Measurable solutions for stochastic evolution equations without uniqueness

Abstract

A method for obtaining measurable solutions to stochastic evolution equations in which there is no uniqueness for the corresponding non-stochastic equation is presented. It involves a technique based on a measurable selection theorem for set-valued functions. No assumptions are needed on the underlying probability space. An application is given to the stochastic Navier–Stokes problem in arbitrary dimensions. We also show the existence of measurable solutions to stochastic ordinary differential equations in which there is no uniqueness. A finite-dimensional generalization is given to adapted solutions in the case of a normal filtration and path uniqueness.

Keywords:

• nonlinear evolution equations
• measurable selections
• measurable multi-functions
• measurable solutions
• non-uniqueness
• stochastic Navier–Stokes

AMS Subject Classifications:

• 35R60
• 60H15
• 35A01
• 35B30
• 35A02
• 35Q35