Invariance and monotonicity for stochastic delay differential equations

Abstract

We study invariance and monotonicity properties of Kunita-type stochastic differential equations in R d with delay. Our first result provides sufficient conditions for the invariance of closed subsets of R d . Then we present a comparison principle and show that under appropriate conditions the stochastic delay system considered generates a monotone (orderpreserving) random dynamical system. Several applications are considered.

A Stochastic Gronwall Lemma and Well-Posedness of Path-Dependent SDEs Driven by Martingale Noise

“…[9,Theorem 5.2.5] under local Lipschitz and linear growth assumptions on f and g and [17] under a one-sided local Lipschitz and a suitable growth condition). Under similar conditions, [19] and [14] show existence and uniqueness even for equations with infinite delay and [12] (see also [2]) proved not only existence and uniqueness but also pathwise continuous dependence of the solution on the initial condition in case g does not depend on the past (otherwise it is known that pathwise continuous dependence on the initial condition does not hold in general, see [11]). Existence and uniqueness results in the jump diffusive case under a local Lipschitz and linear growth condition (even with additional Markovian switching) were obtained in [21].…”

A stochastic Gronwall lemma and well-posedness of path-dependent SDEs driven by martingale noise

“…Mao, 2008, Theorem 5.2.5 under local Lipschitz and linear growth assumptions on f and g and von Renesse and Scheutzow, 2010 under a one-sided local Lipschitz and a suitable growth condition). Under similar conditions, Fengying and Ke (2007) and Ren et al (2008) show existence and uniqueness even for equations with infinite delay and Mohammed and Scheutzow (2003) (see also Chueshov and Scheutzow, 2013) proved not only existence and uniqueness but also pathwise continuous dependence of the solution on the initial condition in case g does not depend on the past (otherwise it is known that pathwise continuous dependence on the initial condition does not hold in general, see Mohammed and Scheutzow, 1997). Existence and uniqueness results in the jump diffusive case under a local Lipschitz and linear growth condition (even with additional Markovian switching) were obtained in Zhu (2017).…”

Ergodicity for Singular-Degenerate Stochastic Porous Media Equations

“…A close connection between random attractors and ergodic and mixing properties of random dynamical systems can be obtained in the case of synchronization (see [14]), which is on hand if the random attractor is a singleton. This case has been investigated in, e. g., [12,13,22,23,43].…”