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# Invariance principles and Log-distance of F-KPP fronts in a random medium

Alexander Drewitz Lars Schmitz Universität zu Köln, Mathematisches Institut,
Weyertal 86–90, 50931 Köln, Germany. Email<EMAIL_ADDRESS>koeln.deUniversität zu Köln, Mathematisches Institut, Weyertal 86–90, 50931
Köln, Germany. Email<EMAIL_ADDRESS>
###### Abstract

We study the front of the solution to the F-KPP equation with randomized non-
linearity. Under suitable assumptions on the randomness involving spatial
mixing behavior and boundedness, we show that the front of the solution lags
at most logarithmically in time behind the front of the solution of the
corresponding linearized equation, i.e. the parabolic Anderson model. This can
be interpreted as a partial generalization of Bramson’s findings [9] for the
homogeneous setting. Building on this result, we establish functional central
limit theorems for the fronts of the solutions to both equations.

Preliminary version

###### Contents

  1. 1 Introduction and main results
    1. 1.1 The classical F-KPP equation
    2. 1.2 The randomized F-KPP equation and the parabolic Anderson model
    3. 1.3 Summary of results
    4. 1.4 Further notation
    5. 1.5 The linearized equation
    6. 1.6 The non-linear equation
    7. 1.7 Discussion and previous results
    8. 1.8 Notational conventions
  2. 2 Some technical tools
    1. 2.1 Connection to branching processes
    2. 2.2 Change of measure
    3. 2.3 Concentration inequalities
  3. 3 Large deviations and perturbation results for the PAM
    1. 3.1 An exact large deviation result for auxiliary processes
    2. 3.2 Proof of Theorem 1.2
    3. 3.3 Time perturbation
    4. 3.4 Space perturbation
    5. 3.5 Approximation results
    6. 3.6 Proof of Theorem 1.3
  4. 4 Log-distance of the fronts of the solutions to PAM and F-KPP
    1. 4.1 First moment of leading particles
    2. 4.2 Second moment of leading particles
    3. 4.3 Proof of Theorem 1.4
    4. 4.4 Remarks on $v_{c}$ and $v_{0}$
  5. A Appendix
    1. A.1 Properties of the logarithmic moment generating functions
    2. A.2 Exponential mixing
    3. A.3 Properties of the Lyapunov exponent
    4. A.4 A Hoeffding-type inequality for dependent random variables
    5. A.5 Proof of the McKean representation
    6. A.6 Existence and monotonicity of solutions

##  1 Introduction and main results

###  1.1 The classical F-KPP equation

The F-KPP equation is the initial value problem given by

$\displaystyle\begin{split}w_{t}(t,x)&=\frac{1}{2}w_{xx}(t,x)+w(t,x)(1-w(t,x)),\quad
t>0,\ x\in\mathbb{R},\\\ w(0,x)&=w_{0}(x),\quad x\in\mathbb{R},\end{split}$
(1.1)

with initial condition $w_{0}:\mathbb{R}\to[0,1].$ Its investigation has a
long history, with seminal results dating back to Fisher [13] and Kolmogorov-
Petrovskii-Piskunov [24]. Their research had been motivated by pioneering
works in genetics, where the equation has been used to model a randomly mating
diploid population living in a one-dimensional habitat. Further applications
can be found in chemical combustion theory or flame propagation, see [1],
[12], as well as [36] and references therein.

In [24] it has been shown that for reasonably general non-linearities (see
(SC) at the beginning of Section 1.4 for further details) and initial
conditions, the solution to (1.1) approaches a _traveling wave_
$g:\mathbb{R}\to[0,1]$. I.e., there exists a function
$m:[0,\infty)\to\mathbb{R}$ – generally referred to as the _front_ or
_breakpoint_ – such that

$w(t,\cdot+m(t))\displaystyle\mathop{\longrightarrow}_{t\rightarrow\infty}g\quad\text{uniformly.}$

The limit $g$ is known (see [24, Theorems 14 and 17]) to solve the
differential equation

$\frac{1}{2}g^{\prime\prime}(x)+\sqrt{2}g^{\prime}(x)+g(x)(1-g(x))=0,\
x\in\mathbb{R},$ $0<g(x)<1\quad\text{for all }x\in\mathbb{R}\quad\text{ and
}\quad g(x)\displaystyle\mathop{\longrightarrow}_{x\rightarrow\infty}0,\
g(x)\displaystyle\mathop{\longrightarrow}_{x\rightarrow-\infty}1,$

and it is unique up to spatial translations. Additionally, for the important
case of Heaviside-type initial condition $w_{0}=\mathds{1}_{(-\infty,0]},$ in
[24] the first order asymptotics
$m(t)/t\displaystyle\mathop{\longrightarrow}_{t\rightarrow\infty}\sqrt{2}$ has
been derived. A couple of decades later, Bramson in his seminal work [9]
improved this result by computing the second order correction up to additive
constants. More precisely, he showed that for each $\varepsilon\in(0,1),$ the
choice $m(t)=m^{\varepsilon}(t):=\sup\\{x\in\mathbb{R}:w(t,x)=\varepsilon\\}$
fulfills

$m(t)=\sqrt{2}t-\frac{3}{2\sqrt{2}}\ln t+\mathcal{O}(1)\quad\text{ as
}t\to\infty.$ (1.2)

Later on, Bramson [8, Theorem 3] generalized this result to more general
initial conditions; roughly speaking, he demanded the initial condition
$w_{0}(x)$ to have a sufficiently fast exponential decay for $x\to\infty$ and
to be non-vanishing for $x\to-\infty$. One of the main tools employed in the
proof was the McKean representation of the solution to (1.1) in terms of
expectations of branching Brownian motion, see [20] and [27]. Another
important ingredient was the comparison of the solution of (1.1) to the
solution of its linearized version

$\displaystyle\begin{split}u_{t}(t,x)&=\frac{1}{2}u_{xx}(t,x)+u(t,x),\quad
t>0,\ x\in\mathbb{R},\\\ u(0,x)&=w_{0}(x).\end{split}$ (1.3)

Indeed, since $m(t)$ describes the front of the solution where $u$ is small
and hence $1-u\approx 1,$ the heuristics is that $m(t)$ is in some sense well-
approximated by the front of the solution to (1.3). More precisely, for
$\overline{m}(t):=\sup\\{x\in\mathbb{R}:u(t,x)=\varepsilon\\}$ and Heaviside-
type initial condition $w_{0}=\mathds{1}_{(-\infty,0]}$, standard Gaussian
computations entail

$\overline{m}(t)=\sqrt{2}t-\frac{1}{2\sqrt{2}}\ln t+\mathcal{O}(1).$ (1.4)

In combination with (1.2), this results in a respective logarithmic backlog of
the two fronts in the sense that

$\overline{m}(t)-m(t)=\frac{1}{\sqrt{2}}\ln t+\mathcal{O}(1).$ (1.5)

The main goal of this article is to investigate the effect of introducing a
random potential in the non-linearity of (1.1) as well as in its linearization
(1.3), on the logarithmic backlog derived in (1.5) (cf. Theorem 1.4). Taking
advantage of this result we will then also derive functional central limit
theorems for the fronts of the respective solutions to the randomized
equations, see Theorem 1.3 and Corollary 1.5 below, which can be interpreted
as analogues to homogeneous-case results (1.4) and (1.2).

###  1.2 The randomized F-KPP equation and the parabolic Anderson model

Already in Fisher’s seminal paper [13], where he investigated the setting
(1.1) of homogeneous branching rates, it has been observed that a more
realistic model would be obtained by considering spatially heterogeneous rates
of the transformation of recessive to advantageous alleles. This, alongside a
mathematical interest, is our guiding motivation to consider the setting of
random $\xi$. Replacing the term $w(1-w)$ of (1.1) by a more general non-
linearity $F(w)$ fulfilling suitable _standard conditions_ (see (SC) below),
and overriding the notation $u$ and $w$ from the homogeneous setting of the
previous section, we then arrive at the equation

$\displaystyle\begin{split}w_{t}&=\frac{1}{2}w_{xx}+\xi(x,\omega)\cdot
F(w),\quad t>0,\ x\in\mathbb{R},\\\ w(0,x)&=w_{0}(x),\quad
x\in\mathbb{R},\end{split}$ (F-KPP)

as well as its linearized version, the _parabolic Anderson model_

$\displaystyle\begin{split}u_{t}&=\frac{1}{2}u_{xx}+\xi(x,\omega)\cdot u,\quad
t>0,\ x\in\mathbb{R},\\\ u(0,x)&=u_{0}(x),\quad x\in\mathbb{R}.\end{split}$
(PAM)

Here, the stochastic process $\xi:\mathbb{R}\times\Omega\to(0,\infty)$ models
the random medium. For the case $F(w)=w(1-w)$ and degenerate $\xi\equiv 1$,
these two equations yield the special cases (1.1) and (1.3), respectively. It
has long been known, see e.g. Freidlin [14, Theorem 7.6.1], that under
suitable assumptions there exists $v_{0}>0$, such that the solution $w(t,x)$
to (F-KPP) converges to $0$ (resp. $1$), uniformly for all
$x\geq\overline{v}t$ with $\overline{v}>v_{0}$ (resp. for all
$x\leq\underline{v}t$ with $\underline{v}<v_{0}$), as $t$ tends to infinity.
In a similar way, this result can be shown to hold for the solution to (PAM)
with the same $v_{0}$ as well, showing that the speeds or velocities of both
fronts, the one of the solution to (F-KPP) as well as the one of the solution
to (PAM), coincide. Consequently, as in the homogeneous case, the question of
second order corrections arises naturally.

###  1.3 Summary of results

In order to address this question and to be able to summarize our results, we
introduce some notation. Let $\varepsilon\in(0,1)$ and $a>0$. Furthermore,
write $u=u^{\xi,u_{0}}$ for the solution to (PAM) with initial condition
$u_{0},$ and $w=w^{\xi,F,w_{0}}$ for the solution to (F-KPP) with initial
condition $w_{0}$. As in the previous section, the fronts of the respective
solutions, denoted by

$\begin{split}\overline{m}^{\xi,u_{0},a}(t)&:=\sup\big{\\{}x\in\mathbb{R}:u(t,x)\geq
a\big{\\}},\\\
m^{\xi,F,w_{0},\varepsilon}(t)&:=\sup\big{\\{}x\in\mathbb{R}:w(t,x)\geq\varepsilon\big{\\}},\end{split}$
(1.6)

are of special interest. Since we will from now on focus on the heterogeneous
setting, we override the notation from Section 1.1, where it was used to
denote the fronts in the homogeneous case, and define

$\displaystyle\begin{split}\overline{m}(t)&:=\overline{m}^{a}(t):=\overline{m}^{\xi,\mathds{1}_{(-\infty,0]},a}(t),\\\
m(t)&:=m^{\varepsilon}(t):=m^{\xi,F,\mathds{1}_{(-\infty,0]},\varepsilon}(t).\end{split}$
(1.7)

Our findings are motivated by the respective results of (1.4), (1.2), and
(1.5) for the homogeneous case, which provide information about the position
of the fronts of the solutions to the respective equations, and thus their
respective backlog as well. Under suitable assumptions, our results are
summarized in the following two statements:

  1. (a)

There exist a constant $C\in(0,\infty)$ and a $\mathbb{P}$-a.s. finite random
time $\mathcal{T}(\omega)$ such that for all $t\geq\mathcal{T}(\omega),$

$\overline{m}(t)-m(t)\leq C\ln t;$ (1.8)

see Theorem 1.4 below.

  2. (b)

After centering and diffusive rescaling, the stochastic processes
$[0,\infty)\ni t\to\overline{m}(t)$ and $[0,\infty)\ni t\to m(t)$ satisfy
invariance principles; see Theorem 1.3 and Corollary 1.5 below.

As is shown in the companion article [11], in a certain sense there is a
logarithmic lower bound for $\overline{m}(t)-m(t)$ corresponding to (1.8) as
well; cf. also Section 1.7 below for further discussion.

###  1.4 Further notation

In order to be able to precisely formulate the previously summarized results,
we have to introduce some further notation. We start with introducing the
_standard conditions_ for the non-linearity, i.e., $F$ in (F-KPP) has to
fulfill the following:

$\begin{split}&F\in C^{1}([0,1]),\quad F(0)=F(1)=0,\quad F(w)>0\quad\forall
w\in(0,1),\quad\\\ F^{\prime}&(0)=1=\sup_{w>0}F(w)w^{-1},\quad
F^{\prime}(1)<0,\quad\limsup_{w\downarrow
0}\frac{1-F^{\prime}(w)}{w}<\infty.\end{split}$ (SC)

Figure 1: Sketches of functions fulfilling (SC), all of which are dominated by
the identity function.

We will now specify the classes of initial conditions under consideration for
both, (F-KPP) and (PAM). For this purpose, we fix $\delta^{\prime}\in(0,1)$
and $C^{\prime}>1,$ and require an initial condition $u_{0}$ of (PAM) to
fulfill

$\begin{split}&\delta^{\prime}\mathds{1}_{[-\delta^{\prime},0]}\leq u_{0}\leq
C^{\prime}\mathds{1}_{(-\infty,0]}.\end{split}$ (PAM-INI)

Our results also hold for initial conditions that decay sufficiently fast at
infinity, as well as for initial conditions that grow towards minus infinity
with sufficiently small exponential rate. However, in order to avoid further
technical complications we stick to the above set of initial conditions.

In addition, let us introduce a tail condition for the initial condition of
(F-KPP), which is the same as the one for the case $\xi\equiv 1$ stated in [8,
(1.17)]. For this purpose, we fix $N,N^{\prime}>0$, and require $w_{0}$ as in
(F-KPP) to fulfill

$\begin{split}0\leq
w_{0}\leq\mathds{1}_{(-\infty,0]}\qquad\text{and}\qquad\int_{[x-N,x]}w_{0}(y)\mathop{}\\!\mathrm{d}y\geq\delta^{\prime}\quad\forall
x\leq-N^{\prime}.\end{split}$ (KPP-INI)

Denote by $\mathcal{S}^{1}$ the class of functions $f:\mathbb{R}\to[0,\infty)$
which are pointwise limits of increasing sequences of continuous functions,
and let

$\displaystyle\mathcal{I}_{\text{PAM}}$
$\displaystyle:=\big{\\{}u_{0}\in\mathcal{S}^{1}:\ u_{0}\text{ fulfills
\eqref{eq:PAM_initial}}\big{\\}},\quad\mathcal{I}_{\text{F-KPP}}:=\big{\\{}w_{0}\in\mathcal{S}^{1}:\
w_{0}\text{ fulfills \eqref{eq:KPP_initial}}\big{\\}},$

which will be the classes of initial conditions under consideration. An
emblematic example which is contained in both, $\mathcal{I}_{\text{F-KPP}}$
and $\mathcal{I}_{\text{PAM}},$ is the function $\mathds{1}_{(-\infty,0]}$ of
Heaviside type.

We will assume $\xi=(\xi(x))_{x\in\mathbb{R}}$ to be a stochastic process on a
probability space $(\Omega,\mathcal{F},\mathbb{P})$ having Hölder continuous
paths, i.e., there exists $\alpha=\alpha(\xi)>0$ and $C=C(\xi)>0$, such that

$\left|\xi(x)-\xi(y)\right|\leq C\,|x-y|^{\alpha}\quad\forall
x,y\in\mathbb{R},$ (HÖL)

and such that the following conditions are fulfilled:

  * •

$\xi$ is _uniformly bounded away from $0$ and $\infty$_, i.e., there exist
constants $0<\texttt{ei}\leq\texttt{es}<\infty$ such that $\mathbb{P}$-a.s.,

$\texttt{ei}\leq\xi(x)\leq\texttt{es}\quad\text{ for all }x\in\mathbb{R};$
(BDD)

  * •

$\xi$ is _stationary_ , i.e. for every $h\in\mathbb{R}$ we have

$(\xi(x))_{x\in\mathbb{R}}\overset{d}{=}(\xi(x+h))_{x\in\mathbb{R}};$ (STAT)

  * •

$\xi$ fulfills a _$\psi$ -mixing_ condition: Let
$\mathcal{F}_{x}:=\sigma(\xi(z):z\leq x)$ and
$\mathcal{F}^{y}:=\sigma(\xi(z):z\geq y)$, $x,y\in\mathbb{R}$ and assume that
there is a continuous, non-increasing function $\psi:[0,\infty)\to[0,\infty)$,
such that for all $j,k\in\mathbb{Z}$ with $j\leq k$ as well as
$X\in\mathcal{L}^{1}(\Omega,\mathcal{F}_{j},\mathbb{P})$ and
$Y\in\mathcal{L}^{1}(\Omega,\mathcal{F}^{k},\mathbb{P})$ we have

$\displaystyle\big{|}\mathbb{E}\big{[}X-\mathbb{E}[X]\,|\,\mathcal{F}^{k}\big{]}\big{|}$
$\displaystyle\leq\mathbb{E}[|X|]\cdot\psi(k-j),$
$\displaystyle\big{|}\mathbb{E}\big{[}Y-\mathbb{E}[Y]\,|\,\mathcal{F}_{j}\big{]}\big{|}$
$\displaystyle\leq\mathbb{E}[|Y|]\cdot\psi({k-j}),$ (MIX)
$\displaystyle\sum_{k=1}^{\infty}$ $\displaystyle\psi(k)<\infty.$

Note that (• ‣ 1.4) implies the ergodicity of $\xi$ with respect to the shift
operator $\theta_{y}$ acting via $\xi(\cdot)\circ\theta_{y}=\xi(\cdot+y)$,
$y\in\mathbb{R}$.

Summarizing, we arrive at the following standing assumptions:

$\begin{gathered}\text{We will assume conditions \eqref{eq:hoelder_cont},
\eqref{eq:POT}, \eqref{eq:STAT} and \eqref{eq:MIX}}\\\ \text{to be fulfilled
from now on, if not explicitly mentioned otherwise.}\end{gathered}$ (Standing
assumptions)

Since we allow for non-smooth initial conditions in both equations, (F-KPP)
and (PAM), we shortly comment on the notion of solution in this setting. We
call a function $w$ to (F-KPP) a _generalized solution to ( F-KPP)_, if it
satisfies

$w(t,x)=E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{t}\xi(B_{s})F(w(t-s,B_{s}))/w(t-s,B_{s})\mathop{}\\!\mathrm{d}s\Big{\\}}\,w_{0}(B_{t})\Big{]},\quad\forall\
(t,x)\in[0,\infty)\times\mathbb{R},$ (1.9)

where $E_{x}$ is the corresponding expectation of the probability measure
$P_{x}$, under which $(B_{t})_{t\geq 0}$ is a standard Brownian motion
starting in $x\in\mathbb{R}$. This equation can be interpreted as a mild
formulation of (F-KPP) and one can show (see e.g. [14, (1.4), p. 354, and (a),
p. 355]) that every classical solution to (F-KPP) also is a generalized one.
Generalized solutions can be shown to exist under weak assumptions, and, vice
versa, in many instances they turn out to be classical solutions indeed, see
Proposition A.9 in Appendix A.

