Torstein Nilssen

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

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.

Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation

Abstract In this work we present a condition for the regularity, in both space and Malliavin sense, of strong solutions to SDEs driven by Brownian motion. We conjecture that this condition is optimal. As a consequence, we are able to improve the regularity of densities of such solutions. We also apply these results to construct a classical solution to the stochastic transport equation when the drift is Lipschitz.

Regularity of strong solutions of one-dimensional SDE’s with discontinuous and unbounded drift

In this paper we develop a method for constructing strong solutions of one-dimensional Stochastic Differential Equations where the drift may be discontinuous and unbounded. The driving noise is the Brownian Motion and we show that the solution is Sobolev-differentiable in the initial condition and Malliavin differentiable. This method is not based on a pathwise uniqueness argument. We will apply these results to the stochastic transport equation. More specifically, we obtain a continuously differentiable solution of the stochastic transport equation when the driving function is a step function.