Forward–backward stochastic differential systems associated to Navier–Stokes equations in the whole space

Abstract

3A coupled forward-backward stochastic differential system (FBSDS) is formulated in spaces of fields 4 for the incompressible Navier-Stokes equation in the whole space. It is shown to have a unique local solu-5 tion, and further if either the Reynolds number is small or the dimension of the forward stochastic differ-6 ential equation is equal to two, it can be shown to have a unique global solution. These results are shown 7 with probabilistic arguments to imply the known existence and uniqueness results for the Navier-Stokes 8 equation, and thus provide probabilistic formulas to the latter. Related results and the maximum principle 9 are also addressed for partial differential equations (PDEs) of Burgers' type. Moreover, from truncating 10 the time interval of the above FBSDS, approximate solution is derived for the Navier-Stokes equation by 11 a new class of FBSDSs and their associated PDEs; our probabilistic formula is also bridged to the prob-12 abilistic Lagrangian representations for the velocity field, given by Constantin and Iyer (2008) and Zhang 13 ✩

A forward-backward probabilistic algorithm for the incompressible Navier-Stokes equations

“…F. Delbaen et al [26] introduced a coupled FBSDEs system (FBSDS) associated to (9) through the nonlinear Feynman-Kac formula ( , ) = , and the probabilistic representation ∇ =̃︀ 0 wherẽ︀ is itself solution to a BSDE involving a Brownian motion independent from the one of the diffusion. The nonlocal operator ∇ (−△) −1 div div is represented by means of the BSDE defined on the infinite time interval (0, ∞).…”

“…Herẽ︀ 0 ( , , ) and ( , + ) means̃︀ , , 0 and , + , respectively. In [26], the infinite interval (0, ∞) of the probabilistic representation for the operator ∇ (−△) −1 div div is restricted to…”

“…for some ∈ Z + , ∈ (0, 1) and > 0 independent of (see Theorem 6.6 in [26]). Then, the PDE (11) is associated through the nonlinear Feynman-Kac formula…”

“…Recently F. Delbaen, J. Qiu and S. Tang introduced in [26] a new class of coupled FBSDEs associated to the incompressible Navier-Stokes equations. Since their probabilistic approach involves an extra BSDE defined on an infinite time interval for the stochastic representation of the pressure term, Delbaen et al deduced an approximated solution to the velocity field by truncating the infinite interval of the associated FBSDEs system.…”

“…Moreover, stochastic particle systems arise from the McKean-Vlasov equations in both two and three dimensional contexts [63,33,32]. Recently the systems of FBSDEs appear as a novel approach [22,26]. The problem of relating boundary conditions on probabilistic representations presents additional challenges [6,18,50,51].…”