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Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda \to \mathbb{R}$ be a continuous function for which we know the following: for all $x\in\Lambda$, it holds that ${\bf E}( h(X_{\tau_r})\mid X_0=x) = h(x)$ for all small enough $r>0$, where $\tau_r$ is the hitting time of the circumference of radius $r$ centred at $x$.

Can one conclude from this that $h(X_t)$ is a local martingale?

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    $\begingroup$ But I think it is still interesting to see if we can apply Ito in the first place. That means we have to extract some regularity from the mean value property stated for $h$ in the spirit of using MVP results for elliptic equations like "a mean value property of elliptic equations..." ams.org/journals/proc/1967-018-06/S0002-9939-1967-0218747-X/… which will depend on the regularity of $f$. $\endgroup$
    – Thomas Kojar
    Commented May 23, 2023 at 16:15
  • $\begingroup$ @ThomasKojar it's fine to assume some regularity for $h$ (maybe it even has to be "nice" if $X$ is a "good" diffusion --- in the example I have in mind $f$ is an analytic function). Btw, I think I've already figured out how to circumvent my specific issue; but nevertheless I'm curious how can one pass from a "sequence of stopping times"-statement to a "fixed $t$"-statement. $\endgroup$
    – Serguei Popov
    Commented May 23, 2023 at 16:33
  • $\begingroup$ @ThomasKojar $X$ is the Doob's $h$-transform (not with that $h$, though) of the Brownian motion (it's a 2-dimensional BM conditioned on not touching a bounded domain), and $h$ is some complicated thingy which involves the expected value of some function on the boundary of that domain wrt the entrance measure from $x$ there (by the BM). So I was a bit in doubt how to differentiate it correctly... On the other hand, that equality ${\bf E}_x (...) = h(x)$ is easy to obtain. But I've already figured out that one can insert $t\wedge \tau_r$ there instead of just $\tau_r$, thus solving my problem. $\endgroup$
    – Serguei Popov
    Commented May 23, 2023 at 18:27

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