-Bloch (sLLB) equation, among others. To overcome the difficulties posed by the interplay between the nonlinearities and the stochastic forcing, we adopt a "truncate-then-discretise" strategy: the nonlinear term is first truncated before discretising the resulting modified problem. We show that the strong solution to the truncated system converges in probability to that of the original problem. A fully discrete numerical scheme is then proposed for the truncated problem. Assuming initial data in H2, we utilise parabolic smoothing estimates and the temporal H & ouml;lder continuity of the solution to establish both convergence in probability and strong convergence (with quantitative rates) for the two fields used in the mixed formulation. Numerical simulations are provided to support the theoretical results.

The incompressible Hall-magnetohydrodynamics (Hall–MHD) system presents substantial analytical and computational challenges due to its stiff, highly nonlinear Hall term and the strict requirement that the magnetic field remains solenoidal. In this paper, we study a Voigt-regularised Hall–MHD system, which is of independent analytical interest and provides a physically consistent, well-posed regularisation of the original model. We propose, analyse, and implement a structure-preserving, linear, fully discrete finite element method for this regularised problem. Using finite element exterior calculus and a mixed formulation, the spatial discretisation enforces the divergence-free condition on the magnetic field exactly, while a skew-symmetric, linearly implicit time discretisation yields unconditional energy stability. We establish optimal convergence rates for the Voigt-regularised problem and, additionally, derive error estimates for the unregularised Hall–MHD system, with the Voigt regularisation playing a crucial role in the non-resistive regime. Finally, numerical simulations in both 2.5D and 3D corroborate the theoretical results and demonstrate the physical fidelity of the scheme.

The Landau-Lifshitz-Baryakhtar (LLBar) equation perturbed by both additive and multiplicative noises is a system of fourth-order stochastic PDEs which models the evolution of magnetic spin fields in ferromagnetic materials at elevated temperatures, taking into account longitudinal damping, long-range interactions, spin current, and noise-induced phenomena at high temperatures. In this paper, we show the existence of a unique pathwise solution (which is analytically strong) to the stochastic LLBar equation posed in a bounded domain D subset of Rd, where d = 1, 2, 3. We also prove the convergence of this pathwise solution to that of the stochastic Landau-Lifshitz-Bloch (LLB) equation in the limit of vanishing exchange relaxation parameter. Finally, we show the Feller property of the Markov semigroup associated with the strong solution, and prove the existence of nontrivial invariant measures. For temperatures above the Curie temperature, exponential stability of the solution and uniqueness of invariant measure are obtained under certain dissipativity conditions. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

We examine a stochastic Landau-Lifshitz-Gilbert equation for a frustrated ferromagnet with competing first and second order exchange interactions exposed to deterministic and random spin transfer torques in form of transport noise. We prove the existence and pathwise uniqueness of weak martingale solutions in the energy space. The result ensures the persistence of topological patterns, occurring in such magnetic systems, under the influence of a fluctuating spin current.

We study the existence of the stochastic flow associated to a linear stochastic evolution equation d ⁡ X = A X d ⁡ t + ∑ k B k X d ⁡ W k , \begin{equation*} \operatorname {d} X= AX\operatorname {d} t +\sum _{k} B_k X\operatorname {d} W_k, \end{equation*} on a Hilbert space. Our first result covers the case where A A is the generator of a C 0 C_0 -semigroup, and ( B k ) (B_k) is a sequence of bounded linear operators such that ∑ k ‖ B k ‖ > + ∞ \sum _k\|B_k\|>+\infty . We also provide sufficient conditions for the existence of stochastic flows in the Schatten classes beyond the space of Hilbert–Schmidt operators. Some new results and examples concerning the so-called commutative case are presented as well.

In this paper we are concerned with the stochastic Landau-Lifshitz-Slonczewski equation (LLS) that describes magnetisation of an inifnite nanowire evolving under current driven spin torque. The current brings into the system a multiplicative gradient noise that appears as a transport term in the equation. We prove the existence, uniqueness and regularity of pathwise solutions to this equation.

We study numerical methods for solving a system of quasilinear stochastic partial differential equations known as the stochastic Landau-Lifshitz-Bloch (LLB) equation on a bounded domain in $\mathbb R^d$ for $d=1,2$. Our main results are estimates of the rate of convergence of the Finite Element Method to the solutions of stochastic LLB. To overcome the lack of regularity of the solution in the case $d=2$, we propose a Finite Element scheme for a regularised version of the equation. We then obtain error estimates of numerical solutions and for the solution of the regularised equation as well as the rate of convergence of this solution to the solution of the stochastic LLB equation. As a consequence, the convergence in probability of the approximate solutions to the solution of the stochastic LLB equation is derived. To the best of our knowledge this is the first result on error estimates for a system of stochastic quasilinear partial differential equations. A stronger result is obtained in the case $d=1$ due to a new regularity result for the LLB equation which allows us to avoid regularisation.

