**Appendix A. Extension of the 1D Solution to** *n***D**

Consider the steady-state solution *p*(*r*) = *C*(*α*) <sup>1</sup> − *<sup>r</sup>*<sup>2</sup> *<sup>α</sup>*/2−<sup>1</sup> , where *C*(*α*) is the normalization factor (cf. Equation (4)). In this Appendix we will use symmetry arguments to show that this form generalizes to higher dimensional setups.

First consider the 1D ball depicted in Figure A1a and imagine a large number of particles distributed according to Equation (4). Next take two small intervals on the right side of *r* = 0: *r*<sup>1</sup> < *r* < *r*<sup>1</sup> + *dr* and *r*<sup>2</sup> < *r* < *r*<sup>2</sup> + *dr*, as depicted. At steady state and within any time interval Δ*t*, there is as much flow from the *r*1-interval to the *r*2-interval as that there is from the *r*2-interval to the *r*1-interval, i.e., *J*<sup>12</sup> = *J*21. This is detailed balance [1]. Next we define a transition rate, *k*12, that is the probability per unit of time for a particle in the *r*1-interval to transit to the *r*2-interval. The rate *k*<sup>21</sup> is analogously defined. Detailed balance implies that *k*<sup>12</sup> *p*(*r*1) = *k*<sup>21</sup> *p*(*r*2) and thus:

$$\frac{k\_{12}}{k\_{21}} = \frac{p(r\_2)}{p(r\_1)} = \left(\frac{1 - r\_2^2}{1 - r\_1^2}\right)^{a/2 - 1}.\tag{A1}$$

**Figure A1.** A Lévy walk in a confined domain. Whenever the particle hits the confinement wall, it comes to a standstill there. The 1D steady-state probability distribution (**a**) is solved in Ref. [23]: *pst*(*r*) ∝  <sup>1</sup> <sup>−</sup> *<sup>r</sup>*<sup>2</sup> *<sup>α</sup>*/2−<sup>1</sup> . Between any two small intervals along the 1D domain, steady state implies *p*(*r*1)*k*<sup>12</sup> = *p*(*r*2)*k*21, where the *k*'s denote transition rates. In 2D (**b**) there is circular symmetry. If we take any narrow bar through the origin and look exclusively at traffic inside that bar, *pst*(*r*) ∝  <sup>1</sup> <sup>−</sup> *<sup>r</sup>*<sup>2</sup> *<sup>α</sup>*/2−<sup>1</sup> applies again. Next, we take a state *<sup>R</sup>*<sup>3</sup> outside the bar (**c**) and include transitions between *r*<sup>1</sup> and *r*<sup>2</sup> that go via any area *R*3. As the circular symmetry implies the absence of vortices, transitions *k* <sup>12</sup> and *k* <sup>21</sup> that go via *R*<sup>3</sup> must also follow *p*(*r*1)*k* <sup>12</sup> = *p*(*r*2)*k* <sup>21</sup>. From here it follows that *pst*(*r*) ∝  <sup>1</sup> <sup>−</sup> *<sup>r</sup>*<sup>2</sup> *<sup>α</sup>*/2−<sup>1</sup> also applies to higher dimensional setups. See the text of this Appendix for more detail.

Next consider the 2D setup depicted in Figure A1b. A bar of width *δ* is going through the center of the circle. We take two little areas at distances *r*<sup>1</sup> and *r*<sup>2</sup> from the center. Consider only trajectories between these two areas that stay within the bar. The traffic inside the bar should mimic the 1D system that was considered in the previous paragraph

and Figure A1a. Now consider also the transitions between the two little areas that proceed through trajectories that are not restricted to the narrow bar. Without loss of generality, we take an area *R*3, cf. Figure A1c, and we consider trajectories between *r*<sup>1</sup> and *r*<sup>2</sup> that pass through *R*3.

It is important to realize that the circular symmetry implies that there can be no vortices within the circular domain. Flow along any simple, closed curve within the unit circle would imply that there are points with net flow in the angular direction. Thus, along the *r*1,*r*2, *R*3-loop there must be as much clockwise flow as there is counterclockwise, i.e., *Jcw* = *Jccw*. This implies *k*12*k*23*k*<sup>31</sup> = *k*13*k*32*k*<sup>21</sup> [49] and thus:

$$\frac{k\_{12}}{k\_{21}} = \frac{k\_{13}k\_{32}}{k\_{23}k\_{31}}.\tag{A2}$$

The "state" *R*<sup>3</sup> can be taken to be anywhere within the circle and be of any size and shape. We can conclude that the ratio *k* 12/*k* <sup>21</sup> for transitions along any path between *r*<sup>1</sup> to *r*<sup>2</sup> within the circle must be equal to the ratio *k*12/*k*<sup>21</sup> for transitions with trajectories inside the bar.

It follows that, for any dimensionality, the probability density at radius *r* must be proportional to  <sup>1</sup> − *<sup>r</sup>*<sup>2</sup> *<sup>α</sup>*/2−<sup>1</sup> . For a normalized probability density in *n* dimensions we derive:

$$p(n,r) = \frac{\Gamma\left(\frac{n+a}{2}\right)}{\pi^{n/2}\Gamma\left(\frac{a}{2}\right)} \left(1-r^2\right)^{a/2-1}.\tag{A3}$$

For *n* = 2 the prefactor reduces to a simple *α*/(2*π*).