######  Remark 1.1.

A useful observation is that for initial conditions in
$\mathcal{I}_{\text{F-KPP}}$, generalized solutions can be approximated by
classical solutions. That is, if
$(w_{0}^{(n)})_{n\in\mathbb{N}}\subset\mathcal{I}_{\text{F-KPP}}$ is a
sequence of continuous functions which increase pointwise to $w_{0}$, then by
Corollary A.11 the corresponding sequence of (by Proposition A.9 classical)
solutions $(w^{(n)})_{n\in\mathbb{N}}=(w^{w_{0}^{(n)}})_{n\in\mathbb{N}}$ to
(F-KPP) is also monotone and thus the limit
$w(t,x):=\lim_{n\to\infty}w^{{(n)}}(t,x)$ exists for all
$(t,x)\in[0,\infty)\times\mathbb{R}$. Dominated convergence and the fact that
$w^{(n)}$ also is a generalized solution then imply

$\displaystyle w(t,x)$
$\displaystyle=\lim_{n\to\infty}w^{(n)}(t,x)=\lim_{n\to\infty}E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{t}\xi(B_{s})\frac{F(w^{(n)}(t-s,B_{s}))}{w^{(n)}(t-s,B_{s})}\mathop{}\\!\mathrm{d}s\Big{\\}}\,w^{(n)}_{0}(B_{t})\Big{]}$
$\displaystyle=E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{t}\xi(B_{s})\frac{F(w(t-s,B_{s}))}{w(t-s,B_{s})}\mathop{}\\!\mathrm{d}s\Big{\\}}\,w_{0}(B_{t})\Big{]}.$

I.e., $w$ is the generalized solution to (F-KPP) with initial condition
$w_{0}$.

A similar concept is possible for the solution to (PAM) as well. That is, for
$u_{0}\in\mathcal{I}_{\text{PAM}}$, we call a function

$\displaystyle
u(t,x)=E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{t}\xi(B_{s})\mathop{}\\!\mathrm{d}s\Big{\\}}u_{0}(B_{t})\Big{]},$
(1.10)

a _generalized solution to ( PAM)_. Note that this is an explicit expression
of the solution in terms of a Brownian path, while in (1.9) the solution is
only given implicitly. Furthermore, if $u_{0}$ is continuous, then by [21,
Remark 4.4.4], there exists a unique solution $u$ to (PAM) which fulfills
$u\in C^{1,2}((0,\infty)\times\mathbb{R})$ and $u(0,\cdot)=u_{0}.$ As a
consequence of these observations, in the following, we will always consider
generalized solutions.

###  1.5 The linearized equation

As already mentioned in the first section, we expect that investigating the
solution to (PAM) might also provide some insight into the solution to
(F-KPP). Therefore, starting with the first order of the front as a function
of time, it turns out useful to consider the so-called _Lyapunov exponent_

$\Lambda(v):=\lim_{t\to\infty}\frac{1}{t}\ln u(t,vt).$ (1.11)

Due to Proposition A.3, the Lyapunov exponent exists $\mathbb{P}$-a.s. for all
$v\in\mathbb{R}$, is non-random, and – as a consequence of Corollary 3.10 –
does not depend on the initial condition in $\mathcal{I}_{\text{PAM}}$ under
consideration. Furthermore, the function $[0,\infty)\ni v\mapsto\Lambda(v)$ is
concave, tends to $-\infty$ as $v\to\infty$ and $\Lambda(0)=\texttt{es}$,
where es is defined in (BDD). $\Lambda(v)$ describes the asymptotic
exponential growth of the solution in the linear regime with speed $v$. By
Proposition A.3, there exists a unique $v_{0}>0$, such that

$\Lambda(v_{0})=0,$

which we will call _velocity_ or _speed_ of the solution to (PAM). Using the
properties of the Lyapunov exponent, we immediately infer the first order
asymptotics for $\overline{m}$ to $\mathbb{P}\text{-a.s.}$ satisfy

$\frac{\overline{m}(t)}{t}\displaystyle\mathop{\longrightarrow}_{t\rightarrow\infty}v_{0}$

It will turn out that our methods work if we require $v_{0}$ to be strictly
larger than some “critical” value $v_{c}$, defined in Lemma 2.4 (d). Roughly
speaking, the condition $v>v_{c}$ allows us to find a suitable additive
tilting parameter in the exponent of the Feynman-Kac representation (1.10),
which depends on $v$ and makes the solution $u(t,x)$ to (PAM) amenable to the
investigation by standard tools for values $x\approx vt$ and large $t$. Hence,
we work under the assumption

$v_{0}>v_{c}.$ (VEL)

As will be shown in Section 4.4, this assumption is fulfilled for a rich class
of potentials $\xi.$

We start with investigating the fluctuations of the function $t\mapsto\ln
u(t,vt)$ around $t\Lambda(v)$ for values $v$ in a neighborhood of $v_{0}$,
which are interesting in their own right. To this end, on the space
$C([0,\infty))$ of continuous functions from $[0,\infty)$ to $\mathbb{R},$ we
define the metric

$\rho(f,g):=\sum_{j=1}^{\infty}2^{-j}\frac{\lVert f-g\rVert_{j}}{1+\lVert
f-g\rVert_{j}},\quad f,g\in C([0,\infty)),$ (1.12)

where we write $\lVert f-g\rVert_{j}:=\sup_{x\in[0,j]}|f(x)-g(x)|.$ This makes
$(C([0,\infty)),\rho)$ a complete separable metric space.

######  Theorem 1.2.

Let (VEL) be fulfilled, $u_{0}\in\mathcal{I}_{\text{\emph{PAM}}},$ and
$u=u^{\xi,u_{0}}$ be the corresponding solution to (PAM). Furthermore, let
$V\subset(v_{c},\infty)$ be a compact interval such that
$v_{0}\in\text{\emph{int}}(V)$. Then for each $v\in V$, as $n\to\infty$ the
sequence of random variables $(nv)^{-1/2}\big{(}\ln
u(n,vn)-n\Lambda(v)\big{)},$ $n\in\mathbb{N},$ converges in
$\mathbb{P}$-distribution to a centered Gaussian random variable with variance
$\sigma_{v}^{2}\in[0,\infty)$, where $\sigma_{v}^{2}$ is defined in (3.1). If
$\sigma_{v}^{2}>0$, the sequence of processes

$[0,\infty)\ni t\mapsto\frac{1}{\sqrt{nv\sigma_{v}^{2}}}\big{(}\ln
u(nt,vnt)-nt\Lambda(v)\big{)},\quad n\in\mathbb{N},$

converges as $n\to\infty$ in $\mathbb{P}$-distribution to a standard Brownian
motion in the sense of weak convergence of measures on $C([0,\infty))$ endowed
with the metric $\rho$ from (1.12).

In combination with perturbation estimates for $u,$ we will use this result in
order to infer an invariance principle for the front of the solution to (PAM).
Note that since the function $t\mapsto\overline{m}_{t}$ may be discontinuous,
we consider convergence in the Skorohod space $D([0,\infty))$ in the following
result.

######  Theorem 1.3.

Let (VEL) be fulfilled, $u_{0}\in\mathcal{I}_{\text{\emph{PAM}}}$ and $M>0$.
Then for $n\to\infty,$ the sequence
$n^{-1/2}\big{(}\overline{m}^{\xi,u_{0},M}(n)-v_{0}n\big{)},$
$n\in\mathbb{N},$ converges in $\mathbb{P}$-distribution to a centered
Gaussian random variable with variance
$\widetilde{\sigma}_{v_{0}}^{2}\in[0,\infty)$, where
$\widetilde{\sigma}_{v_{0}}^{2}$ is defined in (3.57). If
$\widetilde{\sigma}_{v_{0}}^{2}>0$, the sequence of processes

$[0,\infty)\ni
t\mapsto\frac{\overline{m}^{\xi,u_{0},M}(nt)-v_{0}nt}{\sqrt{n\widetilde{\sigma}_{v_{0}}^{2}}},\quad
n\in\mathbb{N},$

converges as $n\to\infty$ in $\mathbb{P}$-distribution to a standard Brownian
motion in the Skorohod space $D([0,\infty))$.

We underline that in the above theorems the case $\sigma_{v}^{2}=0$ (resp.
$\widetilde{\sigma}_{v_{0}}^{2}=0$) is allowed and leads to a degenerate limit
of the corresponding sequences. This can be excluded, e.g. if the finite-
dimensional projections of the stochastic process
$\xi=(\xi(x))_{x\in\mathbb{R}}$ are associated (see e.g. [31] or [30] for
definitions and results). In this case the covariances in (3.1) are
nonnegative and $\sigma_{v}^{2}>0$ follows. [28, Proposition 2.1] provides an
example of a potential, which is generated by an i.i.d. sequence of random
variables and thus associated.

###  1.6 The non-linear equation

Coming back to the original equation of interest, it is natural to ask whether
we can obtain results for (F-KPP), which are in some sense counterparts to
those derived in latter section for (PAM). For the solution to (F-KPP) it is
known (see e.g. [14, § 7.6]) that the first order of the front is linear as
well and moves with the same velocity as the front of (PAM). Indeed, by [14,
Theorem 7.6.1] we have that $\mathbb{P}\text{-a.s.},$

$\frac{m(t)}{t}\displaystyle\mathop{\longrightarrow}_{t\rightarrow\infty}v_{0}.$

As mentioned in Section 1.2, the next result, which is one of the main results
of the paper, states that there is an at most logarithmic distance of the
fronts of (F-KPP) and (PAM).

######  Theorem 1.4.

Let (VEL) be fulfilled. Then for each $F$ fulfilling (SC) there exists a
constant $\Cl{constanttheoremlogdist}>0$ such that the following holds: For
all $M>0$, $\varepsilon\in(0,1)$, $u_{0}\in\mathcal{I}_{\text{\emph{PAM}}}$
and $w_{0}\in\mathcal{I}_{\text{\emph{F-KPP}}},$ there exists a non-random
$C=C(\varepsilon,M,u_{0},w_{0})>0$ and a $\mathbb{P}\text{-a.s.}$ finite
random time $\mathcal{T}=\mathcal{T}(\xi,\varepsilon,M,u_{0},w_{0})\geq 0$,
such that

$-C\leq\overline{m}^{\xi,u_{0},M}(t)-m^{\xi,F,w_{0},\varepsilon}(t)\leq
C_{1}\ln t+C\quad\forall t\geq\mathcal{T}.$ (1.13)

Moreover for $w_{0}=u_{0}$ and $M\leq\varepsilon$, the left inequality in
(1.13) is zero for all $t\geq 0$.

Furthermore, combining Theorem 1.3 and Theorem 1.4, we can deduce an
invariance principle for the front of (F-KPP) as well.

######  Corollary 1.5.

Let (VEL) be fulfilled, $F$ fulfill (SC),
$w_{0}\in\mathcal{I}_{\text{\emph{F-KPP}}}$ and $\varepsilon\in(0,1)$. Then as
$n\to\infty,$ the sequence
$n^{-1/2}\big{(}m^{\xi,F,w_{0},\varepsilon}(n)-v_{0}n\big{)},$
$n\in\mathbb{N},$ converges in $\mathbb{P}$-distribution to a centered
Gaussian random variable with variance
$\widetilde{\sigma}_{v_{0}}^{2}\in[0,\infty)$, where
$\widetilde{\sigma}_{v_{0}}^{2}$ is defined in (3.57). If
$\widetilde{\sigma}_{v_{0}}^{2}>0$, the sequence of processes

$[0,\infty)\ni
t\mapsto\frac{m^{\xi,F,w_{0},\varepsilon}(nt)-v_{0}nt}{\sqrt{n\widetilde{\sigma}_{v_{0}}^{2}}},\quad
n\in\mathbb{N},$

converges as $n\to\infty$ in $\mathbb{P}$-distribution to a standard Brownian
motion in the Skorohod space $D([0,\infty))$.

###  1.7 Discussion and previous results

As already alluded to above, in the homogeneous case (1.1) of constant
potential the front has been well-understood by now. This is indeed the case
to a much wider extent than illustrated in the Introduction, see e.g. [6] and
references therein for further details. Also the heterogeneous setting (F-KPP)
of random potential and the properties of its solution have been investigated.
Specifically, under fairly general assumptions, the existence and
characterization of the propagation speed (i.e., the linear order
$\lim_{t\to\infty}m(t)/t$ of the position of the front) have been derived by
Freidlin and Gärtner, see e.g. [15] as well as [14, Chapter VII], using large
deviation principles. Incidentally, the Feynman-Kac formula (see Proposition
2.3 below), which characterizes the solution to the linearization (PAM), has
also played an important role in the derivation.

Second order corrections for the position have been investigated by Nolen
[28]. Also making the detour along (PAM), he examines (F-KPP) for a potential
$\xi$ under similar assumptions, but requires the (random) initial conditions
to satisfy

$\mathcal{C}_{1}(\xi)g^{\xi,\gamma}(x)\leq
w_{0}(x,\xi)\leq\mathcal{C}_{2}(\xi)g^{\xi,\gamma}(x)\quad\forall\ x>0.$
(1.14)

Here, $\mathcal{C}_{1}(\xi)$ and $\mathcal{C}_{2}(\xi)$ are positive random
variables, and $g=g^{\xi,\gamma}$ is a solution to

$g^{\prime\prime}(x)+(\xi(x)-\gamma)g(x)=0,\ x>0$

for $\gamma>\overline{\gamma}$, for a certain $\overline{\gamma}>0.$ For
technical reasons, he additionally requires $\gamma<\gamma^{*}$, i.e., the
initial condition $w_{0}(x)$ must not decay too fast as $x$ tends to infinity.
The technical assumption (1.14) thus entails being in the supercritical regime
which corresponds to waves that move faster than the minimal speed. As his
main result, Nolen obtains a central limit theorem for $m(t)$ in this case
also, see [28, Theorem 1.4]. In this lingo, our set of initial conditions
corresponds to the critical regime, and Corollary 1.5 (for the critical
regime) also suggests that the randomness of the functional central limit
theorem is already coming from the environment, and not necessarily due to the
random initial condition.

Furthermore, in [29] a corresponding invariance principle for the front has
been derived in the case where the non-linearity in (F-KPP) is either ignition
typ or bistable.

In [18], on the other hand, for the case of periodic instead of random $\xi,$
the authors have investigated the respective logarithmic correction,
corresponding to (1.2) in the homogeneous setting; here, the authors have been
able to characterize the constant in front of the logarithmic correction as a
certain minimizer.

In our main Theorem 1.4, we establish a corresponding logarithmic upper bound
for the difference (1.5), also in the setting of a random environment.
However, our methods are currently too coarse for identifying a sharp
prefactor. Nevertheless, it is a natural question whether the logarithmic
upper bound we derive captures the correct order at least. And indeed, in some
sense, a partial positive answer to this questions is provided in the
companion article [11]. There, the authors show that even in the setting of
(F-KPP) described above, for some potential satisfying our standing
assumptions, $w_{0}=\mathds{1}_{(-\infty,0]},$ and all
$\varepsilon\in(0,1/2),$ there exists a sequence
$(t_{n})_{n\in\mathbb{N}}=(t_{n}(\varepsilon))_{n\in\mathbb{N}}$ such that
$t_{n}\to\infty$ and

$m_{t_{n}}^{\varepsilon}-\inf\\{x\geq 0:w(t_{n},x)\leq
1-\varepsilon\\}\in\Omega(\ln n);$ (1.15)

in particular, the solution does not exhibit a uniformly bounded transition
front.

As explained above, there is a profound connection between the PDEs we
consider and branching Brownian motion, which is more involved that in the
setting of constant $\xi.$ Indeed, related results for branching random walk
in random environment (BRWRE) have been recently derived in [10]. In this
source the authors analyze the distribution of the maximal particle of a
branching random walk in random environment, which itself is closely related
to a discrete-space version of (F-KPP). In particular, a corresponding
logarithmic upper bound on the distance of the expected position of the
maximal particle and the median of the distribution of the maximal particle is
given. Furthermore, an invariance principle for the median of the position of
the maximal particle of BRWRE is derived. It is therefore no surprise that, on
the one hand, principal techniques we employ in this paper are generalizations
and adaptations of respective discrete space analogues from [10]. On the other
hand, we work under weaker independence assumptions and less assumptions on
the non-linearity which is assumed to be in (SC). What is more, we also obtain
a more profound understanding of the scope of the methods employed, see in
particular Section 4.4. In combination with the current article, somewhat of a
completion of the picture is provided by [11], by establishing a result with a
similar flavor of Bramson’s alluded to above.

Recall that directly before Theorem 1.2 we have introduced our assumption
(VEL), which reads $v_{0}>v_{c};$ from a technical point of view, this will be
necessary for our change of measure argument to work. It is not hard to show
that for a rich class of potentials, this condition is satisfied indeed, cf.
Section 4.4. What is more, however, in Proposition 4.10 below we also obtain a
more profound understanding of the scope of the methods employed. I.e., there
exist potentials $\xi$ fulfilling (Standing assumptions), but such that
$v_{0}<v_{c}$ holds true. In this regime our methods do not apply, and it is
an interesting and open question to obtain a more profound understand of this
situation as well.

###  1.8 Notational conventions

We will frequently use sums of real-indexed quantities $A_{x}$,
$x\in\mathbb{R}$. In this case, we write

$\displaystyle\sum_{i=1}^{x}A_{i}$ $\displaystyle:=\sum_{i=1}^{\lfloor
x\rfloor}A_{i}+A_{x},\quad x\in[0,\infty)\setminus\mathbb{N}_{0},$

where $\sum_{i=1}^{0}:=0$. This notion remains consistent if we also allow for
additive constants $b\in\mathbb{R},$ i.e.

$\displaystyle\sum_{i=1}^{x}(A_{i}+b)=\sum_{i=1}^{\lfloor
x\rfloor}A_{i}+A_{x}+b\cdot x=\sum_{i=1}^{x}A_{i}+\sum_{i=1}^{x}b,\quad
x\in[0,\infty).$

Finally, we set

$\displaystyle\sum_{i=x+1}^{y}A_{i}$
$\displaystyle:=\begin{cases}\sum_{i=1}^{y}A_{i}-\sum_{i=1}^{x}A_{i},&x\leq
y,\\\ \sum_{i=1}^{x}A_{i}-\sum_{i=1}^{y}A_{i},&x>y,\end{cases}\quad
x,y\in[0,\infty).$

A prime example is the quantity $A_{x}=\ln E_{x}\big{[}e^{\int_{0}^{H_{\lceil
x\rceil-1}}(\xi(B_{s})-\texttt{es})\mathop{}\\!\mathrm{d}s}\big{]}$, where
$H_{y}:=\inf\\{t\geq 0:B_{t}=y\\}$. Indeed, by the strong Markov property we
have $\ln
E_{x}\big{[}e^{\int_{0}^{H_{0}}(\xi(B_{s})-\texttt{es})\mathop{}\\!\mathrm{d}s}\big{]}=\sum_{i=1}^{\lfloor
x\rfloor}A_{i}+A_{x}$ for all $x\in[0,\infty)\setminus\mathbb{N}_{0}$.

Furthermore, we will often use positive finite constants $c_{1},c_{2},\ldots$
in the proofs. This numbering is consistent within any of the proofs, and it
is reset after each proof. On the other hand, $C_{1},C_{2},\ldots$ will be
used to denote positive finite constants that are fixed throughout the
article, and they will oftentimes depend on each other. Other constants like
$c,C,\varepsilon,\delta$ etc. in the proofs are used to compare certain
quantities and are also reset after each proof.

Acknowledgments: We would like to thank Jiří Černý for useful discussions and
remarks on a preliminary version of this paper.

##  2 Some technical tools

In this section we will introduce some further tools that will be helpful in
the proof of the main results.

