2020/01/31 by Paul Gassiat, Gassiat, Paul
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR) #math.CA #math.PR
paper · pdf · doi:10.48550/arxiv.2001.11914
minor corrections and clarifications
arxiv created 2020/11/13 · arxiv updated 2020/11/16
We give an example of a reflected diffferential equation which may have infinitely many solutions if the driving signal is rough enough (e.g. of infinite p-variation, for some p>2). For this equation, we identify a sharp condition on the modulus of continuity of the signal under which uniqueness holds. Lévy's modulus for Brownian motion turns out to be a boundary case. We further show that in our example, non-uniqueness holds almost surely when the driving signal is a fractional Brownian motion with Hurst index H < (1)/(2). The considered equation is driven by a two-dimensional signal with one component of bounded variation, so that rough path theory is not needed to make sense of the equation.