On Weak Uniqueness for Some Degenerate SDEs by Global L p Estimates

Abstract

We prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant we recover the classical degenerate Ornstein-Uhlenbeck process which only has to satisfy the Hörmander hypoellipticity condition. In the proof we also use global L p -estimates for hypoelliptic Ornstein-Uhlenbeck operators recently proved in Bramanti-Cupini-Lanconelli-Priola (Math. Z. 266 (2010)) and adapt the localization procedure introduced by Stroock and Varadhan. Appendix contains a quite general localization principle for martingale problems.MSC (2010) 60H10, 60J60, 35J70.

Stochastic Hamiltonian flows with singular coefficients

“…However, by comparing with the original proof of Stroock and Varadhan [31,Chapter 7], our proof is quite different from [27,23] as our starting point is a global apriori Krylov's estimate (see Theorem 4.3). In principle, it is reasonable to believe that our argument is applicable for more general equations as studied in [27,23]. In this section we assume that σ satisfies (UE) and for some p > 2(2d+1),…”

“…Here we shall use the freezing coefficient argument and the L p -estimate established in [6] and [5] for degenerate operators with constant coefficients (see also [8] for the case of nonlocal operators). Compared with [6] and [27], we not only consider the optimal regularity of u along the nondegenerate v-direction, but also the optimal regularity of u along the degenerate x-direction.…”

“…)ds is a continuous and bounded functional, we have [27] and [23]. Therein, more general equations were considered.…”

“…By suitable localization techniques as developed in[27], it is possible to weaken the global assumptions inTheorem 4.5 as local ones together with some non-explosion conditions. Proof of Theorem 1.3.…”

L p -Parabolic Regularity and Non-degenerate Ornstein-Uhlenbeck Type Operators

“…The operator A and its parabolic counterpart L = A − ∂ t , which is also called Kolmogorov-Fokker-Planck operator, have recently received much attention (see, for instance, [3], [4], [6], [7], [9], [18], [19], [22] and the references therein). The operator A is the generator of the Ornstein-Uhlenbeck process which solves a linear stochastic differential equation (SDE) describing the random motion of a particle in a fluid (see [20]).…”