Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation

By Salah-Eldin A. Mohammed, Torstein Nilssen, Frank Norbert Proske. 2012

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In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms R d ∋ x 7−→ φ s,t (x) ∈ R d , s, t ∈ R, for a stochastic differential equation (SDE) of the form dXt = b(t, Xt) dt + dBt, s, t ∈ R, Xs = x ∈ R d . The above SDE is driven by a bounded measurable drift coefficientb : R × R d → R d and a d-dimensional Brownian motion B. More specifically, we show that the stochastic flow φs,t(·) of the SDE lives in the space L 2 (; W 1,p (R d , w)) for all s, t and all p > 1, where W 1,p (R d , w) denotes a weighted Sobolev space with weight w possessing a p-th moment with respect to Lebesgue measure on R d . This result is counter-intuitive, since the dominant ‘culture’ in stochastic (and deterministic) dynamical systems is that the flow ‘inherits’ its spatial regularity from the driving vector fields. The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equation � dtu(t, x)+ (b(t, x) · Du(t, x))dt + P d=1 ei · Du(t, x) ◦ dB i t = 0, u(0, x) = u0(x), where b is bounded and measurable, u0 is C 1 b and {ei} d=1 a basis for R d . It is well-known that the deterministic counter part of the above equation does not in general have a solution. Using stochastic perturbations and our analysis of the above SDE, we establish a deterministic flow of Sobolev diffeomorphisms for classical one-dimensional (deterministic) ODE’s driven by discontinuous vector fields. Furthermore, and as a corollary of the latter result, we construct a Sobolev stochastic flow of diffeomorphisms for one-dimensional SDE’s driven by discontinuous diffusion coefficients.

Subfields: Applied Mathematics, Computational Mechanics, Finance

View Paper: https://www.semanticscholar.org/paper/79f91a2a773c04650169bdd27b7b7e77df956428

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Citations in the corpus

Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws (2014, background): “Moreover, it has been proven, see, for example, Fedrizzi and Flandoli [24], Flandoli, Gubinelli and Priola [28] and Mohammed, Nilssen and Proske [44], that solutions to (1.15) satisfy higher regularity than in the deterministic case.”
Ergodicity of stochastic differential equations with jumps and singular coefficients (2017, background): “The interested readers are referred to [17, 18, 31, 34, 45, 52, 54] and references therein.”
Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014, methodology): “Let us mention that regularity under more restrictive assumptions was already proved, see for example [35, 64] or [18] (also for fractional Brownian motion).”
Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients (2014, background): “It should be noticed that in all previous works (see [5, 15, 26]), the argument of mollifying coefficients is used to obtain the Sobolev differentiability of strong solutions.”
Stochastic differential equations with Sobolev diffusion and singular drift and applications (2016, methodology): “It should be noticed that when σt (x) = σt and bt (x) are bounded, (A), (B) and (C) were studied in [18] and [19] by using different arguments.”