Remarks on Stochastic Systems I: Markov properties, local and global uniqueness, and limits of stochastic equations

TL;DR

This paper explores complex behaviors of stochastic differential equations, highlighting issues with Markov properties, uniqueness, and limits, motivated by singular stochastic PDEs and initial condition dependencies.

Contribution

It provides examples illustrating delicate Markov properties, non-uniqueness, and limit behaviors in stochastic differential equations, especially in the context of singular SPDEs.

Findings

Examples of stochastic differential equations with delicate Markov properties.

Illustration of non-uniqueness in solutions based on initial conditions.

Demonstration of limit behaviors of stochastic equations in singular SPDE contexts.

Abstract

In the present paper, we give some examples of stochastic differential equations which have delicateness in the Markov and strong Markov properties, the uniqueness locally in time and globally in time, and initial conditions. Moreover, we show that such stochastic differential equations appear in the limits of stochastic differential equations which have the existence and pathwise uniqueness of solutions. These examples are constructed in motivation to singular stochastic partial differential equations. We also give some examples of shifted equations whose sum of the solutions depends on the choices of the decomposition of the initial condition of the original equation.