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Distribution dependent SDEs driven by additive fractional Brownian motion

Published version
Peer-reviewed

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Authors

Harang, FA 
Mayorcas, A 

Abstract

jats:titleAbstract</jats:title>jats:pWe study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter jats:inline-formulajats:alternativesjats:tex-math$$H\in (0,1)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miH</mml:mi> mml:mo∈</mml:mo> mml:mo(</mml:mo> mml:mn0</mml:mn> mml:mo,</mml:mo> mml:mn1</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. We establish strong well-posedness under a variety of assumptions on the drift; these include the choice jats:disp-formulajats:alternativesjats:tex-math$$\begin{aligned} B(\cdot ,\mu )=(f*\mu )(\cdot ) + g(\cdot ), \quad f,,g\in B^\alpha _{\infty ,\infty },\quad \alpha >1-\frac{1}{2H}, \end{aligned}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mtable mml:mtr mml:mtd mml:mrow mml:miB</mml:mi> mml:mrow mml:mo(</mml:mo> mml:mo·</mml:mo> mml:mo,</mml:mo> mml:miμ</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mo=</mml:mo> mml:mrow mml:mo(</mml:mo> mml:mif</mml:mi> <mml:mrow /> mml:mo∗</mml:mo> mml:miμ</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mrow mml:mo(</mml:mo> mml:mo·</mml:mo> mml:mo)</mml:mo> </mml:mrow> mml:mo+</mml:mo> mml:mig</mml:mi> mml:mrow mml:mo(</mml:mo> mml:mo·</mml:mo> mml:mo)</mml:mo> </mml:mrow> mml:mo,</mml:mo> <mml:mspace /> mml:mif</mml:mi> mml:mo,</mml:mo> <mml:mspace /> mml:mig</mml:mi> mml:mo∈</mml:mo> mml:msubsup mml:miB</mml:mi> mml:mrow mml:mi∞</mml:mi> mml:mo,</mml:mo> mml:mi∞</mml:mi> </mml:mrow> mml:miα</mml:mi> </mml:msubsup> mml:mo,</mml:mo> <mml:mspace /> mml:miα</mml:mi> mml:mo></mml:mo> mml:mn1</mml:mn> mml:mo-</mml:mo> mml:mfrac mml:mn1</mml:mn> mml:mrow mml:mn2</mml:mn> mml:miH</mml:mi> </mml:mrow> </mml:mfrac> mml:mo,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math></jats:alternatives></jats:disp-formula>thus extending the results by Catellier and Gubinelli (Stochast Process Appl 126(8):2323–2366, 2016) to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances. </jats:p>

Description

Keywords

Distribution dependent SDEs, Singular drifts, Regularization by noise, Fractional Brownian motion

Journal Title

Probability Theory and Related Fields

Conference Name

Journal ISSN

0178-8051
1432-2064

Volume Title

185

Publisher

Springer Science and Business Media LLC
Sponsorship
research council of norway (274410)
dfg, german research foundation (390685813)