Stochastic evolution equations with nonlinear diffusivity, recent progress and critical cases

2025/10/23 by Ioana Ciotir, Ciotir, Ioana, Dan Goreac +3
Economics, Econometrics and Finance · Computer Science · Mathematics · #Stochastic processes and financial applications #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2510.20471

Abstract

This short survey article stems from recent progress on critical cases of stochastic evolution equations in variational formulation with additive, multiplicative or gradient noises. Typical examples appear as the limit cases of the stochastic porous medium equation, stochastic fast- and super fast-diffusion equations, self-organized criticality, stochastic singular p-Laplace equations, and the stochastic total variation flow, among others. We present several different notions of solutions, results on convergence of solutions depending on a parameter, and homogenization. Furthermore, we provide some references hinting at the recent progress in regularity results, long-time behavior, ergodicity, and numerical analysis.

Citations

• Conservative stochastic PDEs on the whole space
• Stochastic porous media equation with Robin boundary conditions, gravity-driven infiltration and multiplicative noise
• SVI solutions to stochastic nonlinear diffusion equations on general measure spaces
• The stochastic fast logarithmic equation in ℝ^d with multiplicative Stratonovich noise
• Ergodicity and random dynamical systems for conservative SPDEs
• Porous media equations with nonlinear gradient noise and Dirichlet boundary conditions
• On temporal regularity for strong solutions to stochastic p-Laplace systems
• The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles
• Well-posedness of the Dean--Kawasaki and the nonlinear Dawson--Watanabe equation with correlated noise
• Stochastic Generalized Porous Media Equations over σ-finite Measure Spaces with Non-Continuous Diffusivity Function
• Stochastic partial differential equations arising in self-organized criticality
• The stochastic p-Laplace equation on ℝ^d
• Conservative stochastic PDE and fluctuations of the symmetric simple exclusion process
• Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organised criticality
• Finite time extinction for the 1D stochastic porous medium equation with\n transport noise
• Well-posedness of renormalized solutions for a stochastic p-Laplace\n equation with L¹-initial data
• Well-posedness of nonlinear diffusion equations with nonlinear,\n conservative noise
• Uniqueness for a class of stochastic Fokker-Planck and porous media equations
• Singular-degenerate multivalued stochastic fast diffusion equations
• Stochastic variational inequalities and regularity for degenerate\n stochastic partial differential equations
• A regularity result for quasilinear stochastic partial differential equations of parabolic type
• Finite speed of propagation for stochastic porous media equations
• Strong Solutions for Stochastic Partial Differential Equations of\n Gradient Type
• Existence and Uniqueness of Nonnegative Solutions to the Stochastic Porous Media Equation
• Existence of Strong Solutions for Stochastic Porous Media Equation under General Monotonicity Conditions
• Stochastic Generalized Porous Media and Fast Diffusion Equations
• Strong Solutions of Stochastic Generalized Porous Media Equations: Existence, Uniqueness and Ergodicity

Related

• The total variation flow perturbed by gradient linear multiplicative\n noise
• The critical variational setting for stochastic evolution equations
• Singular-degenerate multivalued stochastic fast diffusion equations
• Stochastic partial differential equations arising in self-organized criticality
• Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence