Distribution dependent SDEs driven by additive fractional Brownian motion

Galeati, Lucio; Harang, Fabian A.; Mayorcas, Avi

doi:10.48550/arxiv.2105.14063

Cited by 5 publications

(7 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such a finite measure b is also in B −1 ∞ . In this space, the existence of a unique strong solution was shown in [18] for H < 1/4 (elaborating on the path-by-path result of [7]). Hence in this case, Theorem 2.10 extends this result to H = 1/4.…”

Section: Existence and Uniqueness Resultsmentioning

confidence: 96%

See 1 more Smart Citation

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

Anzeletti¹,

Richard²,

Tanré³

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Existence and Uniqueness Resultsmentioning

confidence: 96%

“…(i), p.196]) we have B s0 p0,q0 (I) ֒→ B s1 p1,q1 (I). For s ∈ R + \ N and p = q = ∞, Besov spaces coincide with Hölder spaces, see Equations (18) p.38 and (1) p.51 in [35], which we define now for s ∈ (0, 1] only. Definition 1.6.…”

Section: Notations and Definitionsmentioning

confidence: 99%

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

Anzeletti¹,

Richard²,

Tanré³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…On the downside, since our results all apply in the case ε = 0, our noise can be very degenerate and we can never go beyond the classical Vlasov setting. In this regard, another motivation for the current work is to provide a family of "baseline results" to be compared to the regularizing features of suitable nondegenerate choices of Y ; in our second article [29] we will study in detail the well-posedness of (1.1) for highly irregular B when the noise is sampled as a fractional Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

Distribution dependent SDEs driven by additive continuous noise

Galeati

Harang

Mayorcas

2022

Electron. J. Probab.

Self Cite

View full text Add to dashboard Cite

We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [17]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the equation, as well as almost sure convergence of the associated particle system, for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions.

show abstract

“…In the following two theorems, the weak well-posedness of (1.14) and (1.17) follow from Nualart [21] (stated in dimension 1, but generalizes to higher dimensions straightforwardly). Weak well-posedness of the McKean-Vlasov SDE (1.15) and (1.18) are proved in [6], [8].…”

Section: Introductionmentioning

confidence: 99%

Entropic propagation of chaos for mean field diffusion with $L^p$ interactions via hierarchy, linear growth and fractional noise

Han¹

2022

Preprint

View full text Add to dashboard Cite

New quantitative propagation of chaos results for mean field diffusion are proved via local and global entropy estimates. In the first result we work on the torus and consider singular, divergence free interactions K ∈ L p , p > d. We prove a O(k 2 /n 2 ) convergence rate in relative entropy between the k-marginal laws of the particle system and its limiting law at each time t, as long as the same holds at time 0. The proof is based on local estimates via a form of BBGKY hierarchy and exemplifies a method to extend the framework in Lacker [16] to singular interactions.Then we prove quantitative propagation of chaos for interactions that are only assumed to have linear growth. This generalizes to the case where the driving noise is replaced by a fractional Brownian motion B H , for all H ∈ (0, 1). These proofs follow from global estimates and subGaussian concentration inequalities. We obtain O(k/n) convergence rate in relative entropy in each case, yet the rate is only valid on [0, T * ] with T * a fixed finite constant depending on various parameters of the system.

show abstract

Distribution dependent SDEs driven by additive fractional Brownian motion

Cited by 5 publications

References 36 publications

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

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

Distribution dependent SDEs driven by additive continuous noise

Entropic propagation of chaos for mean field diffusion with $L^p$ interactions via hierarchy, linear growth and fractional noise

Contact Info

Product

Resources

About