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The Brownian first-passage time $T_a$ is finite almost surely. Indeed the Brownian reflection principle gives $\mathbb{P}(T_a\le t)=2\mathbb{P}(B_t\ge a)\to 1$ as $t\to \infty$. Continuity of the paths means the first passage to $x+y$ occurs after the first passage to $x$. By the Strong Markov property, the process $B_{T_x+s}-x$ is an independent fresh Brownian motion. Thus $T_{x+y}-T_x$ has the same probability distribution as $T_y$ and is independent of $T_x$. Taking the Laplace transforms of nonnegative random variables givesFor fixed $\lambda \ge 0$, these transforms lie in $(0,1]$. Put $g(a)=-\log {\phi }_a(\lambda )$. It is additive on positive arguments and nondecreasing, because the hitting times increase with the level. Writing $c(\lambda )=g(1)$, additivity gives $g(r)=rc(\lambda )$ for positive rational $r$, and rational upper and lower approximations then give $g(a)=ac(\lambda )$ for every $a>0$. Hence ${\phi }_a(\lambda )=e^{-ac(\lambda )}$, with $c(\lambda )\ge 0$.

The Brownian scaling identity $T_a\stackrel{d}{=}{a}^2{T}_1$ givesFor $\lambda >0$, take $a=\sqrt{\lambda }$ in this relation with the transform parameter one. It follows that $c(\lambda )=C\sqrt{\lambda }$, where $C=c(1)\ge 0$. At zero, finiteness of the hitting time gives $c(0)=0$. Thus $\boxed{{\phi }_a(\lambda )=e^{-aC\sqrt{\lambda }}}$, with the constant determined in part (b).

For the Brownian first-passage time to a line, set $\lambda =\theta >0$ and choose $h=b+\sqrt{b^2+2\lambda }>0$, so $h^2/2-bh=\lambda$. The Exponential martingale for Brownian motion $M_s=e^{h{W}_s-h^{2}s/2}$, stopped at $T_{a,b}$, obeysThe optional stopping theorem at the bounded time $s\wedge T_{a,b}$ gives expectation one. The contribution from $\{T_{a,b}>s\}$ is at most $e^{ha-\lambda s}$, and tends to zero. Taking $e^{-\lambda \infty }=0$ therefore yieldsLetting $\theta \downarrow 0$ shows that the hitting probability is one for $b\le 0$ and $e^{-2ab}$ for $b>0$; in the latter case the hitting-time law has mass at infinity.

To derive the finite-time linear-boundary Brownian first-passage distribution, put $Z_s=W_s-bs$. This is Brownian motion with drift $\mu =-b$, and the event is $\{{\sup }_{s\le t}{Z}_s\ge a\}$. For driftless Brownian motion, the Brownian reflection principle says that the joint endpoint density on this event equals the normal density $p_t(x)$ when $x\ge a$ and the reflected density $p_t(2a-x)$ when $x<a$, where $p_t(x)=(2\pi t{)}^{-1/2}{e}^{-x^2/(2t)}$. The constant-drift change of measure weights these densities by $e^{\mu x-{\mu }^{2}t/2}$. Thus the crossing probability isCompleting the square gives respectively $\Phi ((\mu t-a)/\sqrt{t})$ and $e^{2\mu a}\Phi ((-\mu t-a)/\sqrt{t})$. Substituting $\mu =-b$ provesThe density weighting follows directly from the Exponential martingale for Brownian motion, or the Girsanov theorem, on the fixed finite horizon; it does not require eventual hitting.

For the barrier digital call, under the risk-neutral measure in the Black-Scholes model,Assume $\sigma >0$, put $a=\log (c/S_0)/\sigma >0$ and $b=-(\rho -{\sigma }^2/2)/\sigma$. The upper stock barrier is reached exactly when $W_s^Q$ reaches $a+bs$. The payment is at the fixed maturity $t_0$, not at the random hitting time. Hence its price isIf $\sigma =0$, use the deterministic path $S_s=S_0{e}^{\rho s}$ instead: the value is $e^{-\rho t_0}$ if that path reaches $c$ by $t_0$, and zero otherwise. The nondegenerate formula concerns the usual positive-volatility model.