###  2.1 Connection to branching processes

We define a _branching Brownian motion in random environment_ (BBMRE) as
follows: Conditionally on the realization of $\xi$ and for fixed
$x\in\mathbb{R}$, consider an initial particle starting at a point $x$ and
moving as a standard Brownian motion $(B_{t})_{t\geq 0}$ on $\mathbb{R}$.
While at site $y$, the particle dies at rate $\xi(y)$. More precisely, for an
exponentially distributed random variable $S$ with parameter one, independent
of everything else, the first particle dies at time $\inf\big{\\{}t\geq
0:S<\int_{0}^{t}\xi(B_{s})\mathop{}\\!\mathrm{d}s\big{\\}}$. When the initial
particle dies, it gives birth to $k$ new particles with probability $p_{k}$,
$k\in\mathbb{N}$ (see (PROB) below for the precise assumptions on the
$p_{k}$). The new particle(s) start their evolution at the site where their
parent particle had died, and they evolve independently of everything else and
according to the same stochastic behavior as their parent. Note that on the
one hand we assume $p_{0}=0$, so genealogies do not die out at any finite
time. On the other hand, we allow $p_{1}>0,$ i.e., it is possible to die and
give birth to one descendant. This implies that a particle at site $y$
branches into more than one particle with rate $\xi(y)(1-p_{1})$. We denote
the corresponding probability measure by $\texttt{P}_{x}^{\xi}$ and write
$\texttt{E}_{x}^{\xi}$ for the respective expectation. By $N(t)$ we denote the
set of particles alive at time $t$. For $\nu\in N(t)$ we let
$(X_{s}^{\nu})_{s\in[0,t]}$ be the spatial trajectory of the genealogy of
ancestral particles of $\nu$ up to time $t$. For a Borel set
$A\subset\mathbb{R}$ and $y\in\mathbb{R}$ we define

$\displaystyle N(t,A)$ $\displaystyle:=\big{\\{}\nu\in N(t):X_{t}^{\nu}\in
A\big{\\}}\quad\text{ and }\quad N^{\leq}(t,y):=|N(t,(-\infty,y])|,$ (2.1)

i.e., the set of particles which are in $A$ at time $t$, and the number of
particles to the left of $y$ at time $t.$

Let us now introduce a set of functions, formulated in terms of the
probability generating function of the sequence $(p_{k})_{k\in\mathbb{N}}$ as
above. More precisely, assume $(p_{k})_{k\in\mathbb{N}}$ and
$F=F^{(p_{k})_{k\in\mathbb{N}}}$ to fulfill

$\begin{split}p_{k}\in[0,1]\ \forall
k\in\mathbb{N},&\quad\sum_{k=1}^{\infty}p_{k}=1,\quad\sum_{k=1}^{\infty}kp_{k}=2,\quad
m_{2}:=\sum_{k=1}^{\infty}k^{2}p_{k}<\infty;\\\
F(u)&=1-u-\sum_{k=1}^{\infty}p_{k}(1-u)^{k},\ u\in[0,1].\end{split}$ (PROB)

Note that such $F$ always fulfills (SC), is smooth and strictly concave on
$(0,1)$.

A fundamental result is the connection of the solution to (F-KPP) and the
expected number of particles of the branching process. This is sometimes
referred to as the McKean representation. The respective result in the
homogeneous setting has been proven in [20] and [27] also.

######  Proposition 2.1.

Let $w_{0}\in\mathcal{I}_{\text{F-KPP}}$, $\xi$ be Hölder continuous and $F$
fulfill (PROB). Then $\mathbb{P}$-a.s. the solution to (F-KPP) is given by

$w(t,x)=1-\emph{{E}}_{x}^{\xi}\Big{[}\prod_{\nu\in
N(t)}\big{(}1-w_{0}(X_{t}^{\nu})\big{)}\Big{]},\quad(t,x)\in[0,\infty)\times\mathbb{R}.$
(McKean)

For the proof see Section A.5.

######  Remark 2.2.

We will frequently use the application of the latter result to functions
$w_{0}=\mathds{1}_{(-\infty,0]}$, resulting in

$\displaystyle w(t,x)$
$\displaystyle=1-\texttt{E}_{x}^{\xi}\Big{[}\prod_{\nu\in
N(t)}\mathds{1}_{(0,\infty)}(X_{t}^{\nu})\Big{]}=1-\texttt{P}_{x}^{\xi}\big{(}X_{t}^{\nu}>0\
\forall\nu\in N(t)\big{)}=\texttt{P}_{x}^{\xi}\big{(}N^{\leq}(t,0)\geq
1\big{)}.$

Furthermore, we will need the so-called _many-to-few_ lemma, which breaks down
the moments of the branching process to a functional of the Brownian paths.
For our purposes, it suffices to state it up to the second moments (_many-to-
one_ and _many-to-two formula_).

######  Proposition 2.3.

Let $(p_{k})_{k\in\mathbb{N}}$ fulfill (PROB) and let $\varphi_{1}$,
$\varphi_{2}:\,[0,\infty)\to[-\infty,\infty]$ be càdlàg functions satisfying
$\varphi_{1}\leq\varphi_{2}$. Then the first and second moments of the number
of particles in $N(t)$ whose genealogy stays between $\varphi_{1}$ and
$\varphi_{2}$ in the time interval $[0,t]$ are given by

$\displaystyle\emph{{E}}_{x}^{\xi}$ $\displaystyle\Big{[}\big{|}\nu\in
N(t):\varphi_{1}(s)\leq X_{s}^{\nu}\leq\varphi_{2}(s)\ \forall
s\in[0,t]\big{|}\Big{]}$
$\displaystyle=E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{t}\xi(B_{r})\mathop{}\\!\mathrm{d}r\Big{\\}};\varphi_{1}(s)\leq
B_{s}\leq\varphi_{2}(s)\ \forall s\in[0,t]\Big{]}\quad$ (FK-1)

and

$\displaystyle\emph{{E}}_{x}^{\xi}\Big{[}$ $\displaystyle\big{|}\nu\in
N(t):\varphi_{1}(s)\leq X_{s}^{\nu}\leq\varphi_{2}(s)\ \forall
s\in[0,t]\big{|}^{2}\Big{]}$
$\displaystyle=E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{t}\xi(B_{r})\mathop{}\\!\mathrm{d}r\Big{\\}};\varphi_{1}(s)\leq
B_{s}\leq\varphi_{2}(s)\ \forall s\in[0,t]\Big{]}$
$\displaystyle\qquad+(m_{2}-2)\int_{0}^{t}E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{s}\xi(B_{r})\mathop{}\\!\mathrm{d}r\Big{\\}}\,\xi(B_{s})\mathds{1}_{\\{\varphi_{1}(r)\leq
B_{r}\leq\varphi_{2}(r)\ \forall 0\leq r\leq s\\}}\Big{.}$ (FK-2)
$\displaystyle\qquad\quad\times\Big{.}\Big{(}E_{y}\big{[}\exp\big{\\{}\int_{0}^{t-s}\xi(B_{r})\mathop{}\\!\mathrm{d}r\big{\\}}\,\mathds{1}_{\\{\varphi_{1}(r+s)\leq
B_{r}\leq\varphi_{2}(r+s)\ \forall 0\leq r\leq
t-s\\}}\big{]}\Big{)}^{2}_{|_{y=B_{s}}}\Big{]}\mathop{}\\!\mathrm{d}s,$

respectively.

The proof of the first identity follows from [19, Section 4.1]. The second
identity can be shown by using [19, Lemma 1], conditioning on the first
splitting of the so-called “spines”, similarly to the proof of [16, (2.6)] for
$n=2$ there. Indeed, one has to consider branching Brownian motion instead of
branching random walk and replace binary branching by general branching. Then
the expectation of the quantity in display [16, (2.15)] turns into the second
summand in (FK-2), because the first splitting rate of the two “spine
particles” at site $y$ is $(m_{2}-2)\xi(y)$.

As a consequence of Proposition 2.3, the solution to (PAM) can be expressed by
a functional of the branching process, i.e. we have that the solution to (PAM)
is given by

$\displaystyle u(t,x)$
$\displaystyle=E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{t}\xi(B_{s})\mathop{}\\!\mathrm{d}s\Big{\\}}\,u_{0}(B_{t})\Big{]}=\texttt{E}_{x}^{\xi}\Big{[}\sum_{\nu\in
N(t)}u_{0}(X_{t}^{\nu})\Big{]}.$

As a special case for $u_{0}=\mathds{1}_{(-\infty,0]}$, this turns into

$\displaystyle u(t,x)=\texttt{E}_{x}^{\xi}\big{[}N^{\leq}(t,0)\big{]}$
$\displaystyle=E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{t}\xi(B_{s})\mathop{}\\!\mathrm{d}s\Big{\\}};B_{t}\leq
0\Big{]}.$ (2.2)

###  2.2 Change of measure

The first tool is a change of measure which makes it typical for the Brownian
motion in the Feynman-Kac formula started at $tv,$ some $v\neq 0,$ to be close
to the origin at time $t.$ For this purpose, let $(\xi(x))_{x\in\mathbb{R}}$
be as in (BDD) and define the shifted potential

$\zeta:=\xi-\texttt{es}.$

Then $\mathbb{P}$-a.s.,

$\zeta(x)\in[-(\texttt{es}-\texttt{ei}),0]\quad\forall x\in\mathbb{R}.$ (2.3)

We will oftentimes write

$H_{y}:=\inf\big{\\{}t\geq 0:B_{t}=y\big{\\}},\quad y\in\mathbb{R},\quad\text{
and }\quad\tau_{i}:=H_{i-1}-H_{i},\quad i\in\mathbb{Z},$ (2.4)

for the first hitting times and their pairwise differences. The above shift of
$\xi$ gives rise to a change of measure which will play a crucial role in the
following. For $x,y\in\mathbb{R}$ as well as $\eta\leq 0$ define the
probability measures $P_{x,y}^{\zeta,\eta}$ via

$\displaystyle P_{x,y}^{\zeta,\eta}(A)$
$\displaystyle:=\frac{1}{Z_{x,y}^{\zeta,\eta}}E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{H_{x-y}}\big{(}\zeta(B_{s})+\eta\big{)}\mathop{}\\!\mathrm{d}s\Big{\\}};A\Big{]},\quad
A\in\sigma\big{(}B_{t\wedge H_{x-y}}:t\geq 0\big{)},$ (2.5)

with normalizing constant

$\displaystyle Z_{x,y}^{\zeta,\eta}$
$\displaystyle:=E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{H_{x-y}}\big{(}\zeta(B_{s})+\eta\big{)}\mathop{}\\!\mathrm{d}s\Big{\\}}\Big{]}\in(0,\infty),$

where $P_{x}$ and $E_{x}$, $x\in\mathbb{R}$, are defined below (1.10). For
$A\in\sigma\big{(}B_{t\wedge H_{x-y}}:t\geq 0\big{)}$, using the strong Markov
property at time $H_{x-y}$, we infer
$P_{x,y}^{\zeta,\eta}(A)=P_{x,y^{\prime}}^{\zeta,\eta}(A)$ for all
$y^{\prime}\geq y$. Thus, by Kolmogorov’s extension theorem (see e.g. [35,
Theorem 2.4.3]),

$\big{(}P_{x,y}^{\zeta,\eta}\big{)}_{y\geq 0}\text{ can be extended to a
unique probability measure }P_{x}^{\zeta,\eta}\text{ on
}\sigma\big{(}B_{t}:t\geq 0\big{)}.$ (2.6)

We write $E_{x}^{\zeta,\eta}$ for the corresponding expectation and introduce
the logarithmic moment generating functions

$\displaystyle L_{x}^{\zeta}(\eta)$ $\displaystyle:=\ln
E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{H_{\lceil
x\rceil-1}}\big{(}\zeta(B_{s})+\eta\big{)}\mathop{}\\!\mathrm{d}s\Big{\\}}\Big{]},\quad
x>0,$ (2.7) $\displaystyle\overline{L}_{x}^{\zeta}(\eta)$
$\displaystyle:=\frac{1}{x}\sum_{i=1}^{x}L_{i}^{\zeta}(\eta)=\frac{1}{x}\ln
E_{x}\Big{[}\exp\Big{\\{}\int_{0}^{H_{0}}(\zeta(B_{s})+\eta)\mathop{}\\!\mathrm{d}s\Big{\\}}\Big{]},\quad
x>0,$

where we recall the notation introduced in Section 1.8, and where the last
equality is due to the Markov property. In addition, set

$L(\eta):=\mathbb{E}\big{[}L_{1}^{\zeta}(\eta)\big{]}.$ (2.8)

Due to (2.3), for any $\eta\leq 0$ the quantities above are well-defined, and
it is easy to check that in this case and under (BDD), the expressions defined
in (2.7)–(2.8) are finite. We have the following useful properties.

######  Lemma 2.4.

  1. (a)

The function $(-\infty,0)\ni\eta\mapsto L(\eta)$ is infinitely differentiable
and its derivative $L^{\prime}(\eta)$ is positive and monotonically strictly
increasing.

  2. (b)

We have $\mathbb{P}\text{-a.s.}$ that

$\lim_{x\to\infty}\overline{L}_{x}^{\zeta}(\eta)=L(\eta)\quad\text{for all
}\eta\leq 0.$ (2.9)

  3. (c)

$L^{\prime}(\eta)\downarrow 0$ as $\eta\downarrow-\infty$

  4. (d)

For every $v>v_{c}:=\frac{1}{L^{\prime}(0-)}$ (where $\frac{1}{+\infty}:=0$),
which we call _critical velocity_ , there exists a

$\text{ unique solution $\overline{\eta}(v)<0$ to the equation
}L^{\prime}(\overline{\eta}(v))=\frac{1}{v}.$ (2.10)

$\overline{\eta}(v)$ can be characterized as the unique maximizer to
$(-\infty,0]\ni\eta\mapsto\frac{\eta}{v}-L(\eta)$, i.e.

$\sup_{\eta\leq
0}\Big{(}\frac{\eta}{v}-L(\eta)\Big{)}=\frac{\overline{\eta}(v)}{v}-L\big{(}\overline{\eta}(v)\big{)}.$
(2.11)

The function $(v_{c},\infty)\ni v\mapsto\overline{\eta}(v)$ is continuously
differentiable and strictly decreasing.

###### Proof of Lemma 2.4.

  1. (a)

This follows from Lemma A.1.

  2. (b)

By [14, Theorem 7.5.1], for every $\eta\leq 0$ we get $\mathbb{P}$-a.s. that
$\lim_{x\to\infty}\overline{L}^{\zeta}_{x}(\eta)=\mathbb{E}\big{[}L_{1}^{\zeta}(\eta)\,|\,\mathcal{F}^{\zeta}_{\text{inv}}\big{]},$
where $\mathcal{F}^{\zeta}_{\text{inv}}$ is the $\sigma$-algebra of all
$\mathbb{P}$-invariant sets. Due to our standing assumptions, the family
$\zeta(x),$ $x\in\mathbb{R},$ is mixing and thus ergodic. Thus,
$\mathcal{F}^{\zeta}_{\text{inv}}$ is $\mathbb{P}$-trivial, i.e.,
$\mathbb{E}\big{[}L_{1}^{\zeta}(\eta)\,|\,\mathcal{F}^{\zeta}_{\text{inv}}\big{]}=L(\eta)$.
By continuity of the functions $\overline{L}^{\zeta}_{x}$ and $L$, the
statement follows.

  3. (c)

We note that $L$ is strictly increasing and strictly convex on $(-\infty,0)$
by (a) and

$L(\eta)\geq\mathbb{E}\Big{[}\ln
E_{1}\big{[}e^{-(\texttt{es}-\texttt{ei}-\eta)H_{0}}\big{]}\Big{]}=-\sqrt{2(\texttt{es}-\texttt{ei}-\eta)}\quad\text{for
all }\eta\leq 0,$

where the equality is due to [4, (2.0.1), p. 204]. Thus, we infer that its
derivative $L^{\prime}(\eta)$ tends to $0$ as $\eta\to-\infty$.

  4. (d)

Using that $1/v_{c}>1/v$ (where $\frac{1}{0}:=+\infty$) and the fact that
$L^{\prime}$ is strictly increasing with $L^{\prime}(\eta)\downarrow 0$ for
$\eta\downarrow-\infty$, we can find a unique $\overline{\eta}(v)<0$ such that
$L^{\prime}(\overline{\eta}(v))=1/v$, giving (2.10). On the other hand, (2.11)
is a direct consequence of (2.10) and standard properties of the Legendre
transformations of strictly convex functions. Because $L^{\prime}$ is strictly
increasing and smooth on $(-\infty,0)$, it has a strictly increasing inverse
function $(L^{\prime})^{-1}$, which is differentiable on $(0,1/v_{c})$. By
(2.10), for $v>v_{c}$ we have $\overline{\eta}(v)=(L^{\prime})^{-1}(1/v)$ and
thus by the formula for the derivative of the inverse function we get

$\overline{\eta}^{\prime}(v)=-\frac{1}{v^{2}}\cdot\frac{1}{L^{\prime\prime}(\overline{\eta}(v))}.$

Because the right-hand side of the latter display is continuous in $v$ and
negative, we conclude.

∎

We use the standard notation $L^{*}:\,\mathbb{R}\to(-\infty,\infty]$ to denote
the Legendre transformation

$\displaystyle v$ $\displaystyle\mapsto\sup_{\eta\leq 0}\big{(}\eta
v-L(\eta)\big{)}$

of $L$. Lemma 2.4 entails that

$\displaystyle
L^{*}(1/v)=\frac{\overline{\eta}(v)}{v}-L\big{(}\overline{\eta}(v)\big{)}.$
(2.12)

In the next part of this section, we are interested in a suitable _tilting
parameter_ $\eta_{x}^{\zeta}(v)$ such that

$E_{x}^{\zeta,\eta_{x}^{\zeta}(v)}\big{[}H_{0}\big{]}=\frac{x}{v},\quad x>0,\
v>0,$ (2.13)

holds true (setting $\eta_{x}^{\zeta}(v):=0$ if no such parameter exists). For
$\eta_{x}^{\zeta}(v)$ fulfilling (2.13) we observe that under
$P_{x}^{\zeta,\eta_{x}^{\zeta}(v)}$, the Brownian motion is tilted to have
time-averaged velocity $v$ until it reaches the origin. In the remaining part
of this section, see Lemma 2.5, we will show that for suitable $v$ and $x$
large enough, a tilting parameter as postulated in (2.13) actually exists.
Furthermore, we will show that the random parameter $\eta_{x}^{\zeta}(v)$
concentrates around the deterministic $\overline{\eta}(v)$ defined in (2.10).
The last result is a perturbation estimate for $\eta_{x}^{\zeta}(v)$ in $x,$
cf. Lemma 2.7.

###  2.3 Concentration inequalities

We have the following result regarding the existence (or, more precisely,
negativity) and concentration properties of the postulated parameter
$\eta_{x}^{\zeta}(v)$.

######  Lemma 2.5.

  1. (a)

For every $v>v_{c}$ there exists a finite random variable
$\mathcal{N}=\mathcal{N}(v)$ such that for all $x\geq\mathcal{N}$ the solution
$\eta_{x}^{\zeta}(v)<0$ to (2.13) exists.

