4 research outputs found
Regularisation by fractional noise for one-dimensional differential equations with nonnegative distributional drift
We study existence and uniqueness of solutions to the equation , where is a distribution in some Besov space and is a fractional Brownian motion with Hurst parameter . First, the equation is understood as a nonlinear Young equation. This involves a nonlinear Young integral constructed in the space of functions with finite -variation, which is well suited when is a nonnegative (or nonpositive) distribution. Depending on , a condition on the Besov regularity of is given so that solutions to the equation exist. The construction is deterministic, and can be replaced by a deterministic path with a sufficiently smooth local time.Using this construction we prove the existence of weak solutions (in the probabilistic sense). We also prove that solutions coincide with limits of strong solutions obtained by regularisation of . This is used to establish pathwise uniqueness and existence of a strong solution. In particular when is a finite nonnegative measure, weak solutions exist for , while pathwise uniqueness and strong existence hold when . The proofs involve fine properties of the local time of the fractional Brownian motion, as well as new regularising properties of this process which are established using the stochastic sewing Lemma
Regularisation by fractional noise for one-dimensional differential equations with nonnegative distributional drift
We study existence and uniqueness of solutions to the equation , where is a distribution in some Besov space and is a fractional Brownian motion with Hurst parameter . First, the equation is understood as a nonlinear Young equation. This involves a nonlinear Young integral constructed in the space of functions with finite -variation, which is well suited when is a nonnegative (or nonpositive) distribution. Depending on , a condition on the Besov regularity of is given so that solutions to the equation exist. The construction is deterministic, and can be replaced by a deterministic path with a sufficiently smooth local time.Using this construction we prove the existence of weak solutions (in the probabilistic sense). We also prove that solutions coincide with limits of strong solutions obtained by regularisation of . This is used to establish pathwise uniqueness and existence of a strong solution. In particular when is a finite nonnegative measure, weak solutions exist for , while pathwise uniqueness and strong existence hold when . The proofs involve fine properties of the local time of the fractional Brownian motion, as well as new regularising properties of this process which are established using the stochastic sewing Lemma