A standard Brownian motion is a real stochastic process $(B_t{)}_{t\ge 0}$ with $B_0=0$ almost surely, continuous sample paths almost surely, and independent increments such that $B_t-B_s$ has normal distribution $N(0,t-s)$ whenever $0\le s<t$. With its completed, right-continuous natural filtration $({\mathcal{F}}_t)$, its Strong Markov property is the following: for every almost-surely finite stopping time $T$, the processis a standard Brownian motion independent of ${\mathcal{F}}_T$. Here the stopping-time sigma-algebra isIn particular, for every bounded ${\mathcal{F}}_T$-measurable random variable $H$ and bounded Borel functional $G$ on the continuous path space,Thus the independence is from the entire stopped past, not only from $B_T$, and is a statement about the whole future path, not just a single increment.

To prove the Brownian reflection principle without assuming that a level is hit in finite time, fix $t>0$, $m>0$, and set $T_m=\inf \{s\ge 0:B_s\ge m\}$ and $R=T_m\wedge t$. Continuity of the sample paths makes $T_m$ a stopping time and gives $B_{T_m}=m$ whenever $T_m<\infty$. The stopping time $R$ is bounded. Reflect the path after $R$:By the Strong Markov property, the future increments after $R$ are an independent Brownian motion. Negating them preserves their probability distribution, by symmetry of their joint normal distributions. Hence the reflected process $\hat{B}$ has the same path law as $B$.

If $T_m>t$, reflection changes nothing on $[0,t]$ and ${\hat{B}}_t=B_t<m$. If $T_m\le t$, both paths first hit $m$ at the same time and ${\hat{B}}_t=2m-B_t$. Therefore, for $x\le m$, the following events agree path by path:Indeed, the right-hand endpoint is at least $m$, so the reflected path must have hit $m$ before or at $t$; this also forces the original path to have done so. The reflection relation then makes the endpoint inequality equivalent to $B_t\le x$. Equality cases at $T_m=t$ satisfy the same relation. Since ${\hat{B}}_t$ and $B_t$ have the same probability distribution,For $t=0$, both sides are zero, because $B_0=M_0=0$ and $2m-x\ge m>0$.

For $t>0$, set $x=m$ in this identity. Continuity implies $\{B_t>m\}\subseteq \{M_t\ge m\}$, and the normal distribution of $B_t$ has no atom at $m$. Splitting according to whether $B_t\le m$ givesBoth random variables are nonnegative. Their tails agree for all $m>0$, which determines their probability distributions, including the mass at zero. Thus the Brownian running maximum satisfies

For $x>0$ and $t>0$, continuity again gives $\{T_x\le t\}=\{M_t\ge x\}$. Since $B_t/\sqrt{t}$ is a standard normal random variable, the distribution function of the Brownian first-passage time isIt tends to one as $t\to \infty$ and to zero as $t\downarrow 0$, proving almost-sure finiteness and absence of an atom at zero. For the standard normal random variable $B_1$,The value of this ratio on the zero-probability event $\{B_1=0\}$ may be defined arbitrarily. Matching the distribution functions provesFinally, differentiate the distribution function on $t>0$, using ${\Phi }^{\prime }=\varphi$ and $\frac{d}{dt}(x/\sqrt{t})=-x/(2t^{3/2})$. The first-passage-time density isand is zero for $t\le 0$. Its integral is one by the endpoint limits of $F_{T_x}$, so no missing mass at an infinite hitting time is present.

For zero drift, the reflection principle gives the known Brownian first-passage time densityTo obtain a general drift, on a fixed finite horizon use the exponential martingale $M_t=e^{c{B}_t-c^{2}t/2}$ as change-of-measure density. The Novikov condition holds for constant $c$, and the Girsanov theorem says that the coordinate process $B$ has drift $c$ under the new measure. Thus the hitting law of $B$ under that measure is the law of $B+ct$ under the original measure. Conditional expectation of $M_T$ at a hitting time bounded by $T$ equals its stopped value, by the martingale optional sampling theorem. At $H_a=t$ this value is $e^{ca-c^{2}t/2}$. The drifted Brownian first-passage density is obtained by multiplying $h_0$, giving $\boxed{h_c(t)=a{e}^{-(a-ct{)}^2/(2t)}/\sqrt{2\pi t^3}}$.

