Markovian lifting and asymptotic log-Harnack inequality for stochastic Volterra integral equations

Abstract

We introduce a new framework of Markovian lifts of stochastic Volterra integral equations (SVIEs for short) with completely monotone kernels. We define the state space of the Markovian lift as a separable Hilbert space which incorporates the singularity or regularity of the kernel into the definition. We show that the solution of an SVIE is represented by the solution of a lifted stochastic evolution equation (SEE for short) defined on the Hilbert space and prove the existence, uniqueness and Markov property of the solution of the lifted SEE. Furthermore, we establish an asymptotic log-Harnack inequality and some consequent properties for the Markov semigroup associated with the Markovian lift via the asymptotic coupling method.

47 pages

References in corpus (4)

• SPDE in Hilbert Space with Locally Monotone Coefficients
• Approximation of Stochastic Volterra Equations with kernels of completely monotone type
• Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises
• Asymptotic Log-Harnack Inequality for Monotone SPDE with Multiplicative Noise

Cited by in corpus (3)

• Superposition of interacting stochastic processes with memory and its application to migrating fish counts
• Weak well-posedness of stochastic Volterra equations with completely monotone kernels and non-degenerate noise
• Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts