On Nonlinear Rough Paths

Nualart, David; Xia, Panqiu

doi:10.30757/alea.v17-22

Cited by 4 publications

(3 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To this end, one could hope to use techniques developed on nonlinear rough paths (see e.g. [30,9]), but the exact formulation of the equation in this context is not completely clear.…”

Section: Discussionmentioning

confidence: 99%

Regularization of multiplicative SDEs through additive noise

Galeati¹,

Harang²

2020

Preprint

View full text Add to dashboard Cite

We investigate the regularizing effect of certain additive continuous perturbations on SDEs with multiplicative fractional Brownian motion (fBm). Traditionally, a Lipschitz requirement on the drift and diffusion coefficients is imposed to ensure existence and uniqueness of the SDE. We show that suitable perturbations restore existence, uniqueness and regularity of the flow for the resulting equation, even when both the drift and the diffusion coefficients are distributional, thus extending the program of regularization by noise to the case of multiplicative SDEs. Our method relies on a combination of the non-linear Young formalism developed by Catellier and Gubinelli [8], and stochastic averaging estimates recently obtained by Hairer and Li [20].

show abstract

“…To this end, one could hope to use techniques developed on nonlinear rough paths (see e.g. [30,9]), but the exact formulation of the equation in this context is not completely clear.…”

Section: Discussionmentioning

confidence: 99%

Regularization of multiplicative SDEs through additive noise

Galeati¹,

Harang²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…This then allows us to build an infinite-dimensional rough path Z ε (taking values in a space of vector fields on R d ) associated to (1.4) in a similar way as in [40,Sec. 1.5] (see also the "nonlinear rough paths" of [58] and [26]) and to reformulate (1.4) as an RDE driven by Z ε with nonlinearity given by point evaluation. Section 3.2 provides details of the construction of Z ε , while Sect.…”

Section: The Markov Semigroup Associated To the Process Y Is Strongly...mentioning

confidence: 99%

Generating Diffusions with Fractional Brownian Motion

Hairer

2022

Commun. Math. Phys.

View full text Add to dashboard Cite

We study fast/slow systems driven by a fractional Brownian motion B with Hurst parameter $$H\in (\frac{1}{3}, 1]$$ H ∈ ( 1 3 , 1 ] . Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if $$Y^\varepsilon $$ Y ε denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale $$\varepsilon \ll 1$$ ε ≪ 1 , the solutions of the equation $$\begin{aligned} dX^\varepsilon = {\varepsilon }^{\frac{1}{2}-H} F(X^\varepsilon ,Y^\varepsilon )\,dB+F_0(X^\varepsilon ,Y^{\varepsilon })\,dt\; \end{aligned}$$ d X ε = ε 1 2 - H F ( X ε , Y ε ) d B + F 0 ( X ε , Y ε ) d t converge to a regular diffusion without having to assume that F averages to 0, provided that $$H< \frac{1}{2}$$ H < 1 2 . For $$H > \frac{1}{2}$$ H > 1 2 , a similar result holds, but this time it does require F to average to 0. We also prove that the n-point motions converge to those of a Kunita type SDE. One nice interpretation of this result is that it provides a continuous interpolation between the time homogenisation theorem for random ODEs with rapidly oscillating right-hand sides ($$H=1$$ H = 1 ) and the averaging of diffusion processes ($$H= \frac{1}{2}$$ H = 1 2 ).

show abstract

“…Furthermore, if one can prove that w b ∈ C γ t C η x,loc for general distributions b, one can not hope for a γ > 1 2 , which is required to apply the nonlinear Young formalism employed in this article. To this end, one could hope to use techniques developed on nonlinear rough paths (see, e.g., [9,31]), but the exact formulation of the equation in this context is not completely clear.…”

Section: Further Extensionsmentioning

confidence: 99%

Regularization of multiplicative SDEs through additive noise

Galeati

Harang

2022

Ann. Appl. Probab.

View full text Add to dashboard Cite

We investigate the regularizing effect of certain additive continuous perturbations on SDEs with multiplicative fractional Brownian motion (fBm). Traditionally, a Lipschitz requirement on the drift and diffusion coefficients is imposed to ensure existence and uniqueness of the SDE. We show that suitable perturbations restore existence, uniqueness and regularity of the flow for the resulting equation, even when both the drift and the diffusion coefficients are distributional, thus extending the program of regularization by noise to the case of multiplicative SDEs. Our method relies on a combination of the nonlinear Young formalism developed by Catellier and Gubinelli (Stochastic

show abstract

On Nonlinear Rough Paths

Cited by 4 publications

References 27 publications

Regularization of multiplicative SDEs through additive noise

Regularization of multiplicative SDEs through additive noise

Generating Diffusions with Fractional Brownian Motion

Regularization of multiplicative SDEs through additive noise

Contact Info

Product

Resources

About