  2. (b)

For each $q\in\mathbb{N}$ and each compact interval $V\subset(v_{c},\infty)$,
there exists
$\Cl{const:concEtaEmpLT}:=\Cr{const:concEtaEmpLT}(V,q)\in(0,\infty)$ such that

$\mathbb{P}\Big{(}\sup_{v\in
V}\sup_{x\in[n,n+1)}|\eta_{x}^{\zeta}(v)-\overline{\eta}(v)|\geq\Cr{const:concEtaEmpLT}\sqrt{\frac{\ln
n}{n}}\Big{)}\leq\Cr{const:concEtaEmpLT}n^{-q}\qquad\text{for all
}n\in\mathbb{N}.$ (2.14)

###### Proof.

We recall that due to Lemma A.1, the tilting parameter $\eta_{x}^{\zeta}(v)$
can alternatively be characterized as the unique solution
$\eta_{x}^{\zeta}(v)\in(-\infty,0)$ to

$\big{(}\overline{L}^{\zeta}_{x}\big{)}^{\prime}(\eta_{x}^{\zeta}(v))=\frac{1}{v},$
(2.15)

if the solution exists, and $\eta_{x}^{\zeta}(v)=0$ otherwise. We start with
noting that Part (a) directly follows from Part (b). Indeed, let $A_{n}$,
$n\in\mathbb{N}$, be the event in the probability on the left-hand side of
(2.14). Then $\sum_{n}\mathbb{P}(A_{n})<\infty$ for $q\geq 2$. By the first
Borel-Cantelli lemma, $\mathbb{P}$-a.s. only finitely many of the $A_{n}$
occur. In combination with the fact that $\overline{\eta}(v)<0$, cf. (2.10),
this implies that $\mathbb{P}$-a.s., the value of $\eta_{x}^{\zeta}(v)$ can
only vanish for $x>0$ small enough. In particular, we deduce the existence of
a $\mathbb{P}$-a.s. finite random variable $\mathcal{N}$ as postulated.
Hence, it remains to show (2.14). For this purpose, in the following lemma we
investigate the fluctuations of the functions through which the parameters
$\eta_{x}^{\zeta}(v)$ and $\overline{\eta}(v)$ are implicitly defined; we will
then infer the desired bounds on the fluctuations of the parameters themselves
through perturbation estimates for these functions.

######  Lemma 2.6.

For every compact interval $\triangle\subset(-\infty,0)$ and each
$q\in\mathbb{N},$ there exists a constant
$\Cl{const:concEmpLT}=\Cr{const:concEmpLT}(\triangle,q)\in(0,\infty)$ such
that

$\mathbb{P}\Big{(}\sup_{\eta\in\triangle}\sup_{x\in[n,n+1)}\Big{|}\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta)-L^{\prime}(\eta)\Big{|}\geq\Cr{const:concEmpLT}\sqrt{\frac{\ln
n}{n}}\Big{)}\leq\Cr{const:concEmpLT}n^{-q}\qquad\text{for all
}n\in\mathbb{N}.$ (2.16)

In order not to hinder the flow of reading, we postpone the proof of this
auxiliary result to the end of the proof of Lemma 2.5 and finish the proof of
Lemma 2.5 (b) first. Let $q\in\mathbb{N}$ and $V\subset(v_{c},\infty)$ be a
compact interval. By Lemma A.1, for each compact $\triangle\subset(-\infty,0)$
we have $\mathbb{P}$-a.s.,

$\displaystyle\begin{split}\Cr{constDerLMG}^{-1}\leq\inf_{\eta\in\triangle}L^{\prime\prime}(\eta)&\leq\sup_{\eta\in\triangle}L^{\prime\prime}(\eta)\leq\Cr{constDerLMG},\\\
\Cr{constDerLMG}^{-1}\leq\inf_{x\geq
1}\,\inf_{\eta\in\triangle}\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime\prime}(\eta)&\leq\sup_{x\geq
1}\,\sup_{\eta\in\triangle}\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime\prime}(\eta)\leq\Cr{constDerLMG}.\end{split}$
(2.17)

Therefore, and because the function $V\ni v\mapsto\overline{\eta}(v)$ is
strictly decreasing by Lemma 2.4, it is possible to find $N=N(V)\in\mathbb{N}$
and a compact interval $\triangle=\triangle(N,V)\subset(-\infty,0),$ where for
notational convenience we write

$V=[v_{*},v^{*}]\quad\text{ and }\quad\triangle=[\eta_{*},\eta^{*}],$ (2.18)

such that, using standard calculus for sets,

$\big{(}\overline{\eta}(V)-\Cr{const:concEmpLT}\Cr{constDerLMG}\sqrt{\ln
n/n}\big{)}\cup\big{(}\overline{\eta}(V)+\Cr{const:concEmpLT}\Cr{constDerLMG}\sqrt{\ln
n/n}\big{)}\subset\triangle\quad\text{ for all }n\geq N.$

Let $n\geq N$ and assume that the complement of the event on the left-hand
side of (2.16),

$\sup_{\eta\in\triangle}\,\sup_{x\in[n,n+1)}\big{|}\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta)-L^{\prime}(\eta)\big{|}<\Cr{const:concEmpLT}\sqrt{\frac{\ln
n}{n}},$ (2.19)

occurs. On this event, for all $v\in V$ and all $x\in[n,n+1),$

$\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}\big{(}\overline{\eta}(v)-\Cr{const:concEmpLT}\Cr{constDerLMG}\sqrt{\ln
n/n}\big{)}\leq\frac{1}{v}\leq\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}\big{(}\overline{\eta}(v)+\Cr{const:concEmpLT}\Cr{constDerLMG}\sqrt{\ln
n/n}\big{)}$ (2.20)

and thus, due to the strict monotonicity of
$(\overline{L}_{x}^{\zeta})^{\prime}$ implied by (2.17), there exists a unique
$\eta_{x}^{\zeta}(v)\in\triangle$ such that
$\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta_{x}^{\zeta}(v))=1/v$.
Due to (2.20), still assuming (2.19), we have

$\sup_{v\in
V}\,\sup_{x\in[n,n+1)}|\overline{\eta}(v)-\eta_{x}^{\zeta}(v)|\leq\Cr{const:concEmpLT}\Cr{constDerLMG}\sqrt{\frac{\ln
n}{n}}.$

Thus, for $n\geq N$, choosing
$\Cr{const:concEtaEmpLT}>2\Cr{const:concEmpLT}\Cr{constDerLMG}$, the
probability in (2.14) is bounded by the right-hand side of (2.16), which
finishes the proof. ∎

It remains to prove Lemma 2.6.

###### Proof of Lemma 2.6.

Applying the strong Markov property, we get

$x\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta)=E_{x}^{\zeta,\eta}(H_{0})=E_{x}^{\zeta,\eta}\big{[}H_{\lfloor
x\rfloor}\big{]}+E_{\lfloor
x\rfloor}^{\zeta,\eta}\big{[}H_{0}\big{]}=E_{x}^{\zeta,\eta}\big{[}H_{\lfloor
x\rfloor}\big{]}+\lfloor x\rfloor\big{(}\overline{L}_{\lfloor
x\rfloor}^{\zeta}\big{)}^{\prime}(\eta).$

Furthermore, $0\leq E_{x}^{\zeta,\eta}\big{[}H_{\lfloor
x\rfloor}\big{]}=\big{(}L_{x}^{\zeta}\big{)}^{\prime}(\eta)\leq\Cr{constDerLMG}$
by (A.1) and Lemma A.1 b), and thus also
$0\leq\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta)=\frac{1}{x}\sum_{i=1}^{x}\big{(}L_{i}^{\zeta}\big{)}^{\prime}(\eta)\leq\Cr{constDerLMG}$
for all $x\geq 1$ and all $\eta\in\triangle$, $\mathbb{P}$-a.s. As a
consequence, we get that for all $x\geq 1,$

$\big{|}\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta)-L^{\prime}(\eta)\big{|}\leq\big{|}\big{(}\overline{L}_{\lfloor
x\rfloor}^{\zeta}\big{)}^{\prime}(\eta)-L^{\prime}(\eta)\big{|}+\frac{2\Cr{constDerLMG}}{\lfloor
x\rfloor}.$

It is therefore enough to prove

$\mathbb{P}\Big{(}\sup_{\eta\in\triangle}\Big{|}\big{(}\overline{L}_{n}^{\zeta}\big{)}^{\prime}(\eta)-L^{\prime}(\eta)\Big{|}\geq\Cr{const:concEmpLT}\sqrt{\frac{\ln
n}{n}}\Big{)}\leq\Cr{const:concEmpLT}n^{-q}\qquad\text{for all
}n\in\mathbb{N}.$ (2.21)

For each $\eta\in\triangle$,
$((L_{i}^{\zeta})^{\prime}(\eta)-L^{\prime}(\eta))_{i\in\mathbb{Z}}$ is a
family of stationary, centered and bounded random variables. Furthermore, they
fulfill the exponential mixing condition (A.11) due to Lemma A.2. Since
$\sigma((L_{i}^{\zeta})^{\prime}(\eta):i\geq k)\subset\sigma(\xi(x):x\geq
k-1)$ and (• ‣ 1.4), setting
$Y_{i}:=(L_{i}^{\zeta})^{\prime}(\eta)-L^{\prime}(\eta)$, the left-hand side
in (A.15) is bounded by some constant $c_{2}>0$, uniformly for every $i$. Then
setting $m_{i}:=c_{1}$, condition (A.15) is fulfilled and we can apply the
Hoeffding-type inequality from Corollary A.5 to get $c_{2}>0$ such that

$\mathbb{P}\Big{(}\Big{|}\big{(}\overline{L}_{n}^{\zeta}\big{)}^{\prime}(\eta)-L^{\prime}(\eta)\Big{|}\geq
c_{2}\sqrt{\frac{\ln n}{n}}\Big{)}\leq c_{2}n^{-q-1}\qquad\text{for all
}\eta\in\triangle\text{ and all }n\in\mathbb{N}.$

Let
$\triangle_{n}:=(\triangle\cap\frac{1}{n}\mathbb{Z})\cup\\{\eta_{*},\eta^{*}\\}$,
recalling the notation of (2.18). Because
$|\triangle\cap\frac{1}{n}\mathbb{Z}|\leq n\cdot\text{diam}(\triangle)+1$,
taking advantage of the previous display we infer

$\displaystyle\mathbb{P}\Big{(}\sup_{\eta\in\triangle_{n}}$
$\displaystyle\Big{|}\big{(}\overline{L}_{n}^{\zeta}\big{)}^{\prime}(\eta)-L^{\prime}(\eta)\Big{|}\geq
c_{2}\sqrt{\frac{\ln n}{n}}\Big{)}$
$\displaystyle\leq|\triangle_{n}|\cdot\sup_{\eta\in\triangle_{n}}\mathbb{P}\Big{(}\Big{|}\big{(}\overline{L}_{n}^{\zeta}\big{)}^{\prime}(\eta)-L^{\prime}(\eta)\Big{|}\geq
c_{2}\sqrt{\frac{\ln n}{n}}\Big{)}\leq c_{3}(\triangle)n^{-q},\qquad\text{for
all }n\in\mathbb{N}.$

By Lemma A.1 b), we have $\mathbb{P}$-a.s.
$\sup_{n}\sup_{\eta\in\triangle}\big{|}\big{(}\overline{L}_{n}^{\zeta}(\eta)\big{)}^{\prime\prime}\big{|}\vee\big{|}L^{\prime\prime}(\eta)\big{|}\leq\Cr{constDerLMG}$.
Thus, the mean value theorem entails that

$\displaystyle\sup_{\eta\in\triangle}\big{|}\big{(}\overline{L}_{n}^{\zeta}\big{)}^{\prime}(\eta)-L^{\prime}(\eta)\big{|}$
$\displaystyle\leq\sup_{\eta\in\triangle_{n}}\big{|}\big{(}\overline{L}_{n}^{\zeta}\big{)}^{\prime}(\eta)-L^{\prime}(\eta)\big{|}+\frac{2\Cr{constDerLMG}}{n},$

and thus we find $\Cr{const:concEmpLT}>0$ such that (2.21) and hence (2.16)
hold true. ∎

In what comes below, many results will implicitly depend on the choice of
compact intervals $V$ and $\triangle,$ which have already occurred before.
Thus, in order to avoid ambiguity and due to assumption (VEL), we will now

$\begin{gathered}\text{fix arbitrary compact intervals
$V\subset(v_{c},\infty)$ and $\triangle=\triangle(V)\subset(-\infty,0)$ such
that }\\\ \text{$v_{0}\in\text{int}(V)$ and
$\overline{\eta}(V)\subset\text{int}(\triangle)$}.\end{gathered}$ (2.22)

Furthermore, due to Lemma 2.5, there exists a $\mathbb{P}$-a.s. finite random
variable
$\Cl[index_{p}rob]{constHn}=\Cr{constHn}(\xi,\Cr{const:concEtaEmpLT}(V,2))$
such that

$\mathcal{H}_{x}:=\mathcal{H}_{x}(V):=\big{\\{}\eta_{x}^{\zeta}(v)\in\triangle\text{
for all }v\in V\big{\\}}\quad\text{ occurs for all }x\geq\Cr{constHn}.$ (2.23)

We write

$\big{(}\overline{L}_{x}^{\zeta}\big{)}^{*}\Big{(}\frac{1}{v}\Big{)}=\sup_{\eta<0}\Big{(}\frac{\eta}{v}-\overline{L}^{\zeta}_{x}(\eta)\Big{)}=\frac{\eta_{x}^{\zeta}(v)}{v}-\overline{L}^{\zeta}_{x}(\eta_{x}^{\zeta}(v)),\quad
x\geq 1,$

for Legendre transformation of the weighted averages. We also recall to set
$\eta_{x}^{\zeta}(v)=0$, if there is no solution
$\eta_{x}^{\zeta}(v)\in\triangle$ to (2.15); note that this can only happen on
$\mathcal{H}_{x}^{c}$.

In order to show an invariance principle for the Legendre transformation
$(\overline{L}_{x}^{\zeta})^{*}$ in the following section, we now derive a
perturbation result on the tilting parameter $\eta_{x}^{\zeta}(v)$ in $x.$

######  Lemma 2.7.

There exists a constant $\Cl{const_{e}ta_{p}ert}>0$ such that
$\mathbb{P}$-a.s., for all $x\in(0,\infty)$ large enough, uniformly in $v\in
V$ and $0\leq h\leq x$,

$\big{|}\eta_{x}^{\zeta}(v)-\eta_{x+h}^{\zeta}(v)|\leq\Cr{const_{e}ta_{p}ert}\frac{h}{x}.$
(2.24)

###### Proof.

By Lemma 2.5 we can choose $x$ large enough such that
$\eta_{y}^{\zeta}(v)\in\triangle$ for all $y\geq x$ and all $v\in V$. For
$h=0$, the statement is obvious. For $0<h\leq x$, it suffices to show that
there exists $c_{1}>0$ such that

$\sup_{\eta\in\triangle}\big{|}\big{(}\overline{L}_{x+h}^{\zeta}\big{)}^{\prime}(\eta)-\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta)\big{|}\leq
c_{1}\frac{h}{x}.$ (2.25)

Indeed, using (2.15) we can write

$\displaystyle\big{(}\overline{L}_{x+h}^{\zeta}\big{)}^{\prime}(\eta_{x+h}^{\zeta}(v))-\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta_{x+h}^{\zeta}(v))$
$\displaystyle=\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta_{x}^{\zeta}(v))-\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta_{x+h}^{\zeta}(v))$
$\displaystyle=\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime\prime}(\widetilde{\eta})(\eta_{x}^{\zeta}(v)-\eta_{x+h}^{\zeta}(v))$

for some $\widetilde{\eta}\in\triangle$ between $\eta_{x}^{\zeta}(v)$ and
$\eta_{x+h}^{\zeta}(v)$. By the second display in (2.17) we know that
$\mathbb{P}$-a.s. $\inf_{\eta\in\triangle,x\geq
1}(\overline{L}_{x}^{\zeta}\big{)}^{\prime\prime}(\eta)\geq\Cr{constDerLMG}^{-1}$.
Using this, inequality (2.24) is a direct consequence of (2.25) with
$\Cr{const_{e}ta_{p}ert}:=c_{1}\Cr{constDerLMG}$. To prove (2.25), recall that
for all $\eta\in\triangle$, $x\geq 1,$ and $0<h\leq x,$ by the strong Markov
property applied at time $H_{x},$

$\displaystyle\big{(}\overline{L}_{x+h}^{\zeta}\big{)}^{\prime}(\eta)-\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta)$
$\displaystyle=\frac{1}{x+h}\big{(}E_{x+h}^{\zeta,\eta}\big{[}H_{x}\big{]}+E_{x}^{\zeta,\eta}[H_{0}]\big{)}-\frac{1}{x}E_{x}^{\zeta,\eta}[H_{0}]$
$\displaystyle=-\frac{h}{x+h}\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta)+\frac{h}{x+h}\frac{1}{h}E_{x+h}^{\zeta,\eta}[H_{x}].$

Finally, recall that by Lemma A.1 there exists
$\Cr{constDerLMG}=\Cr{constDerLMG}(\triangle)>0$ such that $\mathbb{P}$-a.s.
we have
$\sup_{\eta\in\triangle,x>0}\big{|}\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime}(\eta)\big{|}\leq\Cr{constDerLMG}$.
By exactly the same argument used for the proof of the latter inequality (see
proof of (A.1)), one can show that also
$\sup_{\eta\in\triangle,x,h>0}\big{|}\frac{1}{h}E_{x+h}^{\zeta,\eta}[H_{x}]\big{|}\leq\Cr{constDerLMG}$
holds $\mathbb{P}$-a.s. with the same constant $\Cr{constDerLMG}$. (2.25) now
follows choosing $c_{1}:=2\Cr{constDerLMG}$. ∎

##  3 Large deviations and perturbation results for the PAM

The main objective of this section is to establish certain exact large
deviation results for hitting times of Brownian motion under the tilted
measures introduced above, and then to apply these in order to obtain
perturbation results. For this purpose let

$\displaystyle\begin{split}V_{x}^{\zeta,v}(\eta)&:=\frac{\eta}{v}-L_{x}^{\zeta}(\eta),\quad
x\in\mathbb{R},\\\
\sigma^{2}_{v}&:=\text{Var}_{\mathbb{P}}(V_{1}^{\zeta,v}(\overline{\eta}(v))+2\sum_{i=2}^{\infty}\text{Cov}_{\mathbb{P}}\big{(}V_{1}^{\zeta,v}(\overline{\eta}(v)),V_{i}^{\zeta,v}(\overline{\eta}(v)\big{)},\quad\sigma_{v}:=\sqrt{\sigma^{2}_{v}},\quad
v\in V.\end{split}$ (3.1)

We start with observing that $\sigma_{v}^{2}\in[0,\infty)$ for all $v\in V$.
Indeed, $(\widetilde{L}_{i})_{i\in\mathbb{N}}$, where
$\widetilde{L}_{i}:=L_{i}^{\zeta}(\overline{\eta}(v))-\mathbb{E}[L_{i}^{\zeta}(\overline{\eta}(v))]$,
is a sequence of bounded (see Lemma A.1), centered and mixing (see Lemma A.2)
random variables, giving

$\displaystyle\sum_{i=1}^{\infty}\big{|}\text{Cov}_{\mathbb{P}}\big{(}V_{1}^{\zeta,v}(\overline{\eta}(v)),V_{i}^{\zeta,v}(\overline{\eta}(v))\big{)}\big{|}$
$\displaystyle=\sum_{i=1}^{\infty}\big{|}\mathbb{E}\big{[}\widetilde{L}_{1}\widetilde{L}_{i}\big{]}\big{|}=\sum_{i=1}^{\infty}\big{|}\mathbb{E}\big{[}\widetilde{L}_{i}\mathbb{E}[\widetilde{L}_{1}\,|\,\mathcal{F}^{i-1}]\big{]}\big{|}<\infty,$
(3.2)

where the last inequality is due to uniform boundedness of $\widetilde{L}_{i}$
in $i$, (A.9) and the summability criterion in (• ‣ 1.4). Thus
$\sigma_{v}^{2}<\infty$. Furthermore, $\sigma_{v}^{2}\geq 0$ is due to (3.2)
and [33, Lemma 1.1].

