The Wikipedia article on the Lévy distribution states that "The time of hitting a single point, at distance $\alpha$ from the starting point, by the Brownian motion has the Lévy distribution with $c=\alpha ^{2}$. (For a Brownian motion with drift, this time may follow an inverse Gaussian distribution, which has the Lévy distribution as a limit.)" where $c$ refers to the scaling parameter of the distribution. Is there an equivalent formula for 2- or even d-dimensional Brownian motion? The Wikipedia page for First-hitting time models derives the formula for 1 dimension:
The probability density function (PDF) for a particle in one dimension is found by solving the one-dimensional diffusion equation. (This equation states that the position probability density diffuses outward over time. It is analogous to say, cream in a cup of coffee if the cream was all contained within some small location initially. After a long time the cream has diffused throughout the entire drink evenly.) Namely,
$\frac {\partial p(x,t\mid x_{0})}{\partial t}=D{\frac {\partial ^{2}p(x,t\mid x_{0})}{\partial x^{2}}},$
given the initial condition $p(x,t={0}\mid x_{0})=\delta (x-x_{0})$; where $x(t)$ is the position of the particle at some given time, $x_{0}$ is the tagged particle's initial position, and $D$ is the diffusion constant with the S.I. units $m^{2}s^{-1}$ (an indirect measure of the particle's speed). The bar in the argument of the instantaneous probability refers to the conditional probability. The diffusion equation states that the rate of change over time in the probability of finding the particle at $x(t)$ position depends on the deceleration over distance of such probability at that position.
It can be shown that the one-dimensional PDF is $p(x,t;x_{0})={\frac {1}{\sqrt {4\pi Dt}}}\exp \left(-{\frac {(x-x_{0})^{2}}{4Dt}}\right).$
This states that the probability of finding the particle at $x(t)$ is Gaussian, and the width of the Gaussian is time dependent. More specifically the Full Width at Half Maximum (FWHM) – technically, this is actually the Full Duration at Half Maximum as the independent variable is time – scales like
${\rm {FWHM}}\sim {\sqrt {t}}$.
Using the PDF one is able to derive the average of a given function, $L$, at time $t$:
$\langle L(t)\rangle \equiv \int _{-\infty }^{\infty }L(x,t)p(x,t)\,dx,$ $\langle L(t)\rangle \equiv \int _{-\infty }^{\infty }L(x,t)p(x,t)\,dx,$
where the average is taken over all space (or any applicable variable).
The First Passage Time Density (FPTD) is the probability that a particle has first reached a point $x_{c}$ at exactly time $t$ (not at some time during the interval up to $t$). This probability density is calculable from the Survival probability (a more common probability measure in statistics). Consider the absorbing boundary condition $p(x_{c},t)=0$ (The subscript c for the absorption point $x_{c}$ is an abbreviation for cliff used in many texts as an analogy to an absorption point). The PDF satisfying this boundary condition is given by
$p(x,t;x_{0},x_{c})={\frac {1}{\sqrt {4\pi Dt}}}\left(\exp \left(-{\frac {(x-x_{0})^{2}}{4Dt}}\right)-\exp \left(-{\frac {(x-(2x_{c}-x_{0}))^{2}}{4Dt}}\right)\right),$
for $x<x_{c}$. The survival probability, the probability that the particle has remained at a position $x$ for all times up to $t$, is given by
$S(t)\equiv \int _{-\infty }^{x_{c}}p(x,t;x_{0},x_{c})\,dx=\operatorname {erf} \left({\frac {x_{c}-x_{0}}{2{\sqrt {Dt}}}}\right),$
where $\operatorname {erf}$ is the error function. The relation between the Survival probability and the FPTD is as follows: the probability that a particle has reached the absorption point between times $t$ and $t + d t$ is $f(t)\,dt=S(t)-S(t+dt)$. If one uses the first-order Taylor approximation, the definition of the FPTD follows):
$f(t)=-{\frac {\partial S(t)}{\partial t}}.$
By using the diffusion equation and integrating, the explicit FPTD is
$f(t)\equiv {\frac {|x_{c}-x_{0}|}{\sqrt {4\pi Dt^{3}}}}\exp \left(-{\frac {(x_{c}-x_{0})^{2}}{4Dt}}\right).$
The first-passage time for a Brownian particle therefore follows a Lévy distribution.
For $t\gg {\frac {(x_{c}-x_{0})^{2}}{4D}}$, it follows from above that
$f(t)={\frac {\Delta x}{\sqrt {4\pi Dt^{3}}}}\sim t^{-3/2},$
where $\Delta x\equiv |x_{c}-x_{0}|$. This equation states that the probability for a Brownian particle achieving a first passage at some long time (defined in the paragraph above) becomes increasingly small, but always finite.
The first moment of the FPTD diverges (as it is a so-called heavy-tailed distribution), therefore one cannot calculate the average FPT, so instead, one can calculate the typical time, the time when the FPTD is at a maximum ($\partial f/\partial t=0$), i.e.,
$\tau _{\rm {ty}}={\frac {\Delta x^{2}}{6D}}.$
When I tried to expand this into 2 dimensions I realised that a 2-dimensional survival function has an absorption line and not a point. I don't know where to take it from here.