To verify the integrated formula, put $u=(a-cT)/\sqrt{T}$ and $v=(-a-cT)/\sqrt{T}$. The identity $e^{2ac}\varphi (v)=\varphi (u)$ for the standard normal density givesBoth terms vanish as $T\downarrow 0$, so $\boxed{\mathbb{P}(H_a\le T)=\overline{\Phi }((a-cT)/\sqrt{T})+e^{2ac}\Phi ((-a-cT)/\sqrt{T})}$. The distribution has total mass one for $c\ge 0$ and mass $e^{2ac}$ for $c<0$; in the latter case the remaining probability is an atom at infinity.

Fix a deterministic $a\ge 0$. By the Strong Markov property at the finite Brownian first-passage time $H_a$, the process ${\tilde{B}}_s=B_{H_a+s}-a$ is a standard Brownian motion started at zero. Such a process takes positive values arbitrarily soon almost surely: for every $\epsilon >0$, the Brownian reflection principle makes its maximum on $[0,\epsilon ]$ have the distribution of $|B_{\epsilon }|$, so the probability that the maximum is zero is zero. Intersecting these events for $\epsilon =1/n$ gives an infimum of positive-crossing times equal to zero. ThereforeThe infimum defining the strict crossing time need not itself be a time when $B_t>a$; continuity gives $B_{T_a}=a$. The quantifier here is fixed-level almost sure equality, not indistinguishability of stochastic processes as the level varies.

For $0<\alpha <1$, use the power transformation $Y=X^{1-\alpha }$ on the positive stochastic interval. The Itô formula, with $d[X]=X^{2\alpha }dt$, givesThus $Y_t\le x_0^{1-\alpha }+(1-\alpha )B_t$ simultaneously for $t<T$. The right side reaches zero almost surely by recurrence of one-dimensional Brownian motion, and the same positive-path contradiction proves $T$ cannot exceed that Brownian first-passage time. This establishes the finite lifetime threshold for a power diffusion throughout $0<\alpha <1$.

At $\alpha =1$, the solution is the geometric Brownian motionIt is finite and positive at every finite time. On every compact time interval it has a positive minimum, so the hitting times of $1/n$ tend to infinity. Although the strong law for Brownian motion implies $X_t\to 0$ as $t\to \infty$, this is not a finite lifetime. ThereforeThe power transformation is a rescaled Lamperti transform; at $\alpha =1$ the corresponding transformation is the logarithm. No claim about pathwise uniqueness after adjoining the boundary zero is needed: the coefficients are locally Lipschitz continuous inside the positive domain, which is the domain of the given maximal local solution of a stochastic differential equation.

For $a>0$, the Brownian reflection principle at the first-passage time of $a$ gives the Brownian running maximum identitywhere $\Phi$ is the distribution function of the standard normal distribution. Indeed, reflection pairs paths that have reached $a$ and end below $a$ with paths ending above $a$; $B_t$ has no atom at $a$.

As $t\to \infty$, this probability tends to one. The events $\{S_n\ge a\}$ increase with integer $n$, so with probability one the path reaches $a$ in finite time. Taking a countable intersection over positive integer $a$ givesHere $S_{\infty }={\sup }_{s\ge 0}{B}_s$. This also proves that every positive-level Brownian first-passage time is finite almost surely.

Taking $x=y=a$ in (c) and using that $B_t$ has no atom at $a$ givesThis is the distribution function of the Brownian first-passage time; (b) ensures that it is a proper probability distribution. Comparing these distribution functions gives the Brownian scaling identity $T_{ca}\stackrel{d}{=}{c}^2{T}_a$ for $c>0$.

For an integer $n\ge 1$, set $T_0=0$ and successively hit the levels $a,2a,\dots ,na$. The Strong Markov property at these finite stopping times and spatial translation imply thatare independent random variables, each with the probability distribution of $T_a$. Thus, for independent copies $T_a^{(j)}$,ThereforeThe paper's definition is the strictly stable distribution convention, with no centering term, a special case of a stable distribution.

Interpret $P(k)$ in the variance integral as the unsmoothed cosmological density power spectrum of $\delta$. A linear smoothing window gives $P_R(k)=P(k)|\tilde{W}(kR){|}^2$ for the smoothed field; the PDF's description of $P(k)$ as already belonging to ${\delta }_R$ would otherwise count the window twice.