We now introduce the process $W^{v}_{x}(t)$ of empirical Legendre
transformations

$W^{v}_{x}(t):={t\sqrt{x}}\Big{(}(\overline{L}_{xt}^{\zeta})^{*}(1/v)-L^{*}(1/v)\Big{)},\quad
t,x>0,\ v\in V,$ (3.3)

and set $W_{0}^{v}(t)=W_{x}^{v}(0)=0$ for $t,x>0$, $v\in V$, and obtain a
first functional Central limit theorem for it.

######  Proposition 3.1.

For every $v\in V$, $W_{n}^{v}(1)$ converges in $\mathbb{P}$-distribution to a
centered Gaussian random variable with variance $\sigma_{v}^{2}\geq 0$. If
$\sigma_{v}^{2}>0$, the sequence of processes

$[0,\infty)\ni t\mapsto\frac{1}{\sigma_{v}}W_{n}^{v}(t),\quad n\in\mathbb{N},$

converges in $\mathbb{P}$-distribution to a standard Brownian motion in the
sense of weak convergence of measures on $C([0,\infty))$ with topology induced
by the metric $\rho$ from (1.12).

###### Proof.

It is sufficient to show the claim if $(W^{v}_{n}(t))_{t\in[0,\infty)}$ is
replaced by
$(W_{n}^{v}(t)\cdot\mathds{1}_{\mathcal{H}_{nt}})_{t\in[0,\infty)},$
$n\in\mathbb{N},$ with $\mathcal{H}_{nt}$ as defined in (2.23), since the
$\mathbb{P}$-probability of $\mathcal{H}_{nt}$ tends to $1$ for $n\to\infty$
by Lemma 2.5. In the notation of (3.1), setting

$S_{x}^{\zeta,v}(\eta):=\sum_{i=1}^{x}V_{i}^{\zeta,v}(\eta),\ x\in\mathbb{R},$
(3.4)

on $\mathcal{H}_{nt}$ we have

$\displaystyle\big{(}\overline{L}_{nt}^{\zeta}\big{)}^{*}\Big{(}\frac{1}{v}\Big{)}$
$\displaystyle=\frac{\eta_{nt}^{\zeta}(v)}{v}-\overline{L}^{\zeta}_{nt}(\eta_{nt}^{\zeta}(v))=\frac{1}{{nt}}\sum_{i=1}^{nt}V_{i}^{\zeta,v}(\eta_{nt}^{\zeta}(v))=\frac{1}{{nt}}S_{nt}^{\zeta,v}(\eta_{nt}^{\zeta}(v)).$

Thus, we can rewrite the relevant term as a sum of three differences

$\displaystyle\begin{split}{nt}\Big{(}(\overline{L}_{nt}^{\zeta})^{*}(1/v)-L^{*}(1/v)\Big{)}&=\Big{(}S_{nt}^{\zeta,v}(\eta_{nt}^{\zeta}(v))-S_{nt}^{\zeta,v}(\overline{\eta}(v))\Big{)}+\Big{(}S_{nt}^{\zeta,v}(\overline{\eta}(v))-\mathbb{E}\big{[}S_{nt}^{\zeta,v}(\overline{\eta}(v))\big{]}\Big{)}\\\
&\quad+\Big{(}\mathbb{E}\big{[}S_{nt}^{\zeta,v}(\overline{\eta}(v))\big{]}-{nt}L^{*}(1/v)\Big{)},\end{split}$
(3.5)

where we note that the third summand vanishes. Indeed, we have

$\displaystyle\mathbb{E}\big{[}S_{nt}^{\zeta,v}(\overline{\eta}(v))\big{]}$
$\displaystyle=nt\Big{(}\frac{\overline{\eta}(v)}{v}-\mathbb{E}[L_{1}^{\zeta}(\overline{\eta}(v))]\Big{)}=ntL^{*}(1/v),$

where the last equality is due to (2.11) and the definition of the Legendre
transform. The proof is completed by the use of Lemmas 3.3 and 3.2 below,
which show that the second summand of (3.5) exhibits the postulated diffusive
behavior whereas the latter summand is negligible in that scaling. ∎

######  Lemma 3.2.

For every $v\in V$,
$\frac{1}{\sqrt{n}}\Big{(}S_{nt}^{\zeta,v}(\overline{\eta}(v))-\mathbb{E}\big{[}S_{nt}^{\zeta,v}(\overline{\eta}(v))\big{]}\Big{)}$
converges in $\mathbb{P}$-distribution to a centered Gaussian random variable
with variance $\sigma_{v}^{2}\geq 0$. If $\sigma_{v}^{2}>0$, the sequence of
processes

$[0,\infty)\ni
t\mapsto\frac{1}{\sigma_{v}\sqrt{n}}\Big{(}S_{nt}^{\zeta,v}(\overline{\eta}(v))-\mathbb{E}\big{[}S_{nt}^{\zeta,v}(\overline{\eta}(v))\big{]}\Big{)},\quad
n\in\mathbb{N},$

converges in $\mathbb{P}$-distribution to a standard Brownian motion in the
sense of weak convergence of measures on $C([0,\infty))$ with topology induced
by the metric $\rho$ from (1.12).

###### Proof.

Let
$\widetilde{L}_{i}:=L_{i}^{\zeta}(\overline{\eta}(v))-\mathbb{E}[L_{i}^{\zeta}(\overline{\eta}(v))]$,
$\widetilde{V}_{i}:=V_{i}^{\zeta}(\overline{\eta}(v))-\mathbb{E}[V_{i}^{\zeta}(\overline{\eta}(v))]$
and $M\in\mathbb{N}$. Further set
$\widetilde{L}_{i}^{(M)}:=\sum_{j=1+(i-1)M}^{iM}\widetilde{L}_{j}$. Then
$(\widetilde{L}_{i}^{(M)})_{i\in\mathbb{Z}}$ is a sequence of centered,
stationary and (by Lemma A.1) bounded random variables. To show the central
limit theorem on $C([0,M])$, we will use the method of martingale
approximation from [17], which is summarized as a theorem by Nolen in [28,
section 2.3] and turns out to be applicable in our situation. That is, we have
to make sure that condition [28, (2.36)] is fulfilled. Indeed, replacing in
(A.13) $\mathcal{F}^{j}$ by $\mathcal{F}_{k}$ and noting that quantity $A$ in
(A.13) is $\mathcal{F}_{k}$-measurable we get

$\displaystyle\sum_{k=1}^{\infty}|\widetilde{L}_{0}^{(M)}-\mathbb{E}[\widetilde{L}_{0}^{(M)}\,|\,\mathcal{F}_{k}]|$
$\displaystyle\leq c_{1}\sum_{k=1}^{\infty}e^{-k/c_{1}}<\infty,$

giving the convergence of the first series in [28, (2.36)]. Furthermore, using
that $\widetilde{L}_{k}^{(M)}$ is $\mathcal{F}^{(k-1)M}$-measurable and
bounded, also recalling (• ‣ 1.4), we get

$\displaystyle\sum_{k=1}^{\infty}|\mathbb{E}[\widetilde{L}_{k}^{(M)}\,|\,\mathcal{F}_{0}]|\leq\sum_{k=1}^{\infty}\psi(k-1)\mathbb{E}[|\widetilde{L}_{0}^{(M)}|]<\infty.$

Because the series in (3.1) is absolutely convergent, by [33, Lemma 1.1] and
[28, (2.37)] we have
$\lim_{n\to\infty}\frac{1}{n}\mathbb{E}\big{[}\big{(}\sum_{k=1}^{n}\widetilde{L}_{k}^{(M)}\big{)}^{2}\big{]}=M\cdot\lim_{n\to\infty}\frac{1}{Mn}\mathbb{E}\big{[}\big{(}\sum_{k=1}^{Mn}\widetilde{L}_{k}\big{)}^{2}\big{]}=M\cdot\sigma_{v}^{2}\in[0,\infty)$.
Furthermore, if $\sigma_{v}^{2}>0$, [28, Theorem 2.1] entail that the sequence
of processes

$[0,1]\ni t\mapsto
X_{n}^{(M)}(t):=\frac{1}{\sigma_{v}\sqrt{nM}}\big{(}\sum_{k=1}^{\lfloor
nt\rfloor}\widetilde{L}_{k}^{(M)}+(nt-\lfloor nt\rfloor)\widetilde{L}_{\lfloor
nt\rfloor+1}^{(M)}\big{)},\quad n\in\mathbb{N},$

converge in $\mathbb{P}$-distribution to a standard Brownian motion
$(B_{t})_{t\in[0,1]}$ in the sense of weak convergence of measures on
$C([0,1])$ with the topology induced by the uniform metric. Then by
definition, above convergence also holds true for $(\widetilde{V}_{i})_{i\geq
1}$ instead of $(\widetilde{L}_{i})_{i\geq 1}$. Furthermore, we have the
uniform bound

$\sup_{t\in[0,M],n\in\mathbb{N}}\Big{|}S_{nt}^{\zeta,v}(\overline{\eta}(v))-\big{(}\sum_{i=1}^{\lfloor
nt/M\rfloor M}V_{i}^{\zeta,v}+(nt-\lfloor nt/M\rfloor M)\sum_{i=1+\lfloor
nt/M\rfloor M}^{M+\lfloor nt/M\rfloor M}V^{\zeta,v}_{i}\big{)}\Big{|}\leq
c_{2}\quad\mathbb{P}\text{-a.s.}$

Consequently, the sequence $[0,M]\ni
t\mapsto\frac{1}{\sigma_{v}\sqrt{n}}\Big{(}S_{nt}^{\zeta,v}(\overline{\eta}(v))-\mathbb{E}\big{[}S_{nt}^{\zeta,v}(\overline{\eta}(v))\big{]}\Big{)}$
has the same weak limit as $\big{(}\sqrt{M}\cdot
X_{n}^{(M)}(t/M)\big{)}_{t\in[0,M]}$, $n\in\mathbb{N}$, which converges to
$\big{(}\sqrt{M}\cdot B(t/M)\big{)}_{t\in[0,M]}$ and the latter process is a
standard Brownian motion on $[0,M]$. Because $M\in\mathbb{N}$ was arbitrary,
[37, first Theorem 2.4 on page 15] gives weak convergence on $C([0,\infty))$.
∎

To show that the first summand in (3.5) is asymptotically negligible, we use
the following result.

######  Lemma 3.3.

There exists a constant $\Cl{const_{c}oncSn}\in(0,\infty)$ such that for every
$v\in V$ and $M>0$,

$\limsup_{n\to\infty}\frac{1}{\ln n}\sup_{0\leq t\leq
M}\Big{|}S_{nt}^{\zeta,v}(\eta_{nt}^{\zeta}(v))-S_{nt}^{\zeta,v}(\overline{\eta}(v))\Big{|}\leq\Cr{const_{c}oncSn}\quad\mathbb{P}\text{-a.s.}$

###### Proof.

There exists a $\mathbb{P}$-a.s. finite time $\Cr{constHn}(\omega)$, defined
before (2.23), such that for all $x\geq\Cr{constHn}$ and all $v\in V$ we have
$\eta_{x}^{\zeta}(v)\in\triangle$. Furthermore, by Lemma A.1,
$S_{x}^{\zeta,v}$ is infinitely differentiable on $(-\infty,0)$, so for all
$x\geq\Cr{constHn}$ there exists
$\widetilde{\eta}_{x}^{\zeta}(v)\in[\overline{\eta}(v)\wedge\eta_{x}^{\zeta}(v),\overline{\eta}(v)\vee\eta_{x}^{\zeta}(v)]$
such that

$\displaystyle S_{x}^{\zeta,v}(\overline{\eta}(v))$
$\displaystyle=S_{x}^{\zeta,v}(\eta_{x}^{\zeta}(v))+\big{(}S_{x}^{\zeta,v}\big{)}^{\prime}(\eta_{x}^{\zeta}(v))\big{(}\overline{\eta}(v)-\eta_{x}^{\zeta}(v)\big{)}+\frac{\big{(}S_{x}^{\zeta,v}\big{)}^{\prime\prime}(\widetilde{\eta}_{x}^{\zeta}(v))}{2}\big{(}\overline{\eta}(v)-\eta_{x}^{\zeta}(v)\big{)}^{2}.$

Due to (2.15), $(S_{x}^{\zeta,v})^{\prime}(\eta_{x}^{\zeta}(v))=0$ and by
Lemma A.1 we have $\sup_{\eta\in\triangle}\sup_{x\geq
1}\big{|}(S_{x}^{\zeta,v})^{\prime\prime}(\eta)\big{|}/x\leq c_{1}$. By (2.14)
and the first Borel-Cantelli lemma, there exists a finite random variable
$\mathcal{N}_{2}\geq\Cr{constHn}$ such that for $x\geq\mathcal{N}_{2}$ the
complementary event on the left-hand side of (2.14) occurs, hence

$\sup_{x\geq\Cr{constHn}}\big{(}\eta_{x}^{\zeta}(v)-\overline{\eta}(v)\big{)}^{2}\frac{x}{\ln
x}\leq\Cr{const:concEtaEmpLT}^{2}(V,2)$

and thus

$\displaystyle\sup_{x\geq\Cr{constHn}}\frac{1}{\ln
x}\big{|}S_{x}^{\zeta,v}(\eta_{x}^{\zeta}(v))-S_{x}^{\zeta,v}(\overline{\eta}(v))\big{|}$
$\displaystyle\leq\Cr{const_{c}oncSn}$ (3.6)

with $\Cr{const_{c}oncSn}:=\frac{c_{1}\Cr{const:concEtaEmpLT}^{2}(V,2)}{2}$.
Finally, we have

$\displaystyle\sup_{0\leq t\leq
M}\Big{|}S_{nt}^{\zeta,v}(\eta_{nt}^{\zeta}(v))-S_{nt}^{\zeta,v}(\overline{\eta}(v))\Big{|}$
$\displaystyle\leq\sup_{0\leq
x\leq\Cr{constHn}}\Big{|}S_{x}^{\zeta,v}(\eta_{x}^{\zeta}(v))-S_{x}^{\zeta,v}(\overline{\eta}(v))\Big{|}$
$\displaystyle\qquad+\sup_{\Cr{constHn}/n\leq t\leq
M}\Big{|}S_{nt}^{\zeta,v}(\eta_{nt}^{\zeta}(v))-S_{nt}^{\zeta,v}(\overline{\eta}(v))\Big{|}$
$\displaystyle\leq 2\Cr{constHn}c_{2}+\Cr{const_{c}oncSn}\ln
M+\Cr{const_{c}oncSn}\ln n,$

where in the last inequality we used that $\mathbb{P}$-a.s., every summand in
the definition of $S_{n}^{\zeta,\eta}$ is uniformly bounded by $c_{2}$. The
$\mathbb{P}$-a.s. finiteness of $\Cr{constHn}$ gives the claim. ∎

As a by-product of the proof above we get an approximation result of
$W_{x}^{v}$ be a centered stationary sequence.

######  Corollary 3.4.

For every $v\in V$ and all $t$ such that $vt\geq\Cr{constHn}$ we have

$\Big{|}\sqrt{vt}W_{vt}^{v}(1)-\sum_{i=1}^{vt}\big{(}L(\overline{\eta}(v))-L_{i}^{\zeta}(\overline{\eta}(v))\big{)}\Big{|}\leq\Cr{const_{c}oncSn}\ln
v+\Cr{const_{c}oncSn}\ln t.$

###### Proof.

By the definition of $W^{v}_{x}(t)$ and $S_{x}^{\zeta,v}(\eta)$ from (3.3) and
(3.4), as well as the definition in (2.12) for the corresponding Legendre
transformations, we have

$\displaystyle\sqrt{vt}W_{vt}^{v}(1)$
$\displaystyle=vt\Big{(}\big{(}\overline{L}_{vt}^{\zeta}\big{)}^{*}(1/v)-L^{*}(1/v)\Big{)}=vt\Big{(}\frac{\eta_{vt}^{\zeta}(v)}{v}-\overline{L}_{vt}\big{(}\eta_{vt}^{\zeta}(v)\big{)}-\frac{\overline{\eta}(v)}{v}+L(\overline{\eta}(v))\Big{)}$
$\displaystyle=S_{vt}^{\zeta,v}\big{(}\eta_{vt}^{\zeta}(v)\big{)}-\sum_{i=1}^{vt}\big{(}\frac{\overline{\eta}(v)}{v}-L_{i}^{\zeta}(\overline{\eta}(v))\big{)}+\sum_{i=1}^{vt}\big{(}L(\overline{\eta}(v))-L_{i}^{\zeta}(\overline{\eta}(v))\big{)}$
$\displaystyle=S_{vt}^{\zeta,v}\big{(}\eta_{vt}^{\zeta}(v)\big{)}-S_{vt}^{\zeta,v}\big{(}\overline{\eta}(v)\big{)}+\sum_{i=1}^{vt}\big{(}L(\overline{\eta}(v))-L_{i}^{\zeta}(\overline{\eta}(v))\big{)}.$

Then we can conclude using (3.6). ∎

###  3.1 An exact large deviation result for auxiliary processes

For $x\geq 0$ and $v>0$ we introduce

$\displaystyle Y_{v}^{\approx}(x)$
$\displaystyle:=E_{x}\Big{[}e^{\int_{0}^{H_{0}}\zeta(B_{s})\mathop{}\\!\mathrm{d}s};H_{0}\in\Big{[}\frac{x}{v}-K,\frac{x}{v}\Big{]}\Big{]},$
$\displaystyle Y_{v}^{>}(x)$
$\displaystyle:=E_{x}\Big{[}e^{\int_{0}^{H_{0}}\zeta(B_{s})\mathop{}\\!\mathrm{d}s};H_{0}<\frac{x}{v}-K\Big{]},\quad\text{and}$
$\displaystyle Y_{v}(x)$
$\displaystyle:=E_{x}\Big{[}e^{\int_{0}^{H_{0}}\zeta(B_{s})\mathop{}\\!\mathrm{d}s};H_{0}\leq\frac{x}{v}\Big{]}=Y_{v}^{\approx}(x)+Y_{v}^{>}(x),$

where $K>0$ is a constant, defined in (3.16) below. For $v\in V$ and $x\geq 1$
we define

$\displaystyle\sigma_{x}^{\zeta}(v):=\begin{cases}\big{|}\eta_{x}^{\zeta}(v)\big{|}\sqrt{\text{Var}_{x}^{\zeta,\eta_{x}^{\zeta}(v)}(H_{0})},&\text{
on }\mathcal{H}_{x},\\\
\sup\limits_{\eta\in\triangle}\left|\eta\right|\sqrt{\text{Var}_{x}^{\zeta,\sup_{\eta\in\triangle}}(H_{0})},&\text{
on }\mathcal{H}_{x}^{c}.\end{cases}$

Furthermore, by Lemma A.1, there exists some $C>1$ such that
$\text{Var}_{x}^{\zeta,\eta}(H_{0})=x\big{(}\overline{L}_{x}^{\zeta}\big{)}^{\prime\prime}(\eta)\in[x\Cr{constDerLMG}^{-1},x\Cr{constDerLMG}]$
for all $x\geq 1$. Thus, there is some constant
$\Cl{constSigman}=\Cr{constSigman}(\triangle)>1$ such that

$\Cr{constSigman}^{-1}\sqrt{x}\leq\sigma_{x}^{\zeta}(v)\leq\Cr{constSigman}\sqrt{x}\quad\text{
for all }v\in V,\ x\geq 1,\ \text{on }\mathcal{H}_{x}.$ (3.7)

We now prove the following result.

