Weak well-posedness of multidimensional stable driven SDEs in the critical case

Raynal, Paul-Éric Chaudru de; Menozzi, Stéphane; Priola, Enrico

doi:10.1142/s0219493720400043

Cited by 12 publications

(13 citation statements)

References 57 publications

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“…which coincide with (C0) and (C0 S ) in [17]. In this case, under the same Besov regularity on b, (A) and (B) were established by Chaudru de Raynal, Jabir and Menozzi in [17].…”

Section: Introductionsupporting

confidence: 70%

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Second order fractional mean-field SDEs with singular kernels and measure initial data

Hao¹,

Röckner²,

Zhang³

2023

Preprint

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In this paper we establish the local and global well-posedness of weak and strong solutions to second order fractional mean-field SDEs with singular/distribution interaction kernels and measure initial value, where the kernel can be Newton or Coulomb potential, Riesz potential, Biot-Savart law, etc. Moreover, we also show the stability, smoothness and the short time singularity and large time decay estimates of the distribution density. Our results reveal a phenomenon that for nonlinear mean-field equations, the regularity of the initial distribution could balance the singularity of the kernel. The precise relationship between the singularity of kernels and the regularity of initial value are calculated, which belongs to the subcritical regime in scaling sense. In particular, our results provide microscopic probability explanation and establish a unified treatment for many physical models such as fractional Vlasov-Poisson-Fokker-Planck system, the vorticity formulation of 2D-fractal Navier-Stokes equations, surface quasi-geostrophic models, fractional porous medium equation with viscosity, etc.

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“…which coincide with (C0) and (C0 S ) in [17]. In this case, under the same Besov regularity on b, (A) and (B) were established by Chaudru de Raynal, Jabir and Menozzi in [17].…”

Section: Introductionsupporting

confidence: 70%

“…which coincide with (C0) and (C0 S ) in [17]. In this case, under the same Besov regularity on b, (A) and (B) were established by Chaudru de Raynal, Jabir and Menozzi in [17]. We would like to point out that the interesting physical models usually need to assume p 0 > 1.…”

Section: Introductionsupporting

confidence: 65%

Second order fractional mean-field SDEs with singular kernels and measure initial data

Hao¹,

Röckner²,

Zhang³

2023

Preprint

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“…Furthermore, a large set of theoretical tools is available to analyse the effect of Brownian perturbations to deterministic evolutions and this topic has a long and extensive literature [8,16,17,23,37,58,59]. Other classes of random perturbations, like fractional Brownian motion (fBm) have been more recently analysed, or even more exotic variants (e.g., ˛-stable and log regular processes) [1,6,13,38,45,49]. Let us finally mention the remarkable results from [10] concerning rates of convergence of numerical schemes for (1.1).…”

Section: Introductionmentioning

confidence: 99%

Noiseless regularisation by noise

Galeati¹,

Gubinelli²

2021

Rev. Mat. Iberoam.

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We analyse the effect of a generic continuous additive perturbation to the well-posedness of ordinary differential equations. Genericity here is understood in the sense of prevalence. This allows us to discuss these problems in a setting where we do not have to commit ourselves to any restrictive assumption on the statistical properties of the perturbation. The main result is that a generic continuous perturbation renders the Cauchy problem well-posed for arbitrarily irregular vector fields. Therefore we establish regularisation by noise "without probability".

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“…When γ = 1/α, α in (0, 1] we refer to e.g. [CdRMP20a,CdRMP20b] for results in that direction and to [DD15, CC18, FIR17, ZZ17, CdRM19, LZ19] for results related to the case where α in (1, 2] so that the above rule allows β to take negative values, the drift being then a distribution.…”

Section: Related Discussion and Perspectivesmentioning

confidence: 99%

Strong regularization by Brownian noise propagating through a weak Hörmander structure

Raynal

Honoré

Menozzi

2022

Probab. Theory Relat. Fields

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We establish strong uniqueness for a class of degenerate SDEs of weak Hörmander type under suitable Hölder regularity conditions for the associated drift term. Our approach relies on the Zvonkin transform which requires to exhibit good smoothing properties of the underlying parabolic PDE with rough, here Hölder, drift coefficients and source term. Such regularizing effects are established through a perturbation technique (forward parametrix approach) which also heavily relies on appropriate duality properties on Besov spaces.For the method employed, we exhibit some sharp thresholds on the Hölder exponents for the strong uniqueness to hold.

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Weak well-posedness of multidimensional stable driven SDEs in the critical case

Cited by 12 publications

References 57 publications

Second order fractional mean-field SDEs with singular kernels and measure initial data

Second order fractional mean-field SDEs with singular kernels and measure initial data

Noiseless regularisation by noise

Strong regularization by Brownian noise propagating through a weak Hörmander structure

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