For a scale-free matter power spectrum $P(k)=A{k}^{n_{\mathrm{eff}}}$, substitute $q=kR$ to obtainWhenever the dimensionless integral is finite and nonzero, this proves the scale-free smoothed density varianceThe printed condition $n_{\mathrm{eff}}<3$ is insufficient by itself. For a normalized window with $\tilde{W}(0)=1$, infrared convergence requires $n_{\mathrm{eff}}>-3$. Ultraviolet convergence depends on the filter: a spherical top-hat window function requires $n_{\mathrm{eff}}<1$, a Gaussian filter suppresses every ultraviolet power, and a sharp-k smoothing filter has finite Fourier support. In particular, the monotone relation $R\to \infty \Rightarrow S={\sigma }_R^2\to 0$ requires $n_{\mathrm{eff}}>-3$ in the scale-free model.

At a single variance $S_{\ast }$, the Gaussian random field has ${\delta }_{S_{\ast }}\sim N(0,S_{\ast })$. Thus its endpoint tail isHere $\mathrm{erfc}$ is the complementary error function.

For the crossing probability, the key assumption is independent, symmetric increments as the smoothing scale changes. A sharp-k smoothing filter, $\tilde{W}(kR)=1_{k<1/R}$, supplies this property: decreasing $R$ adds independent Fourier shells of the underlying Gaussian field. Parameterizing their accumulated variance by $S$ gives a Brownian motion with covariance $\min (S,S^{\prime })$. An arbitrary real-space window gives correlated increments and does not justify this argument.

Reflect a trajectory after its first hit of ${\delta }_c$. The Strong Markov property and symmetric future increments preserve its probability, and reflection maps an endpoint below the barrier to one above it. Every endpoint above the barrier has already crossed; the reflected paths give an equally probable set that crossed but ended below. The Brownian reflection principle therefore yields the excursion-set description of halo formation:Its derivative is the Brownian first-passage time densityThe factor of two resolves trajectories that crossed a larger-scale barrier but finish below it on the chosen smaller scale.

Define the halo peak height $\nu ={\delta }_c/{\sigma }_R$. The cumulative probability $\mathcal{P}(>M)$ is a dimensionless mass fraction. With $V_M=M/\bar{\rho }$, its conversion to halo number density givesConsequently the Press-Schechter halo mass function isIt is positive when ${\sigma }_R$ decreases with mass. A chosen mass-radius convention $M\propto R^3$ gives $d\log {\sigma }_R/d\log M=-(n_{\mathrm{eff}}+3)/6$. For a sharp Fourier filter that mass assignment needs a convention, since its real-space kernel is not a localized top-hat volume.

For warm dark matter, the cutoff introduces a new length and invalidates the pure power-law variance scaling near that length. With the same sharp-k filter and $n_{\mathrm{eff}}>-3$, the variance is exactlyThe Brownian walk runs only up to $S_{\max }$ and then stops. The cutoff does not destroy independence of the Fourier shells that are still present. Thus the first-crossing density and derivative mass-function formula remain valid on the invertible part of $S(M)$, using the actual cutoff variance. For $R<k_{\mathrm{WDM}}^{-1}$, $d{\sigma }_R/dM=0$ and this idealized sharp-filter model has no new crossings or new low-mass haloes. Its total collapsed fraction isThis is halo first crossing with a finite variance cutoff: some trajectories never cross, so one must not force the collapsed fraction to one. The uncut power-law abundance cannot be extrapolated to small masses. For other windows, a variance plateau still occurs but the Markov/reflection derivation is not exact; the same mass function is then a modelling approximation, rather than a result established by this calculation.

For planar Brownian motion started at $1$, continuous winding angles sampled at the displayed radii have independent stationary increments. At $T_r$, apply the Strong Markov property, rotate by the current angle, and rescale space by $e^r$ and time by $e^{-2r}$. The future radial-level winding increment has the original law and is independent of the past. The logarithmic radius and winding angle are independent Brownian coordinates on the common clock $\int |B_s{|}^{-2}ds$. At radial passage they are therefore $U_{{\eta }_r}=-r$ and $V_{{\eta }_r}$, where ${\eta }_r$ is a Brownian first-passage time. Its Laplace transform gives the displayed characteristic function. The level process is a symmetric Cauchy process; replacing the passage clock by its right-continuous version supplies a càdlàg modification without changing any fixed-level law.