[Paper Review] On Stochastic Evolution Equations with non-Lipschitz Coefficients

Xicheng Zhang|ArXiv.org|Mar 9, 2007

Stochastic processes and financial applications26 references4 citations

TL;DR

This paper establishes existence and uniqueness of solutions for stochastic evolution equations with non-Lipschitz coefficients, including backward, functional, and Volterra-type equations, using an evolution triple framework. The key contribution is extending Krylov-Rozovskii's theory to handle degenerate and quasi-linear SPDEs like stochastic porous medium and reaction-diffusion equations with random coefficients, validated via stopping time techniques and monotonicity estimates.

ABSTRACT

In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that is, backward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equations. In particular, the results can be used to treat a large class of quasi-linear stochastic equations, which includes the reaction diffusion and porous medium equations.

Motivation & Objective

• To extend the classical Krylov-Rozovskii theory for stochastic evolution equations to non-Lipschitz coefficients.
• To establish existence and uniqueness of solutions for backward stochastic evolution equations with non-Lipschitz drift and diffusion coefficients.
• To develop a framework for stochastic functional integral evolution equations, including Volterra-type equations with non-Lipschitz nonlinearities.
• To apply the abstract results to quasi-linear SPDEs such as stochastic porous medium and reaction-diffusion equations with random degeneracy.
• To overcome the challenge of random coercivity coefficients in SPDEs where standard Lipschitz conditions fail.

Proposed method

• Utilizes the evolution triple framework (X1 ⊂ H ⊂ X1*) to handle quasi-linear SPDEs with non-Lipschitz nonlinearities.
• Applies stopping time techniques to manage random coercivity coefficients in the stochastic evolution equation setting.
• Employs Galerkin approximation and Picard iteration for proving existence and uniqueness in backward and functional evolution equations.
• Imposes monotonicity and coercivity conditions (H1–H4) on the operators A and B, with coefficients depending on time, space, and randomness.
• Uses Itô’s formula and energy estimates in weighted Lp and Sobolev spaces to control nonlinear terms.
• Verifies that the abstract conditions (H1)–(H4) hold for specific SPDEs by checking integrability and monotonicity via Hölder and Young’s inequalities.

Experimental results

Research questions

• RQ1 Can existence and uniqueness be established for stochastic evolution equations with non-Lipschitz coefficients when standard Lipschitz conditions fail?
• RQ2 How can backward stochastic evolution equations with non-Lipschitz coefficients be solved in the framework of evolution triples?
• RQ3 Can stochastic Volterra-type and functional integral evolution equations with non-Lipschitz nonlinearities be treated using Picard iteration and monotonicity?
• RQ4 Does the abstract framework extend to degenerate SPDEs such as stochastic porous medium equations with random diffusion coefficients?
• RQ5 Can the results be applied to stochastic reaction-diffusion equations with multiplicative noise and degenerate diffusion depending on the Brownian path?

Key findings

• The abstract framework ensures existence and uniqueness of solutions for stochastic evolution equations with non-Lipschitz coefficients under monotonicity and coercivity conditions (H1)–(H4).
• The stochastic porous medium equation with random diffusion coefficient |wt|·Δ(|u|^{p-2}u) admits a unique solution in the generalized sense for almost all paths of the Brownian motion.
• The stochastic reaction-diffusion equation with multiplicative noise √|wt|·u·dwt also admits a unique solution satisfying u ∈ L²([0,T]; H¹₀) ∩ Lᵖ([0,T]×O) ∩ C([0,T]; L²) a.s.
• The solution map is continuous in the L²(O) norm, and the solution satisfies energy estimates involving random time-dependent coercivity coefficients.
• The framework successfully handles degenerate SPDEs where the diffusion coefficient vanishes almost surely at some times, by using stopping times and pathwise analysis.
• The results generalize prior works by Krylov-Rozovskii and Pardoux by removing the need for Lipschitz continuity and allowing arbitrary growth in the drift and diffusion coefficients.