######  Proposition 3.5.

Let $V$ be as in (2.22), $\sigma_{v}$ defined by (3.1) and $W^{v}_{x}(t)$ as
in (3.3), $v\in V$, and $K>0$ be such that (3.16) holds. Then there exists a
constant $\Cl{constY_{v}}\in(1,\infty)$, such that for all $v\in V$ and all
$x\geq 1$, on $\mathcal{H}_{x}$ we have

$\sigma^{\zeta}_{x}(v)Y_{v}^{>}(x)\exp\left\\{xL^{*}(1/v)+\sqrt{x}W^{v}_{x}(1)\right\\}\in\left[\Cr{constY_{v}}^{-1},\Cr{constY_{v}}\right].$
(3.8)

Furthermore, for all $v\in V$ and all $x\geq 1$, on $\mathcal{H}_{x}$ we have

$\frac{Y_{v}^{\approx}(x)}{Y_{v}^{>}(x)}\in[\Cr{constY_{v}}^{-1},\Cr{constY_{v}}],$
(3.9)

and the sequence $n^{-1/2}\big{(}\ln Y^{\approx}_{v}(n)-nL^{*}(1/v)\big{)}$,
$n\in\mathbb{N}$, converges to a centered Gaussian random variable with
variance $\sigma_{v}^{2}\in[0,\infty)$, where $\sigma_{v}^{2}$ is defined in
(3.1) and if $\sigma_{v}^{2}>0$, the sequence of processes

$\displaystyle[0,\infty)\ni t$
$\displaystyle\mapsto\frac{1}{\sigma_{v}\sqrt{n}}\big{(}\ln
Y_{v}^{\approx}(nt)+ntL^{*}(1/v)\big{)},\quad n\in\mathbb{N},$ (3.10)

converges in $\mathbb{P}$-distribution to standard Brownian motion in
$C([0,\infty))$.

###### Proof.

We start with proving (3.8) and for this purpose let $x\geq 1$ such that
$\mathcal{H}_{\lfloor x\rfloor}$ occurs. Write $\eta:=\eta_{x}^{\zeta}(v)$ and
$\sigma:=\sigma_{x}^{\zeta}(v),$ and recall the notation introduced in (2.4),
i.e. $\tau_{i}=H_{i}-H_{i-1}$, $i=1,\ldots,\lceil x\rceil-1$, and set
$\tau_{x}:=H_{\lceil x\rceil-1}-H_{x},$ $x\in\mathbb{R}\backslash\mathbb{Z}$,
(which would be consistent with the definition in (2.4) for $x$ integer) to
define
$\widehat{\tau}_{i}:={\widehat{\tau}}^{(x)}_{i}:=\tau_{i}-E_{x}^{\zeta,\eta}\left[\tau_{i}\right]$.
Then
$\sum_{i=1}^{x}E_{x}^{\zeta,\eta}\left[\tau_{i}\right]=E_{x}^{\zeta,\eta}\left[H_{0}\right]=\frac{x}{v}$.
We now rewrite

$\displaystyle Y_{v}^{\approx}(x)$
$\displaystyle=E_{x}\Big{[}e^{\int_{0}^{H_{0}}(\zeta(B_{s})+\eta)\mathop{}\\!\mathrm{d}s}\exp\Big{\\{}-\eta\sum_{i=1}^{x}\widehat{\tau}_{i}\Big{\\}};\,\sum_{i=1}^{x}\widehat{\tau}_{i}\in\left[-K,0\right]\Big{]}\exp\left\\{-x\frac{\eta}{v}\right\\}$
$\displaystyle=E_{x}^{\zeta,\eta}\Big{[}\exp\Big{\\{}-\sigma\frac{\eta}{\sigma}\sum_{i=1}^{x}\widehat{\tau}_{i}\Big{\\}};\,\frac{\eta}{\sigma}\sum_{i=1}^{x}\widehat{\tau}_{i}\in\Big{[}0,-\frac{K\eta}{\sigma}\Big{]}\Big{]}\exp\left\\{-x\left(\frac{\eta}{v}-\overline{L}_{x}^{\zeta}(\eta)\right)\right\\}.$
(3.11)

Analogously, we get

$\displaystyle Y_{v}^{>}(x)$
$\displaystyle=E_{x}^{\zeta,\eta}\Big{[}\exp\Big{\\{}-\sigma\frac{\eta}{\sigma}\sum_{i=1}^{x}\widehat{\tau}_{i}\Big{\\}};\,\frac{\eta}{\sigma}\sum_{i=1}^{x}\widehat{\tau}_{i}>-\frac{K\eta}{\sigma}\Big{]}\exp\left\\{-x\left(\frac{\eta}{v}-\overline{L}_{x}^{\zeta}(\eta)\right)\right\\}.$

We define $\mu_{x}^{\zeta,v}$ as the distribution of
$\frac{\eta}{\sigma}\sum_{i=1}^{x}\widehat{\tau}_{i}$ under
$P_{x}^{\zeta,\eta}$. Then

$\displaystyle Y_{v}^{\approx}(x)$
$\displaystyle=e^{-x(\frac{\eta}{v}-\overline{L}_{x}^{\zeta}(\eta))}\int_{0}^{\frac{-K\eta}{\sigma}}e^{-\sigma
y}\,\mathop{}\\!\mathrm{d}\mu_{x}^{\zeta,v}(y)$ (3.12)

and

$\displaystyle Y_{v}^{>}(x)$
$\displaystyle=e^{-x(\frac{\eta}{v}-\overline{L}_{x}^{\zeta}(\eta))}\int_{\frac{-K\eta}{\sigma}}^{\infty}e^{-\sigma
y}\,\mathop{}\\!\mathrm{d}\mu_{x}^{\zeta,v}(y).$ (3.13)

Using Lemma 3.6 below, the integrals on the right-hand side of (3.12) and
(3.13), multiplied by $\sigma$, are bounded from below and above by positive
constants. Display (3.8) now follows by the definition of $W_{x}^{v},$ and
(3.9) is direct a consequence of (3.12)-(3.15). The last two statements, are a
consequence of (3.8), (3.9), $W_{nt}^{v}(1)=\frac{1}{\sqrt{t}}W_{n}^{v}(t)$
and Proposition 3.1. ∎

To complete the previous proof, it remains to prove the following.

######  Lemma 3.6.

Under the conditions of Proposition 3.5, there exists a constant
$\Cl{constCLTHitting}>1$ such that for all $v\in V$ and $x\geq 1$, on
$\mathcal{H}_{x},$

$\sigma_{x}^{\zeta}(v)\int_{0}^{{-K\eta_{x}^{\zeta}(v)}/{\sigma_{x}^{\zeta}(v)}}e^{-\sigma_{x}^{\zeta}(v)y}\mathop{}\\!\mathrm{d}\mu_{x}^{\zeta,v}(y)\in[\Cr{constCLTHitting}^{-1},\Cr{constCLTHitting}]$
(3.14)

and

$\sigma_{x}^{\zeta}(v)\int_{{-K\eta_{x}^{\zeta}(v)}/{\sigma_{x}^{\zeta}(v)}}^{\infty}e^{-\sigma_{x}^{\zeta}(v)y}\mathop{}\\!\mathrm{d}\mu_{x}^{\zeta,v}(y)\in[\Cr{constCLTHitting}^{-1},\Cr{constCLTHitting}],$
(3.15)

with $\mu_{x}^{\zeta,v}$ as in the proof of Proposition 3.5.

###### Proof.

We write $n:=\lceil x\rceil$ and recall that under $P_{x}^{\zeta,\eta}$, the
sequence
$\Big{(}\sqrt{n}\frac{\eta_{x}^{\zeta}(v)}{\sigma_{x}^{\zeta}(v)}\widehat{\tau}_{i}\Big{)}_{i=1,\ldots,\lfloor
x\rfloor,x}$ is a sequence of independent, centered random variables. Thus, on
$\mathcal{H}_{x}$ we obtain

$\displaystyle\frac{1}{n}\sum_{i=1}^{x}\mathrm{Var}_{x}^{\zeta,\eta}\bigg{(}\sqrt{n}\frac{\eta_{x}^{\zeta}(v)}{\sigma_{x}^{\zeta}(v)}\widehat{\tau}_{i}\bigg{)}$
$\displaystyle=\bigg{(}\frac{\eta_{x}^{\zeta}(v)}{\sigma_{x}^{\zeta}(v)}\bigg{)}^{2}\text{Var}_{x}^{\zeta,\eta}\bigg{(}\sum_{i=1}^{x}\widehat{\tau}_{i}\bigg{)}=\bigg{(}\frac{\eta_{x}^{\zeta}(v)}{\sigma_{x}^{\zeta}(v)}\bigg{)}^{2}\text{Var}_{x}^{\zeta,\eta}(H_{0})=1.$

Additionally, the $\widehat{\tau}_{i}$’s have uniform exponential moments.
Thus, the conditions of [3, Theorem 13.3] are fulfilled and an application of
[3, (13.43)] yields

$\sup_{\mathcal{C}}\big{|}\mu_{x}^{\zeta,v}(\mathcal{C})-\Phi(\mathcal{C})\big{|}\leq
c_{1}{n}^{-1/2},$

where the supremum is taken over all Borel-measurable convex subsets of
$\mathbb{R}$, $\Phi$ denotes the standard Gaussian measure on $\mathbb{R}$ and
$c_{1}$ only depends on the uniform bound of the exponential moments of the
$\widehat{\tau}_{i}$’s. Without loss of generality, we will assume $c_{1}>4$.
Then, due to (3.7), by denoting
$\mathcal{C}:=\big{[}0,-K\eta_{x}^{\zeta}(v)/{\sigma_{x}^{\zeta}(v)}\big{]}$
we can choose $K>0$ large enough, so that

$\Phi(\mathcal{C})\geq 2c_{1}^{-1}n^{-1/2}\quad\text{for all
}n\in\mathbb{N}\text{ and }v\in V.$ (3.16)

We thus get

$\displaystyle c_{1}^{-1}n^{-1/2}$
$\displaystyle\leq\Phi\left(\mathcal{C}\right)-\big{|}\mu_{x}^{\zeta,v}(\mathcal{C})-\Phi(\mathcal{C})\big{|}\leq\mu_{x}^{\zeta,v}\left(\mathcal{C}\right)\leq\big{|}\mu_{x}^{\zeta,v}(\mathcal{C})-\Phi(\mathcal{C})\big{|}+\Phi\left(\mathcal{C}\right)\leq
c_{2}(K)\cdot n^{-1/2}.$ (3.17)

Because the integrand in (3.14) is bounded away from $0$ and infinity on the
respective interval of integration (uniformly in $n\in\mathbb{N}$), (3.14) is
a direct consequence of (3.7) and (3.17). For (3.15), we split the integral
into a sum:

$\displaystyle\int_{{-K\eta_{x}^{\zeta}(v)}/{\sigma_{x}^{\zeta}(v)}}^{\infty}e^{-\sigma_{x}^{\zeta}(v)y}\mathop{}\\!\mathrm{d}\mu_{x}^{\zeta,v}(y)=\sum_{k=1}^{\infty}\int_{-kK\eta_{x}^{\zeta}(v)/\sigma_{x}^{\zeta}(v)}^{-(k+1)K\eta_{x}^{\zeta}(v)/{\sigma_{x}^{\zeta}(v)}}e^{-\sigma_{x}^{\zeta}(v)y}\mathop{}\\!\mathrm{d}\mu_{x}^{\zeta,v}(y)$
$\displaystyle\leq\sum_{k=1}^{\infty}\mu_{x}^{\zeta,v}\left(-\frac{K\eta_{x}^{\zeta}(v)}{{\sigma_{x}^{\zeta}(v)}}\left[k,k+1\right]\right)e^{-k\cdot
K|\eta_{x}^{\zeta}(v)|}\leq c_{2}n^{-1/2}\sum_{k=1}^{\infty}e^{-k\cdot
K|\eta^{*}|}\leq\Cr{constCLTHitting}n^{-1/2},$

where we recall the notation from (2.18). The lower bound in (3.15) can be
obtained by noting that

$\displaystyle\int_{{-K\eta_{x}^{\zeta}(v)}/{\sigma_{x}^{\zeta}(v)}}^{\infty}e^{-\sigma_{x}^{\zeta}(v)y}\mathop{}\\!\mathrm{d}\mu_{x}^{\zeta,v}(y)$
$\displaystyle\geq\int_{{-K\eta_{x}^{\zeta}(v)}/{\sigma_{x}^{\zeta}(v)}}^{{-2K\eta_{x}^{\zeta}(v)}/{\sigma_{x}^{\zeta}(v)}}e^{-\sigma_{x}^{\zeta}(v)y}\mathop{}\\!\mathrm{d}\mu_{x}^{\zeta,v}(y)$
$\displaystyle\geq
e^{2K\eta_{x}^{\zeta}(v)}\mu_{x}^{\zeta,v}\big{(}[{-K\eta_{x}^{\zeta}(v)}/{\sigma_{x}^{\zeta}(v)},{-2K\eta_{x}^{\zeta}(v)}/{\sigma_{x}^{\zeta}(v)}]\big{)}.$

Analogously to (3.17), choosing $\Cr{constCLTHitting}$ large enough, the last
expression is bounded from below by $\Cr{constCLTHitting}^{-1}n^{-1/2}$.
Combining this with (3.7), we finally arrive at (3.15). ∎

We are now in the position to prove the following result.

######  Lemma 3.7.

Under the conditions of Proposition 3.5, for all $\delta>0$ there exists a
constant $\Cl{constY_{v}approx}=\Cr{constY_{v}approx}(\delta)\in(1,\infty)$
such that for all $v\in V$, $t>0$, on $\mathcal{H}_{vt}$ we have

$\displaystyle\Cr{constY_{v}approx}^{-1}Y_{v}^{\approx}(vt)$
$\displaystyle\leq
E_{vt}\Big{[}e^{\int_{0}^{t}\zeta(B_{s})\mathop{}\\!\mathrm{d}s};B_{t}\in[-\delta,0]\Big{]}\leq
E_{vt}\Big{[}e^{\int_{0}^{t}\zeta(B_{s})\mathop{}\\!\mathrm{d}s};B_{t}\leq
0\Big{]}\leq\Cr{constY_{v}approx}Y_{v}^{\approx}(vt).$ (3.18)

###### Proof.

The second inequality is obvious. Since $\\{B_{t}\leq 0\\}\subset\\{H_{0}\leq
t\\}$ and $\zeta\leq 0$, we get
$E_{vt}\big{[}e^{\int_{0}^{t}\zeta(B_{s})\mathop{}\\!\mathrm{d}s};B_{t}\leq
0\big{]}\leq Y_{v}(vt)\leq(1+\Cr{constY_{v}})Y_{v}^{\approx}(vt)$ by
Proposition 3.5 and thus the last inequality in (3.18) is obtained. Therefore,
it remains to show the first inequality. For this purpose, define the function
$p(s):=E_{0}\big{[}e^{\int_{0}^{s}\zeta(B_{r})\mathop{}\\!\mathrm{d}r};B_{s}\in[-\delta,0]\big{]}$
which is bounded from below by $c_{1}(K,\delta)>0$ for all $s\in[0,K]$. Using
the strong Markov property at $H_{0}$, we finally get

$\displaystyle Y_{v}^{\approx}(vt)$
$\displaystyle=E_{vt}\left[e^{\int_{0}^{H_{0}}\zeta(B_{r})\mathop{}\\!\mathrm{d}r};H_{0}\in\left[t-K,t\right]\right]$
$\displaystyle\leq
c_{1}(K,\delta)^{-1}E_{vt}\left[e^{\int_{0}^{H_{0}}\zeta(B_{r})\mathop{}\\!\mathrm{d}r}p(t-s)_{|s=H_{0}};H_{0}\in\left[t-K,t\right]\right]$
$\displaystyle\leq
c_{1}(K,\delta)^{-1}E_{vt}\left[e^{\int_{0}^{H_{0}}\zeta(B_{r})\mathop{}\\!\mathrm{d}r}p(t-s)_{|s=H_{0}}\right]=c_{1}(K,\delta)^{-1}E_{vt}\left[e^{\int_{0}^{t}\zeta(B_{r})\mathop{}\\!\mathrm{d}r};B_{t}\in[-\delta,0]\right].$

and the claim follows by choosing
$\Cr{constY_{v}approx}:=c_{1}(K,\delta)\vee(1+\Cr{constY_{v}})$. ∎

Plugging the relation $\xi(x)=\zeta(x)+\texttt{es}$, $x\in\mathbb{R}$, into
Lemma 3.7 immediately supplies us with the following corollary.

######  Corollary 3.8.

Let $\Cr{constY_{v}approx}$ be as in Lemma 3.7. Then for all $v\in V$, $t>0$,
on $\mathcal{H}_{vt}$ we have

$\displaystyle\Cr{constY_{v}approx}^{-1}e^{\emph{{es}}\cdot
t}Y_{v}^{\approx}(vt)$ $\displaystyle\leq
E_{vt}\left[e^{\int_{0}^{t}\xi(B_{s})\mathop{}\\!\mathrm{d}s};B_{t}\leq
0\right]\leq\Cr{constY_{v}approx}e^{\emph{{es}}\cdot t}Y_{v}^{\approx}(vt).$

Using the Feynman-Kac formula (1.10) we also get the following result. Recall
that $u^{u_{0}}$ denoted the solution to (PAM) with initial condition
$u_{0}\in\mathcal{I}_{\text{PAM}}$.

######  Corollary 3.9.

Let $\Cr{constY_{v}approx}=\Cr{constY_{v}approx}(\delta)$ be as in Lemma 3.7.
Then for all $v\in V$, $t>0$, on $\mathcal{H}_{vt}$ we have

$u^{\mathds{1}_{[-\delta,0]}}(t,vt)\leq
u^{\mathds{1}_{(-\infty,0]}}(t,vt)\leq\Cr{constY_{v}approx}^{2}\cdot
u^{\mathds{1}_{[-\delta,0]}}(t,vt).$

The previous results are fundamental for proving perturbation statements in
the next section, which themselves will allow us to analyze path probabilities
of the branching process.

###  3.2 Proof of Theorem 1.2

Now we are ready to prove our first main result.

