[Paper Review] Semilinear Stochastic Evolution Equations with Lévy Noise and Monotone Nonlinearity

Erfan Salavati, Bijan Z. Zangeneh|arXiv (Cornell University)|Apr 8, 2013

Stochastic processes and financial applications28 references3 citations

TL;DR

This paper establishes existence, uniqueness, and stability of mild solutions for semilinear stochastic evolution equations driven by Lévy noise with monotone nonlinear drift, using an iterative method and a novel Itô-type inequality for stochastic convolution integrals. The key contribution is proving these results without requiring coercivity conditions on coefficients, extending applicability to hyperbolic-type SPDEs and delay equations.

ABSTRACT

Semilinear stochastic evolution equations with multiplicative Lévy noise and monotone nonlinear drift are considered. Unlike other similar work we do not impose coercivity conditions on coefficients. Existence and uniqueness of the mild solution is proved using an iterative method. The continuity of the solution with respect to initial conditions and coefficients is proved and a sufficient condition for exponential asymptotic stability of the solutions has been derived. The solutions are proved to have a Markov property. Examples on stochastic partial differential equations and stochastic delay equations are provided to demonstrate the theory developed. The main tool in our study is an Itô type inequality which gives a pathwise bound for the norm of stochastic convolution integrals.

Motivation & Objective

• To establish existence and uniqueness of mild solutions for semilinear stochastic evolution equations with multiplicative Lévy noise and monotone nonlinear drift.
• To remove the need for coercivity conditions on coefficients, which restricts applicability in previous works.
• To prove continuity of solutions with respect to initial conditions and coefficients.
• To derive a sufficient condition for exponential asymptotic stability of solutions.
• To demonstrate the Markov property of the solution process and validate the theory with examples on SPDEs and stochastic delay equations.

Proposed method

• Employs an iterative method to construct the mild solution under monotonicity and Lipschitz conditions on the nonlinear drift and diffusion coefficients.
• Introduces a novel Itô-type inequality to provide a pathwise bound for the norm of stochastic convolution integrals driven by Lévy noise.
• Applies the semigroup approach to stochastic evolution equations, using the infinitesimal generator of a $C_0$ semigroup on a Hilbert space.
• Uses the variational framework with dense embeddings $B \subset H \subset B^*$ to handle unbounded operators and ensure monotonicity of drift.
• Reduces second-order hyperbolic SPDEs and stochastic delay equations to first-order systems in product spaces to apply the main theorem.
• Verifies that the transformed coefficients satisfy the required monotonicity and Lipschitz conditions under appropriate Sobolev and Lebesgue space embeddings.

Experimental results

Research questions

• RQ1 Can mild solutions be established for semilinear SPDEs with Lévy noise and monotone drift without imposing coercivity on coefficients?
• RQ2 What pathwise estimates are possible for stochastic convolution integrals driven by Lévy noise, and how can they be used to prove existence and stability?
• RQ3 How does the solution depend continuously on initial conditions and coefficients in the presence of multiplicative Lévy noise?
• RQ4 Under what conditions is the solution exponentially asymptotically stable?
• RQ5 Does the solution process possess the Markov property in this general setting with Lévy noise and monotone drift?

Key findings

• Existence and uniqueness of the mild solution are established for equations with Lévy noise and monotone nonlinear drift, even without coercivity conditions.
• The solution is continuous with respect to initial conditions and coefficients, ensuring robustness under perturbations.
• A sufficient condition for exponential asymptotic stability is derived, depending on the monotonicity constant and noise intensity.
• The solution process is shown to possess the Markov property, enabling the use of probabilistic tools like transition kernels.
• The theory is validated through two examples: a stochastic parabolic SPDE and a second-order stochastic hyperbolic equation, both in Hilbert spaces with appropriate Sobolev embeddings.
• The Itô-type inequality provides a crucial pathwise bound for the stochastic convolution integral, enabling the iterative proof of existence and stability.