Strong solutions of a stochastic differential equation with irregular random drift
Peer reviewed, Journal article
Published version
Date
2022Metadata
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- Institutt for matematiske fag [2474]
- Publikasjoner fra CRIStin - NTNU [38289]
Original version
10.1016/j.spa.2022.05.006Abstract
We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form
where the drift coefficient is random and irregular, with a weak derivative satisfying for some , . The random and regular noise coefficient may vanish. The main contribution is a pathwise uniqueness result under the assumptions that as , and satisfies the one-sided gradient bound , where the process exhibits an exponential moment bound of the form for small times , for some . This study is motivated by ongoing work on the well-posedness of the stochastic Hunter–Saxton equation, a stochastic perturbation of a nonlinear transport equation that arises in the modelling of the director field of a nematic liquid crystal. In this context, the one-sided bound acts as a selection principle for dissipative weak solutions of the stochastic partial differential equation.