Regularisation by fractional noise for one-dimensional differential equations with nonnegative distributional drift

Abstract

We study existence and uniqueness of solutions to the equation dXt = b(Xt)dt + dBt, where b is a distribution in some Besov space and B is a fractional Brownian motion with Hurst parameter H 1/2. First, the equation is understood as a nonlinear Young integral equation. The integral is constructed in a p-variation space, which is well suited when b is a nonnegative (or nonpositive) distribution. Based on the Besov regularity of b, a condition on H is given so that solutions to the equation exist. The construction is deterministic, and B can be replaced by a deterministic path w which has a sufficiently smooth local time.We then consider probabilistic notions of weak and strong solution to this SDE. In particular, when b is a nonnegative finite measure, we show the existence of weak solutions for H < √ 2 − 1. We prove that solutions can also be understood as limits of strong solutions when b is regularised, which is used to establish pathwise uniqueness and existence of a strong solution when H 1/4. 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.

Comparison of classical and path-by-path solutions to SDEs

“…The drift is constructed in a way such that for a solution X the following must hold: X 1 is forced to be equal to 1 or −1. On the time interval [1,2] we let a Brownian motion evolve freely without any drift. On the time interval [2,3] the drift is constructed in a way such that there exists no adapted solution if X 2 ∈ (−1, 1).…”

“…On the time interval [1,2] we let a Brownian motion evolve freely without any drift. On the time interval [2,3] the drift is constructed in a way such that there exists no adapted solution if X 2 ∈ (−1, 1). However if |X 2 | > 1, then X can be extended to the interval [0, 3] with |X t | > 1 for all t ∈ [2, 3] while still being adapted.…”

“…For results on "path-bypath" uniqueness for SPDEs see [4,5]. There are also results on existence and uniqueness of pathby-path solutions available for SDEs with fractional Brownian noise, allowing for distributional drifts (see [2,6,14]).…”

“…Hence, for ω ∈ C 1 ∩ {τ 3} and t ∈ [τ, 4], we can choose X to be the pathwise unique nonpositive solution todX t = 1 X t dt + dB t , t ∈ [τ, 4]and therefore we can construct X :C 1 → C([0, 4]) fulfilling equation (3.2). Let b(t, x) := b(t, x) − ½ {t 1,x<0}1x+ ½ {t 3,x 0} 1 x − For ω ∈ C 2 and t ∈ [0, 4], let X fulfill Xt (ω) = t 0 b(s, Xs (ω))ds + B t (ω)such that X coincides with the two Bessel bridges on [0, 1] and[1,2] so that X1 = 2 and X2 = 1. This is possible by the same arguments as in the proof of Proposition 3.1.…”

Stochastic equations with singular drift driven by fractional Brownian motion

“…Recently, several interesting works studying regularization by fractional Brownian noise have appeared: [CG16,ART21,GG22]. Unfortunately, these works do not allow to deduce well-posedness for b ∈ L p (R d ) or for b being a Radon measure under condition (1.3).…”