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Let $X$ be the solution to the multidimensional SDE

$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$

with $W$ a Brownian motion, and $\mu, \sigma$ Lipschitz continuous with $\sigma$ nowhere zero. I'm looking for sources on the following result:

Almost surely, $X$ is Hölder continuous of every order less than $\frac{1}{2}$, and nowhere locally Hölder continuous of order $\frac{1}{2}$.

Is this result true? If so, is it proven/written down anywhere?

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    $\begingroup$ Yes, sorry for the self-advertisement but the Holder continuity in time for solutions of SDEs is indeed a standard result, see for instance my blog fabricebaudoin.blog/2012/10/16/… $\endgroup$
    – Fabrice Baudoin
    Commented Jun 16 at 7:11
  • $\begingroup$ @FabriceBaudoin Thanks for the reference! I think this gives the Holder continuity of all orders less than $\frac{1}{2}$. Is the second part of the statement on the sharpness of $\frac{1}{2}^-$ written anywhere? $\endgroup$
    – Nate River
    Commented Jun 16 at 7:17
  • $\begingroup$ @FabriceBaudoin Hm, interesting that $(1+\varepsilon)$-Lipschitz continuity is required. Has any progress been made on the merely Lipschitz case that they mention? Or maybe this result, which is less sharp, has already been proven for the Lipschitz case? $\endgroup$
    – Nate River
    Commented Jun 16 at 7:33

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Not (yet) a complete answer. As pointed out by Fabrice Baudoin in the comments, the Holder continuity of all orders less than $\frac{1}{2}$ follows easily from the BDG inequality, and the Kolmogorov continuity theorem.

The second part of the statement on the sharpness of the $\frac{1}{2}^-$ exponent appears to be answered affirmatively by Theorem 2 in the following paper by Baldi and Chaleyat-Maurel, but as I cannot read French, I am unable to verify this... in particular I would like to know if his definition and assumptions on the diffusion process matches my own.

If anyone would like to take a look it would be greatly appreciated.

Edit: The paper does indeed answer the question affirmatively.

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    $\begingroup$ Yes, the paper you link to provides optimality of the Holder's exponent in the even more general setting of elliptic SDEs on Riemannian manifolds $\endgroup$
    – Fabrice Baudoin
    Commented Jun 16 at 7:57

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