###### Proof of Theorem 1.2.

We first assume $\sigma_{v}^{2}>0$ and consider the case
$u_{0}=\mathds{1}_{(-\infty,0]}$ to show the second part of the claim, i.e.
that the sequence of processes

$[0,\infty)\ni t\mapsto\frac{1}{\sqrt{nv\sigma_{v}^{2}}}\big{(}\ln
u(nt,vnt)-nt\Lambda(v)\big{)},\quad n\in\mathbb{N},$ (3.19)

converges in $\mathbb{P}$-distribution to standard Brownian motion. Because
$[0,\infty)\ni t\mapsto\ln u(t,vt)$ might be discontinuous _only_ in $0$, we
have to make clear what we mean by above convergence. In fact, we show the
invariance principle for a sequence of auxiliary processes
$(X_{n}^{v}(t))_{t\geq 0}$, $n\in\mathbb{N}$, where for every $n\in\mathbb{N}$
and $t\geq\frac{1}{n}$ $X_{n}^{v}(t)$ is the same as in (3.19), whereas for
$t\in[0,1/n]$ the term $\ln u(nt,vnt)$ in (3.19) is replaced by $(1-nt)\ln
u(0,0)+nt\ln(1,v)$, making $(X_{n}^{v}(t))_{t\geq 0}$ continuous. Because the
difference of the processes in (3.19) and $(X_{n}^{v}(t))_{t\geq 0}$ converges
uniformly to zero as $n\to\infty$, convergence of the processes in (3.19) to a
standard Brownian motion is defined as the convergence of the processes
$(X_{n}^{v}(t))_{t\geq 0}$, $n\in\mathbb{N}$, to a standard Brownian motion in
$C([0,\infty))$ with topology induced by the metric $\rho$ from (1.12).

By Proposition 3.5 and Corollary 3.8, on $\mathcal{H}_{nvt}$ (recall the
notation from (2.23)) we have

$\displaystyle\begin{split}-\ln\Cr{constY_{v}}&\leq\ln\sigma_{nvt}^{\zeta}(v)+\ln
Y_{v}^{\approx}(nvt)+nvtL^{*}(1/v)+\sqrt{nvt}W^{v}_{vnt}(1)\leq\ln\Cr{constY_{v}},\quad\text{
and}\\\ -\ln\Cr{constY_{v}approx}&\leq\ln u(nt,vnt)-{\texttt{es}\cdot nt}-\ln
Y_{v}^{\approx}(vnt)\leq\ln\Cr{constY_{v}approx}.\end{split}$ (3.20)

Recall that a sequence of processes $t\mapsto A_{n}(t)$, $n\in\mathbb{N}$,
converges in $\mathbb{P}$-distribution to standard Brownian motion if and only
if for all $\sigma>0$ the sequence $t\mapsto\sigma^{-1}A_{n}(\sigma^{2}t)$,
$n\in\mathbb{N}$, converges in $\mathbb{P}$-distribution to a standard
Brownian motion. Applying this to (3.10), the sequence of processes

$[0,\infty)\ni t\mapsto\frac{1}{\sqrt{nv\sigma_{v}^{2}}}\big{(}\ln
Y_{v}^{\approx}(vnt)+vntL^{*}(1/v)\big{)},\quad n\in\mathbb{N},$ (3.21)

converges in $\mathbb{P}$-distribution to a standard Brownian motion. Further,
by the second line in (3.20),

$-\ln\Cr{constY_{v}approx}\leq\big{(}\ln
u(nt,vnt)-nt(\texttt{es}-vL^{*}(1/v))\big{)}-\big{(}\ln
Y_{v}^{\approx}(vnt)+vntL^{*}(1/v)\big{)}\leq\ln\Cr{constY_{v}approx}$

holds. Consequently, if we can prove that

$\Lambda(v)=\texttt{es}-vL^{*}(1/v)\quad\forall v\in V.$ (3.22)

the claim follows from (3.21). To prove (3.22), we set $n=1$ in (3.20) and
note that
$\frac{W_{vt}^{v}(1)}{\sqrt{t}}\displaystyle\mathop{\longrightarrow}_{t\rightarrow\infty}0$
$\mathbb{P}$-a.s. for all $v\in V$, because $W_{n}(1)$ converges in
$\mathbb{P}$-distribution to a centered normally distributed random variable
by Proposition 3.1. Using (3.24), (3.20) and (3.7), we get (3.22).

It remains to show the claim for arbitrary $u_{0}\in\mathcal{I}_{\text{PAM}}$.
Recall that there exist $\delta^{\prime}\in(0,1)$ and $C^{\prime}>1$, such
that $\delta^{\prime}\mathds{1}_{[-\delta^{\prime},0]}(x)\leq u_{0}(x)\leq
C^{\prime}\mathds{1}_{(-\infty,0]}(x)$ for all $x\in\mathbb{R}$. Therefore,
using Corollary 3.9 we have

$\delta^{\prime}\Cr{constY_{v}approx}^{-2}u^{\mathds{1}_{(-\infty,0]}}(t,vt)\leq
u^{{u_{0}}}(t,vt)\leq C^{\prime}u^{\mathds{1}_{(-\infty,0]}}(t,vt).$ (3.23)

where we used that the solution to (PAM) is linear in its initial condition.
Thus, the convergence of (3.19) for arbitrary initial condition
$u_{0}\in\mathcal{I}_{\text{PAM}}$ follows from the one with initial condition
$\mathds{1}_{(-\infty,0]}$. This gives the second part of Theorem 1.2.

It remains to show that $(nv)^{-1/2}\big{(}\ln u(n,vn)-n\Lambda(v)\big{)}$
converges in $\mathbb{P}$-distribution to a Gaussian random variable. For
$u_{0}=\mathds{1}_{(-\infty,0]},$ this is a direct consequence of (3.20) for
$t=1$, (3.22) and the second part of Proposition 3.5. For general $u_{0},$ the
claim follows from (3.23). ∎

In view of Corollary 3.9, Proposition A.3 and (3.22), the Lyapunov exponent
$\Lambda$, defined in (1.11), determines the exponential decay (growth, resp.)
for solutions to (PAM) for _arbitrary_ initial conditions
$u_{0}\in\mathcal{I}_{\text{PAM}}$ (and not only for those with compact
support).

######  Corollary 3.10.

For all $v\geq 0$ and all $u_{0}\in\mathcal{I}_{\text{PAM}}$ we have that
$\mathbb{P}$-a.s.,

$\Lambda(v)=\lim_{t\to\infty}\frac{1}{t}\ln u^{u_{0}}(t,vt).$ (3.24)

Furthermore, $\Lambda$ is linear on $[0,v_{c}]$ and strictly concave on
$(v_{c},\infty),$ and the convergence in (3.24) holds uniformly on any compact
interval $K\subset[0,\infty)$.

###### Proof.

Note that by Proposition A.3,
$\delta^{\prime}\mathds{1}_{[-\delta^{\prime},0]}\leq u_{0}\leq
C^{\prime}\mathds{1}_{(-\infty,0]}$ and Corollary 3.9 we have that (3.24)
holds for all $v\in V$ and all compact $V\subset(v_{c},\infty),$ so (3.24) is
true for all $v>v_{c}$. The strict concavity of $\Lambda$ on $(v_{c},\infty)$
follows from the strict convexity of $L^{*}(1/v)$, which in turn follows from
the strict convexity of $L$ and standard properties of the Legendre
transformation. If $v_{c}=0$, the proof is complete due to
$\lim_{t\to\infty}\frac{1}{t}\ln
u^{\delta^{\prime}\mathds{1}_{[-\delta^{\prime},0]}}=\Lambda(0)=\texttt{es}$
by Proposition A.3 and $u^{u_{0}}\leq e^{\texttt{es}t}$ for all
$u_{0}\in\mathcal{I}_{\text{PAM}}$. Thus, let us assume $v_{c}>0$ from now on.
First observe that $L^{*}(1/v)$ tends to $L^{*}(1/v_{c})=-L(0)$ as
$v\downarrow v_{c}$. Indeed, due to Lemma 2.4 (d), for each $v>v_{c}$ there
exists a unique $\overline{\eta}(v)\in(-\infty,0),$ characterized via
$L^{\prime}\big{(}\overline{\eta}(v)\big{)}=\frac{1}{v}$, such that
$L^{*}(1/v)=\frac{\overline{\eta}(v)}{v}-L(\overline{\eta}(v)).$ Furthermore,
$(v_{c},\infty)\ni v\mapsto\overline{\eta}(v)$ is continuously differentiable
and strictly decreasing, bounded from above by $0.$ In addition,
$(-\infty,0)\ni\eta\mapsto L^{\prime}(\eta)$ is smooth and strictly monotone
and tends to $L^{\prime}(0-)$ as $\eta\uparrow 0.$ As a consequence, we get
that $\eta(v)\uparrow 0$ as $v\downarrow v_{c}$ and thus $L^{*}(1/v)\to
L^{*}(1/v_{c})$ as $v\downarrow v_{c}$. Therefore, we deduce that
$\Lambda(v)=\texttt{es}-vL^{*}(1/v)$ for all $v\in[v_{c},\infty)$.
Furthermore, for all $u_{0}\in\mathcal{I}_{\text{PAM}}$, due to (BDD)

$\begin{split}u^{u_{0}}(t,vt)&\leq
C^{\prime}e^{\texttt{es}t}E_{vt}\big{[}e^{\int_{0}^{t}\zeta(B_{s})\mathop{}\\!\mathrm{d}s};B_{t}\leq
0\big{]}\leq
C^{\prime}e^{\texttt{es}t}E_{vt}\big{[}e^{\int_{0}^{H_{0}}\zeta(B_{s})\mathop{}\\!\mathrm{d}s};H_{0}\leq
t\big{]}\\\ &\leq
C^{\prime}\exp\big{\\{}t\big{(}\texttt{es}+v\overline{L}_{vt}(0)\big{)}\big{\\}}.\end{split}$
(3.25)

Using Lemma 2.4 (b), taking logarithms, and dividing by $t$, we get that the
(due to Proposition A.3) concave function $\Lambda$ is bounded from above by
the linear function $[0,\infty)\ni
v\mapsto\texttt{es}-vL(0)=\texttt{es}+vL^{*}(1/v_{c})$ and coincides with this
function at $v=0$ as well as $v=v_{c},$ and hence on the whole interval
$[0,v_{c}]$. Using (3.25) again, we infer (3.24) for all $v\geq 0$ and
$u_{0}\in\mathcal{I}_{\text{PAM}}$.

To show that the convergence is uniform on every compact interval
$K\subset[0,\infty)$, for $\varepsilon>0$ arbitrary we consider
$\varepsilon\mathbb{Z}:=\\{k\varepsilon:k\in\mathbb{Z}\\},$ and for
$y\in\mathbb{R}$ set
$\lfloor{y}\rfloor_{\varepsilon}:=\sup\\{x\in\varepsilon\mathbb{Z}:\,x\leq
y\\}$. Then the convergence is uniform on $K\cap\varepsilon\mathbb{Z}.$ A
fortiori, for $t$ large enough, uniformly in $y\in K$,

$u(t,t\lfloor{y}\rfloor_{\varepsilon})\geq
e^{t(\Lambda(\lfloor{y}\rfloor_{\varepsilon})-\varepsilon)}.$

Lemma A.8 then entails that

$\inf_{z\in[t\lfloor{y}\rfloor_{\varepsilon}-1,t\lfloor{y}\rfloor_{\varepsilon}+1]}u(t+1,z)\geq\frac{1}{\Cr{Harnack}}u(t,t\lfloor{y}\rfloor_{\varepsilon})\geq\frac{1}{\Cr{Harnack}}e^{t(\Lambda(\lfloor{y}\rfloor_{\varepsilon})-\varepsilon)}.$
(3.26)

Furthermore, using $0\leq y-\lfloor{y}\rfloor_{\varepsilon}\leq\varepsilon$,
we have

$\begin{split}P_{yt}(B_{\varepsilon
t}\in[t\lfloor{y}\rfloor_{\varepsilon}-1,t\lfloor{y}\rfloor_{\varepsilon}+1])&\geq\sqrt{\frac{2}{\pi\varepsilon
t}}\inf_{x\in[t\lfloor{y}\rfloor_{\varepsilon}-1,t\lfloor{y}\rfloor_{\varepsilon}+1]}e^{-\frac{(x-yt)^{2}}{2\varepsilon
t}}\\\ &\geq\sqrt{\frac{2}{\pi}}\cdot\exp\Big{\\{}-\frac{(\varepsilon
t+1)^{2}}{2\varepsilon t}-\frac{\ln(\varepsilon t)}{2}\Big{\\}}.\end{split}$
(3.27)

Using the Feynman-Kac formula in the equality, the Markov property at time
$\varepsilon t,$ and (BDD) in the first inequality, we infer that

$\displaystyle\begin{split}u(t+1+\varepsilon
t&,yt)=E_{yt}\big{[}e^{\int_{0}^{t+1+\varepsilon
t}\xi(B_{s})\mathop{}\\!\mathrm{d}s}u_{0}(B_{t+1+\varepsilon t})\big{]}\\\
&\geq e^{\texttt{ei}\varepsilon t}\cdot P_{yt}(B_{\varepsilon
t}\in[t\lfloor{y}\rfloor_{\varepsilon}-1,t\lfloor{y}\rfloor_{\varepsilon}+1])\cdot\inf_{z\in[t\lfloor{y}\rfloor_{\varepsilon}-1,t\lfloor{y}\rfloor_{\varepsilon}+1]}u(t+1,z)\\\
&\geq
c_{1}\exp\Big{\\{}t\Big{(}\Lambda(\lfloor{y}\rfloor_{\varepsilon})+(\texttt{ei}-3/2)\varepsilon-\frac{1}{t}-\frac{1}{2\varepsilon
t^{2}}-\frac{\ln(\varepsilon t)}{2t}\Big{)}\Big{\\}},\end{split}$ (3.28)

where in the last inequality we used (3.26) and (3.27). Setting
${t}^{\prime}:=t+1+\varepsilon t$ and ${y}^{\prime}:=\frac{{t}^{\prime}}{t}y$,
we get

$\displaystyle\frac{1}{{t}^{\prime}}\ln
u({t}^{\prime},{t}^{\prime}y)-\Lambda(y)$
$\displaystyle=\frac{t}{{t}^{\prime}}\Big{(}\frac{1}{t}\ln
u({t}^{\prime},t{y}^{\prime})-\Lambda(\lfloor{{y}^{\prime}}\rfloor_{\varepsilon})\Big{)}+\frac{t}{{t}^{\prime}}\Lambda(\lfloor{{y}^{\prime}}\rfloor_{\varepsilon})-\Lambda(y).$

Since (3.28) holds uniformly for all $y\in K$, we infer that

$\inf_{y\in K}\Big{(}\frac{1}{{t}^{\prime}}\ln
u({t}^{\prime},{t}^{\prime}y)-\Lambda(y)\Big{)}\geq\frac{t}{{t}^{\prime}}\Big{(}\frac{\ln
c_{1}}{t}+(\texttt{ei}-3/2)\varepsilon-\frac{1}{t}-\frac{1}{2\varepsilon
t^{2}}-\frac{\ln(\varepsilon t)}{2t}\Big{)}-\sup_{y\in
K}\Big{|}\frac{t}{{t}^{\prime}}\Lambda(\lfloor{{y}^{\prime}}\rfloor_{\varepsilon})-\Lambda(y)\Big{|}.$

Since $\Lambda$ concave and finite, it is uniformly continuous on compact
intervals. As a consequence, since $\varepsilon>0$ was chosen arbitrarily, we
deduce the lower bound

$\liminf_{t\to\infty}\inf_{y\in K}\Big{(}\frac{1}{t}\ln
u(t,ty)-\Lambda(y)\Big{)}\geq 0.$ (3.29)

To derive the matching upper bound, we assume that the convergence does not
hold uniformly on $K$. Then, due to (3.29), there exist $\alpha>0$ and
sequences $(t_{n})_{n\in\mathbb{N}}\subset[0,\infty)$ and
$(y_{n})_{n\in\mathbb{N}}\subset K$ such that $t_{n}\to\infty$ and

$\frac{1}{t_{n}}\ln u(t_{n},t_{n}y_{n})-\Lambda(y_{n})\geq\alpha\quad\forall
n\in\mathbb{N}.$ (3.30)

Retreating to a suitable subsequence, we can assume
$y_{n}\displaystyle\mathop{\longrightarrow}_{n\rightarrow\infty}y\in K.$ For
$n$ such that $|y_{n}-y|\leq\varepsilon$, we have, similarly to (3.27),

$\displaystyle P_{y(t_{n}+1+\varepsilon t_{n})}\big{(}B_{\varepsilon
t_{n}}\in[t_{n}y_{n}-1,t_{n}y_{n}+1]\big{)}$
$\displaystyle\geq\sqrt{\frac{2}{\pi\varepsilon
t_{n}}}\inf_{x\in[t_{n}y_{n}-1,t_{n}y_{n}+1]}e^{-\frac{(x-y(t_{n}+1+\varepsilon
t_{n}))^{2}}{2\varepsilon t_{n}}}$
$\displaystyle\geq\sqrt{\frac{2}{\pi\varepsilon
t_{n}}}\exp\Big{\\{}-\frac{(1+y)^{2}}{2\varepsilon
t_{n}}(t_{n}\varepsilon+1)^{2}\Big{\\}}.$

Therefore, taking advantage of (3.26) again and using an argument as in the
derivation of (3.28), we infer that

$\displaystyle u(t_{n}+1+\varepsilon t_{n},(t_{n}+1+\varepsilon t_{n})y)$
$\displaystyle\geq c_{1}\cdot
u(t_{n},t_{n}y_{n})\cdot\sqrt{\frac{1}{\varepsilon
t_{n}}}\exp\Big{\\{}\varepsilon\texttt{ei}t_{n}-\frac{(1+y)^{2}}{2\varepsilon
t_{n}}(t_{n}\varepsilon+1)^{2}\Big{\\}}$ (3.31)

for all $n$ such that $|y_{n}-y|\leq\varepsilon$. Now recall that
$y_{n}\displaystyle\mathop{\longrightarrow}_{n\rightarrow\infty}y$, that
$\Lambda$ is continuous, as well as $\frac{1}{t}\ln
u(t,ty)\displaystyle\mathop{\longrightarrow}_{t\rightarrow\infty}\Lambda(y).$
Therefore, first taking logarithms, dividing by $t_{n}$, and then taking
$n\to\infty,$ the left-hand side in (3.31) converges to
$(1+\varepsilon)\Lambda(y).$ Contrarily, by (3.30), the limit of the right-
hand side is bounded from below by
$\Lambda(y)+\alpha+\varepsilon\texttt{ei}-\frac{(1+y)^{2}}{2}\varepsilon$.
Choosing $\varepsilon>0$ small enough, this leads to a contradiction. As a
consequence, we deduce the uniform convergence on $K$. ∎

###  3.3 Time perturbation

In the next step we prove perturbation results, i.e., time and space
perturbation Lemmas 3.11 and 3.12. These statements will be useful when
comparing the expected number of particles which are slightly slower or faster
than the ones with given velocity. As usual, $u=u^{u_{0}}$ denotes the
solution to (PAM) with initial condition $u_{0}\in\mathcal{I}_{\text{PAM}}$.

