Abstract
In this paper, weak uniqueness of hypoelliptic stochastic differential equation with Holder drift is proved when the Holder exponent is strictly greater than 1/3. This result then extends to a weak framework the previous works [ 4 , 23 , 10 ], where strong uniqueness was proved when the regularity index of the drift is strictly greater than 2/3. Part of the result is also shown to be almost sharp thanks to a counter example when the Holder exponent of the degenerate component is just below 1/3. The approach is based on martingale problem formulation of Stroock and Varadhan and so on smoothing properties of the associated PDE which is, in the current setting, degenerate.
Highlights
Let d be a positive integer and Md(R) be the set of d × d matrices with real coefficients
Where x1 and x2 belong to Rd and where the diffusion matrix a := σσ∗ is1 supposed to be uniformly elliptic
We mean that there exists a threshold for the Holder-continuity of the drift with respect to (w.r.t.) the degenerate argument. This Holder-exponent is supposed to be strictly greater than 1/3. We show that this threshold is almost sharp thanks to a counter-example when the Holder
Summary
Let d be a positive integer and Md(R) be the set of d × d matrices with real coefficients. We prove that our assumptions are (almost) minimal by giving a counter-example in the case where the drift F2 is Holder continuous with Holder exponent just below 1/3 This example concerns our degenerate case, we feel that the method could be adapted in order to obtain the optimal threshold (for the weak well posedness) in other settings. We have to ask same kind of assumptions on the drift function F1 and the linearized drift of F2 (see system (5.2) below) This is the reason why both β11 and η are strictly positive and β12 and β22 are supposed to be strictly greater than 1/3 our counter example only holds for the Holder exponent of the drift of the second component in the degenerate variable.
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