Well-posedness of the transport equation by stochastic perturbation

Abstract

We consider the linear transport equation with a globally Hölder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influence of noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of Itô-Tanaka type.

Uniqueness for a stochastic inviscid dyadic model

“…An example in this direction is known for the linear transport equation with poor regularity of coefficients; see [12]. The model of the present paper seems to be the first nonlinear example of this regularization phenomenon (in the area of equations of fluid dynamic type; otherwise see [15], [16] and related works, based on completely different methods).…”

“…The noise of this equation is multiplicative as in [12] and linearly dependent on the first derivatives of the solution. For a Lagrangian motivation of such a noise, in the case of stochastic Navier-Stokes equations, see [22].…”

Global Schauder estimates for a class of degenerate Kolmogorov equations

“…We obtain optimal regularity results in Hölder spaces for both (1.2) λu(x) − Au(x) = f (x), x ∈ R n , and (1.3) ∂ t v(t, x) = Av(t, x) + H(t, x), t ∈ (0, T ], x ∈ R n , v(0, x) = g(x), x ∈ R n , where λ > 0 and the functions f , g and H are given. These results are deduced by sharp L ∞ -estimates on the spatial derivatives of the solution of (1.3) when H = 0, involving Hölder norms of the initial datum g. Global Schauder estimates have been used recently in connection with stochastic differential equations (see [1,6,11]). In [6] Schauder estimates for degenerate elliptic operators L in non-smooth domains are a key ingredient to investigate well-posedness of the martingale problem associated to L. In [11] parabolic Schauder estimates are used to prove the existence of a differentiable stochastic flow in the case of stochastic differential equations with Hölder continuous drift term.…”

Lp-solutions of the stochastic transport equation

“…This equation has been treated for the case u 0 (x) ∈ L ∞ (R d ) (see [12] and [16]) via the stochastic characteristic method. Our aim here is to prove the existence, uniqueness and regularity when the initial data u 0 (x) ∈ L p (R d ) for p ∈ [1, ∞).…”

“…We give a Wong-Zakai principle for the stochastic transport equation (1), this principle is proved via stability properties of the deterministic transport linear equation. We would like to mention that our approach clearly differs from that one in [12], however this article has been a source of inspiration for us.…”