We consider the valuation of contingent claims with delayed dynamics in a Samuelson complete market model. We find a pricing formula that can be decomposed into terms reflecting the current market values of the past and the future, showing how the valuation of prospective cashflows cannot abstract away from the contribution of the past. As a practical application, we provide an explicit expression for the market value of human capital in a setting with wage rigidity. The formula we derive has successfully been used to explicitly solve the infinite dimensional stochastic control problems addressed in Biffis et al. (SIAM J Control Optim 58(4):1906–1938, 2020), Djeiche et al. (Stoch Process Appl 145:48–85, 2022) and Biagini et al. (SIAM J Financial Math 13(3):1004–1039, 2022).

The gradient discretisation method (GDM) is a generic framework, covering many classical methods (Finite Elements, Finite Volumes, Discontinuous Galerkin, etc.), for designing and analysing numerical schemes for diffusion models. In this paper, we study the GDM for a general stochastic evolution problem based on a Leray--Lions type operator. The problem contains the stochastic $p$-Laplace equation as a particular case. The convergence of the Gradient Scheme (GS) solutions is proved by using Discrete Functional Analysis techniques, Skorohod theorem and the Kolmogorov test. In particular, we provide an independent proof of the existence of weak martingale solutions for the problem. In this way, we lay foundations and provide techniques for proving convergence of the GS approximating stochastic partial differential equations.

We investigate a class of zero-sum linear-quadratic stochastic differential games on a finite time horizon governed by slow and fast state equations. The slow-fast nature of the problem can be leveraged to reformulate the associated generalized Riccati equation as a deterministic singular perturbation problem. In doing so, we show that as the evolutionary speed of the slow and fast processes diverge, the existence of a solution to the associated generalized Riccati equation is guaranteed by the existence of a solution to a decoupled pair of differential and algebraic Riccati equations with a reduced order of dimensionality. Furthermore, we are able to formulate a pair of asymptotic estimates to the value function of the game problem by constructing an approximate feedback strategy and observing the limiting value function.

The Landau–Lifshitz–Bloch equation perturbed by a space-dependent noise was proposed in [10] as a model for evolution of spins in ferromagnetic materials at the full range of temperatures, including the temperatures higher than the Curie temperature. In the case of a ferromagnet filling a bounded domain D⊂Rd, d=1,2,3, we show the existence of strong (in the sense of PDEs) martingale solutions. Furthermore, in cases d=1,2 we prove uniqueness of pathwise solutions and the existence of invariant measures.1

We show regularization effect of nonlinear gradient noise to the solution of 1D stochastic parabolic equation. We demonstrate convergence to a martingale (independent upon space variable) when we rescale noise at the extremum points of the process.

We consider 3D stochastic Landau-Lifshitz-Gilbert equations coupled with the Maxwell equations with the full energy. We have proved the existence and some further regularities of the weak solution.

We investigate the asymptotic properties of a finite-time horizon linear-quadratic optimal control problem driven by a multiscale stochastic process with multiplicative Brownian noise. We approach the problem by considering the associated differential Riccati equation and reformulating it as a classical and deterministic singular perturbation problem. Asymptotic properties of this deterministic problem can be gathered from the well-known Tikhonov Theorem. Consequently, we are able to propose two approximation methods to the value function of the stochastic optimal control problem. The first is by constructing an approximately optimal control process whilst the second is by finding the direct limit to the value function. Both approximation methods rely on the existence of a solution to a coupled differential-algebraic Riccati equation with certain stability properties - this is the main difficulty of the paper.

We propose an unconditionally convergent linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert (LLG) equation with multi-dimensional noise. By using the Doss–Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent θ-linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation.

A class of singular stochastic control problems whose value functions satisfy an invariance property was studied by Lasry and Lions (2000). They have shown that, within this class, any singular control problem is equivalent to the corresponding standard stochastic control problem. The equivalence is in the sense that their value functions are equal. In this work, we clarify their idea and extend their work to allow Lévy type noise. In addition, for the purpose of application, we apply our result to an optimal trade execution problem studied by Lasry and Lions (2007).

The stochastic Landau--Lifshitz--Gilbert (LLG) equation coupled with the Maxwell equations (the so called stochastic MLLG system) describes the creation of domain walls and vortices (fundamental objects for the novel nanostructured magnetic memories). We first reformulate the stochastic LLG equation into an equation with time-differentiable solutions. We then propose a convergent $\theta$-linear scheme to approximate the solutions of the reformulated system. As a consequence, we prove convergence of the approximate solutions, with no or minor conditions on time and space steps (depending on the value of $\theta$). Hence, we prove the existence of weak martingale solutions of the stochastic MLLG system. Numerical results are presented to show applicability of the method.

We propose an unconditionally convergent linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert (LLG) equation with multi-dimensional noise. By using the Doss-Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent θ-linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation.

The stochastic Landau--Lifshitz--Gilbert (LLG) equation describes the behaviour of the magnetization under the influence of the effective field consisting of random fluctuations. We first reformulate the equation into an equation the unknown of which is differentiable with respect to the time variable. We then propose a convergent $\theta$-linear scheme for the numerical solution of the reformulated equation. As a consequence, we show the existence of weak martingale solutions to the stochastic LLG equation. A salient feature of this scheme is that it does not involve a nonlinear system, and that no condition on time and space steps is required when $\theta\in(\frac{1}{2},1]$. Numerical results are presented to show the applicability of the method.