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For $g(x)=\log |x|$ on ${\mathbb{R}}^2\setminus \{0\}$,The Itô formula, stopped in a compact annulus, therefore givesThis proves the local martingale property up to a possible hitting of the origin. For the annulus exit time $T$, $M_{t\wedge T}$ is bounded between $\log r$ and $\log R$, so it is a true uniformly integrable martingale.

The exit is finite almost surely. Indeed $|B_t{|}^2-2t$ is a martingale, so stopping at $T\wedge t$ gives $2\mathbb{E}(T\wedge t)=\mathbb{E}|B_{T\wedge t}{|}^2-1\le R^2-1$. Letting $t\to \infty$ proves $\mathbb{E}T<\infty$. Bounded convergence in the logarithmic martingale now yieldsIf $p$ is the probability of reaching radius $r$ before radius $R$, continuity givesHitting zero before reaching radius $R$ would require reaching every positive radius $r$ first. Its probability is bounded by $p$, which tends to zero as $r\downarrow 0$. Any finite-time path hitting zero is bounded before that time and hence hits zero before exiting some integer-radius disk. A countable union over those disks proves $B_t\ne 0$ for every $t\ge 0$, almost surely. This is planar Brownian motion avoids a fixed point; it also makes $M$ a globally defined continuous local martingale.

To decide whether it is a true martingale, compute its expectation directly. Condition on $B_0$ if necessary and use rotation to take $B_0=(1,0)$ and write $B_t=B_0+W_t$. Conditional on $\rho =|W_t|$, the angle of $W_t$ is uniform. The angular average of a logarithmic potential isFor $\rho <1$ this follows by taking the real part of the convergent power series for $\log (1+\rho e^{i\theta })$: each nonconstant Fourier term integrates to zero. For $\rho >1$ factor out $\rho$ and apply the same argument to $1/\rho$. The equality at one follows by the integrable limiting logarithmic singularity. All interchanges are valid: the shifted Gaussian density is bounded near the origin, where ${\int }_0^{1}r|\log r|dr<\infty$, and its Gaussian tail controls logarithmic growth.

The radius $\rho$ has density $(\rho /t)e^{-{\rho }^2/(2t)}$. Consequently, for $t>0$,Since $M_0=0$, $M$ is not a martingale, despite integrability at each deterministic time. This is the logarithmic radius of planar Brownian motion example.

The optional hint is consistent with this conclusion. The upper-radius exit $T(2,0)$ is finite and never occurs at zero. After that exit the stopped square is $(\log 2{)}^2$; before it, a radius in $[1/2,2]$ also gives a logarithmic square no larger than $(\log 2{)}^2$, while a radius below $1/2$ contributes precisely the extra indicated term. This proves the hinted inequality without assuming that the unstopped process is a true martingale.

On a compact time interval a continuous path avoiding the origin has a strictly positive minimum radius. Thus the times $S_{1/n}$ tend to infinity almost surely. Part (a) localizes $M_t=\log |B_t|$, so $M$ is a continuous local martingale.

To show that it is not a martingale, compute its expectation at a positive deterministic time. In distribution $B_t=(1,0)+W_t$, where $W_t$ is a centered isotropic Gaussian vector. Conditional on $\rho =|W_t|$, its angle is uniform. The angular average of a logarithmic potential isFor $\rho <1$ this follows by averaging the real part of the power series for $\log (1+\rho e^{i\theta })$; for $\rho >1$ factor out $\rho$ and apply the same argument to ${\rho }^{-1}$. The value at $1$ follows by the integrable logarithmic singularity, or is irrelevant for the continuously distributed radius.

The radius has density $(\rho /t)e^{-{\rho }^2/(2t)}$. The logarithm of $|B_t|$ is absolutely integrable: near the origin its Gaussian density is bounded and ${\int }_0^{1}r|\log r|dr<\infty$; away from the origin its positive part is bounded by a constant plus $|B_t|$. Therefore conditioning is legitimate, and integration by parts yieldsSince $M_0=0$, this contradicts preservation of expectation by a martingale. Thus the logarithmic radius of planar Brownian motion is a strict local martingale. It is not a nonnegative local martingale; its negative values allow its expectation to increase.