Regularisation by regular noise

Abstract

AbstractWe show that perturbing ill-posed differential equations with (potentially very) smooth random processes can restore well-posedness—even if the perturbation is (potentially much) more regular than the drift component of the solution. The noise considered is of fractional Brownian type, and the familiar regularity condition $$\alpha >1-1/(2H)$$ α > 1 - 1 / ( 2 H ) is recovered for all non-integer $$H>1$$ H > 1 .