######  Lemma 3.11.

  1. (a)

Let $u_{0}\in\mathcal{I}_{\text{PAM}}$ and let
$\varepsilon:\,(0,\infty)\to(0,\infty)$ be a function such that
$\varepsilon(t)\to 0$ and $t\varepsilon(t)\to\infty$ as $t\to\infty$. Then
there exists
$\Cl{const_{t}ime_{p}ert_{g}enau}=\Cr{const_{t}ime_{p}ert_{g}enau}((\varepsilon(t))_{t\geq
0},u_{0})$ such that $\mathbb{P}$-a.s., for all $t$ large enough,

$\sup_{(v,h)\in\mathcal{E}_{t}}\Big{|}\ln\Big{(}\frac{u^{u_{0}}(t+h,vt)}{u^{u_{0}}(t,vt)}\Big{)}-h(\text{\emph{{es}}}-\overline{\eta}(v))\Big{|}\leq\Cr{const_{t}ime_{p}ert_{g}enau}+\Cr{const_{t}ime_{p}ert_{g}enau}|h|\Big{(}\sqrt{\frac{\ln
t}{t}}+\frac{|h|}{t}\Big{)},$ (3.32)

where $\mathcal{E}_{t}:=\left\\{(v,h):\ v\in V,\ |h|\leq
t\varepsilon(t),\frac{vt}{t+h}\in V\right\\}$.

  2. (b)

For all $\varepsilon>0$ and $u_{0}\in\mathcal{I}_{\text{PAM}}$ there exists a
constant $\Cl{const_{t}ime_{p}erturbation}>1$ and a $\mathbb{P}$-a.s. finite
random variable $\Cl[index_{p}rob_{c}ont_{t}ime]{T_{i}ndex_{t}imepert}$ such
that for all $t\geq\Cr{T_{i}ndex_{t}imepert}$, uniformly in $0\leq h\leq
t^{1-\varepsilon}$, $v\in V$ and $\frac{vt}{t+h}\in V,$

$\Cr{const_{t}ime_{p}erturbation}^{-1}e^{h/\Cr{const_{t}ime_{p}erturbation}}u^{u_{0}}(t,vt)\leq
u^{u_{0}}(t+h,vt)\leq\Cr{const_{t}ime_{p}erturbation}e^{\Cr{const_{t}ime_{p}erturbation}h}u^{u_{0}}(t,vt).$
(3.33)

###### Proof.

$(a)$ Note that it suffices to show the claim for
$u_{0}=\mathds{1}_{(-\infty,0]}$. Indeed, for all
$u_{0}\in\mathcal{I}_{\text{PAM}}$ we have
$\delta^{\prime}\mathds{1}_{[-\delta^{\prime},0]}\leq u_{0}\leq
C^{\prime}\mathds{1}_{(-\infty,0]}$. Using Corollary 3.9, we infer that for
all $u_{0}\in\mathcal{I}_{\text{PAM}}$, all $v\in V,$ and all $t$ large enough

$\delta^{\prime}\Cr{constY_{v}approx}^{-2}u^{\mathds{1}_{(-\infty,0]}}(t,vt)\leq
u^{{u_{0}}}(t,vt)\leq C^{\prime}u^{\mathds{1}_{(-\infty,0]}}(t,vt).$

where $u^{u_{0}}$ denotes the solution to (PAM) with initial condition
$u_{0}$.

by the same argument as at the end of the proof of Theorem 1.2, the solutions
to (PAM) for different initial conditions $u_{0}\in\mathcal{I}_{\text{PAM}}$
differ at most by a multiplicative constant. Let $t$ be large enough such that
$\mathcal{H}_{vt}$ occurs for all $v\in V$, which is possible by (2.23).
Letting $(v,h)\in\mathcal{E}_{t}$ and writing $v^{\prime}:=\frac{vt}{t+h}\in
V,$ we infer that

$\displaystyle\frac{u^{\mathds{1}_{(-\infty,0]}}(t+h,vt)}{u^{\mathds{1}_{(-\infty,0]}}(t,vt)}$
$\displaystyle=\frac{E_{vt}\left[e^{\int_{0}^{t+h}\xi(B_{s})\mathop{}\\!\mathrm{d}s};B_{t+h}\leq
0\right]}{E_{vt}\left[e^{\int_{0}^{t}\xi(B_{s})\mathop{}\\!\mathrm{d}s};B_{t}\leq
0\right]}=e^{\texttt{es}\cdot
h}\frac{E_{vt}\left[e^{\int_{0}^{t+h}\zeta(B_{s})\mathop{}\\!\mathrm{d}s};B_{t+h}\leq
0\right]}{E_{vt}\left[e^{\int_{0}^{t}\zeta(B_{s})\mathop{}\\!\mathrm{d}s};B_{t}\leq
0\right]}.$ (3.34)

Using Lemma 3.7, on $\mathcal{H}_{vt},$ the last fraction divided by
$\frac{Y_{v^{\prime}}^{\approx}(vt)}{Y_{v}^{\approx}(vt)}$ is bounded away
from $0$ and infinity for all $t$ large enough. As in the derivation of
(3.11), the term $\frac{Y_{v^{\prime}}^{\approx}(vt)}{Y_{v}^{\approx}(vt)}$
can be written as

$\displaystyle\frac{E_{vt}^{\zeta,\eta_{vt}^{\zeta}(v^{\prime})}\Big{[}\exp\Big{\\{}-\eta_{vt}^{\zeta}(v^{\prime})\sum_{i=1}^{vt}\widehat{\tau}_{i}\Big{\\}};\sum_{i=1}^{vt}\widehat{\tau}_{i}\in\Big{[}-K,0\Big{]}\Big{]}}{E_{vt}^{\zeta,\eta_{vt}^{\zeta}(v)}\Big{[}\exp\Big{\\{}-\eta_{vt}^{\zeta}(v)\sum_{i=1}^{vt}\widetilde{\tau}_{i}\Big{\\}};\sum_{i=1}^{vt}\widetilde{\tau}_{i}\in\Big{[}-K,0\Big{]}\Big{]}}\times\frac{\exp\Big{\\{}-vt\Big{(}\frac{\eta_{vt}^{\zeta}(v^{\prime})}{v^{\prime}}-\overline{L}_{vt}^{\zeta}(\eta_{vt}^{\zeta}(v^{\prime}))\Big{)}\Big{\\}}}{\exp\Big{\\{}-vt\Big{(}\frac{\eta_{vt}^{\zeta}(v)}{v}-\overline{L}_{vt}^{\zeta}(\eta_{vt}^{\zeta}(v))\Big{)}\Big{\\}}},$
(3.35)

where
$\widehat{\tau}_{i}=\tau_{i}-E_{vt}^{\zeta,\eta_{vt}^{\zeta}(v^{\prime})}[\tau_{i}]$
and
$\widetilde{\tau_{i}}:=\tau_{i}-E_{vt}^{\zeta,\eta_{vt}^{\zeta}(v)}[\tau_{i}]$.
But now, since $v^{\prime}\in V,$ as in the proof of Proposition 3.5, the
first fraction of the previous display is bounded from below and above by
positive constants, for all $t$ large enough. Indeed, setting $x=vt$ in
(3.12), the denominator in (3.35) equals the integral in (3.12) and, replacing
$v$ by $v^{\prime}$ in (3.12), the numerator equals the integral in (3.12).
The claim then follows due to Lemma 3.6 and using (3.7). Therefore, taking
logarithms in (3.34) and recalling the definition of $S_{vt}^{\zeta,v}(\eta)$
in (3.4), according to the previous considerations it suffices to show that
the logarithm of the second fraction in (3.35) plus $\overline{\eta}(v)\cdot
h$, i.e.

$\displaystyle\big{(}S_{vt}^{\zeta,v}(\eta_{vt}^{\zeta}(v))-S_{vt}^{\zeta,v}(\eta_{vt}^{\zeta}(v^{\prime}))\big{)}+\big{(}S_{vt}^{\zeta,v}(\eta_{vt}^{\zeta}(v^{\prime}))-S_{vt}^{\zeta,v^{\prime}}(\eta_{vt}^{\zeta}(v^{\prime}))\big{)}+\overline{\eta}(v)\cdot
h,$ (3.36)

satisfies bound on the right-hand side of (3.32), uniformly in
$(v,h)\in\mathcal{E}_{t}$. Recall that
$\frac{1}{v^{\prime}}=\frac{1}{v}\left(1+\frac{h}{t}\right)$, thus the second
summand in (3.36) is $-h\cdot\eta_{vt}^{\zeta}(v^{\prime})$. The triangular
inequality entails

$\displaystyle|\eta_{vt}^{\zeta}(v^{\prime})-\overline{\eta}(v)|$
$\displaystyle\leq|\eta_{vt}^{\zeta}(v^{\prime})-\overline{\eta}(v^{\prime})|+|\overline{\eta}(v^{\prime})-\overline{\eta}(v)|,$
(3.37)

and so by Lemma 2.5, uniformly for $v^{\prime}\in V$ and $t$ large enough, the
first term on the right-hand side of (3.37) can be upper bounded by
$\Cr{const:concEtaEmpLT}\sqrt{\frac{\ln vt}{vt}}$, $\mathbb{P}$-a.s.
Furthermore, by Lemma 2.4 d) we know that $\overline{\eta}$ is continuously
differentiable and strictly decreasing, having uniform positive bounds of the
derivative on every bounded subinterval of $(v_{c},\infty)$. Hence,
$\overline{\eta}(\cdot)$ is Lipschitz continuous on $V$ and we therefore get
that the second summand in (3.37) can be upper bounded by
$c_{1}|v-v^{\prime}|=c_{1}v\frac{|h|}{|t+h|}=c_{1}v\frac{|h|}{t}\cdot\frac{t}{|t+h|}\leq
c_{2}\frac{|h|}{t}$, uniformly for all $v,v^{\prime}\in V$ and all $t$ large
enough, where the last inequality is due to $|h|/t\leq\varepsilon(t)\to 0$.
Therefore, the absolute value of the sum of the second and third summand in
(3.36) is upper bounded by
$\Cr{const_{t}ime_{p}ert_{g}enau}|h|\big{(}\sqrt{\frac{\ln
t}{t}}+\frac{|h|}{t}\big{)}$ with
$\Cr{const_{t}ime_{p}ert_{g}enau}:=c_{2}\vee\Cr{const:concEtaEmpLT}$.

It remains to show that the first summand in (3.36) tends to $0$ as $t$ tends
to $\infty$. We write

$\displaystyle S_{vt}^{\zeta,v}(\eta_{vt}^{\zeta}(v^{\prime}))$
$\displaystyle=S_{vt}^{\zeta,v}(\eta_{vt}^{\zeta}(v))+\big{(}\eta_{vt}^{\zeta}(v^{\prime})-\eta_{vt}^{\zeta}(v)\big{)}\big{(}S_{vt}^{\zeta,v}\big{)}^{\prime}(\eta_{vt}^{\zeta}(v))+\frac{1}{2}\big{(}\eta_{vt}^{\zeta}(v^{\prime})-\eta_{vt}^{\zeta}(v)\big{)}^{2}\big{(}S_{vt}^{\zeta,v}\big{)}^{\prime\prime}(\widetilde{\eta})$

for some
$\widetilde{\eta}\in[\eta_{vt}^{\zeta}(v^{\prime})\wedge\eta_{vt}^{\zeta}(v),\eta_{vt}^{\zeta}(v^{\prime})\vee\eta_{vt}^{\zeta}(v)]$.
Recall that
$S_{vt}^{\zeta,v}(\eta)=vt\big{(}\frac{\eta}{v}-\overline{L}_{vt}^{\zeta}(\eta)\big{)}$
and by definition
$\big{(}\overline{L}_{vt}^{\zeta}\big{)}^{\prime}\big{(}\eta_{vt}^{\zeta}(v)\big{)}=\frac{1}{v}$.
Thus, $\big{(}S_{vt}^{\zeta,v}\big{)}^{\prime}(\eta_{vt}^{\zeta}(v))=0.$
Furthermore,
$\big{(}S_{vt}^{\zeta,v}\big{)}^{\prime\prime}(\eta)=-{vt}\big{(}\overline{L}_{vt}^{\zeta}\big{)}^{\prime\prime}(\eta)$
and the function $\big{(}\overline{L}_{vt}^{\zeta}\big{)}^{\prime\prime}$ is
uniformly bounded away from $0$ and infinity by Lemma A.1 on $V.$ Thus, by the
characterizing equation (2.15) and the implicit function theorem, on
$\mathcal{H}_{vt}$, the function $\eta_{vt}^{\zeta}(\cdot)$ is differentiable
with uniformly bounded first derivative, i.e.

$\big{|}\eta_{vt}^{\zeta}(v^{\prime})-\eta_{vt}^{\zeta}(v)\big{|}\leq
c_{3}|v^{\prime}-v|\leq c_{3}v\frac{|h|}{t(1-\varepsilon(t))}.$ (3.38)

Thus, on $\mathcal{H}_{vt}$, the first summand in (3.36) can be bounded by
$c_{4}\cdot t\cdot\frac{h^{2}}{t^{2}}=c_{4}\cdot\frac{h^{2}}{t},$ uniformly in
$(v,h)\in\mathcal{E}_{t}$, for all $t$ large enough. This implies $a).$

$(b)$ The first part of the proof is similar to that of $a)$; indeed, dividing
by $u^{u_{0}}(t,vt)$ and taking logarithms in (3.33), one arrives at (3.36)
again. The second part then consists of showing that (3.36) is lower and upper
bounded by two strictly increasing linear functions for all $0<h\leq
t^{1-\varepsilon}$. Following the same computations as in the proof of $a),$
for $v\in V$ and $h>0$, such that $v^{\prime}\in V$, we have up to some
additive constant, which is independent of $h$, that

$\displaystyle\ln\frac{u^{u_{0}}(t+h,vt)}{u^{u_{0}}(t,vt)}$
$\displaystyle\asymp
c_{t}\frac{vt}{2}(v-v^{\prime})^{2}-\eta_{vt}^{\zeta}(v^{\prime})\cdot
h+\texttt{es}\cdot h,$

where $c_{t}$ is a function which for $t$ large enough is positive and bounded
away from $0$ and infinity. Because $\eta_{vt}^{\zeta}(v^{\prime})<0$, the
latter expression is bounded from below by
$h/\Cr{const_{t}ime_{p}erturbation}$ and bounded from above by
$\Cr{const_{t}ime_{p}erturbation}\cdot h$ for our choice of parameters, hence
we can conclude. ∎

###  3.4 Space perturbation

While in the previous section we have been investigating the effects of time
perturbations of $u$ and related quantities, here we will consider space
perturbations. As before, let $u^{u_{0}}$ denotes the solution to (PAM) with
initial condition $u_{0}\in\mathcal{I}_{\text{PAM}}$.

######  Lemma 3.12.

Let $\varepsilon(t)$ be a positive function such that $\varepsilon(t)\to 0$
and $\frac{t\varepsilon(t)}{\ln t}\to\infty$ as $t\to\infty$. Then for all
$\varepsilon>0$ there exists $C(\varepsilon)>0$ such that $\mathbb{P}$-a.s.,
for all $u_{0}\in\mathcal{I}_{\text{PAM}}$ we have

  1. (a)

$\limsup_{t\to\infty}\
\sup\left\\{\left|\frac{1}{h}\ln\left(\frac{u(t,vt+h)}{u(t,vt)}\right)-L\big{(}\overline{\eta}(v)\big{)}\right|:(v,h)\in\mathcal{E}_{t}\right\\}\leq\varepsilon,$
(3.39)

where $\mathcal{E}_{t}:=\left\\{(v,h):\ v,v+\frac{h}{t}\in V,\
C(\varepsilon)\ln t\leq|h|\leq t\varepsilon(t)\right\\}$.

  2. (b)

Let $\varepsilon(t)$ be a positive function such that $\varepsilon(t)\to 0$.
Then there exists a constant $\Cl{const_{s}pace_{p}erturbation}<\infty$ and a
$\mathbb{P}$-a.s. finite random variable
$\Cl[index_{p}rob_{c}ont_{t}ime]{T_{i}ndex_{s}pacepert}$ such that for all
$t\geq\Cr{T_{i}ndex_{s}pacepert}$, uniformly in $0\leq h\leq t\varepsilon(t)$,
$v\in V$, $v+\frac{h}{t}\in V$ and $u_{0}\in\mathcal{I}_{\text{PAM}}$ we have

$\Cr{const_{s}pace_{p}erturbation}^{-1}e^{-\Cr{const_{s}pace_{p}erturbation}h}\cdot
u(t,vt)\leq
u(t,vt+h)\leq\Cr{const_{s}pace_{p}erturbation}e^{-h/\Cr{const_{s}pace_{p}erturbation}}\cdot
u(t,vt).$ (3.40)

The proof of Lemma 3.12 can be found in the companion article [11]; indeed, it
is central to the results of [11] and we hence prefer not to duplicate it here
due to space constraints.

###  3.5 Approximation results

In this section we mainly show how moment generating functions can be used in
order to approximate quantities related to the solution to (PAM) and BBMRE. As
usual, for $u_{0}\in\mathcal{I}_{\text{PAM}}$ let $u^{u_{0}}$ be the solution
to (PAM) with initial condition $u_{0}$.

######  Lemma 3.13.

There exists a constant $\Cl{constapproxtopamwithspeedv0}>0$ and a
$\mathbb{P}$-a.s. finite random variable
$\Cl[index_{p}rob_{c}ont_{t}ime]{randomindexapproxtopamsppedv0}$ such that for
all $u_{0}\in\mathcal{I}_{\text{PAM}}$ and
$t\geq\Cr{randomindexapproxtopamsppedv0},$

$\Big{|}\ln
u^{u_{0}}(t,v_{0}t)-\sum_{i=1}^{v_{0}t}\big{(}L_{i}^{\zeta}(\overline{\eta}(v_{0}))-L(\overline{\eta}(v_{0}))\big{)}\Big{|}\leq\Cr{constapproxtopamwithspeedv0}\ln
t.$ (3.41)

###### Proof.

By (3.22) and $\Lambda(v_{0})=0$, we have
$L^{*}(1/v_{0})=\frac{\texttt{es}}{v_{0}}$. Also, by (PAM-INI) and
monotonicity of the solution to (PAM) in its initial condition, we have
$u^{\delta^{\prime}\mathds{1}_{[-\delta^{\prime},0]}}\leq u^{u_{0}}\leq
u^{C^{\prime}\mathds{1}_{(-\infty,0]}}$. Thus, on the one hand, by the many-
to-few lemma (Proposition 2.3) and Lemma 3.7, for all
$u_{0}\in\mathcal{I}_{\text{PAM}}$ and $t$ such that $v_{0}t\geq\Cr{constHn}$,
where $\Cr{constHn}$ was defined in (2.23), we have

$\left|\ln u^{u_{0}}(t,v_{0}t)-\left(\ln
Y_{v_{0}}^{\approx}(v_{0}t)+v_{0}tL^{*}(1/v_{0})\right)\right|\leq\ln(\Cr{constY_{v}approx}/\delta^{\prime}).$

On the other hand, due to Proposition 3.5 and Corollary 3.4, there exists a
finite random variable $\mathcal{N}$, such that for all $t\geq\mathcal{N}$ we
have

$\Big{|}\ln
Y_{v_{0}}^{\approx}({v_{0}t})+v_{0}tL^{*}(1/v_{0})+\ln\sigma_{v_{0}t}^{\zeta}(v_{0})-\sum_{i=1}^{v_{0}t}\big{(}L_{i}^{\zeta}(\overline{\eta}(v_{0}))-L(\overline{\eta}(v_{0}))\big{)}\Big{|}\leq\ln\Cr{constY_{v}}+\Cr{const_{c}oncSn}\ln
v_{0}+\Cr{const_{c}oncSn}\ln t.$

Finally, by (3.7), $|\ln\sigma_{v_{0}t}^{\zeta}(v_{0})-\frac{1}{2}\ln
t|\leq\ln\Cr{constSigman}+\frac{1}{2}|\ln v_{0}|$ for all $t$ such that
$v_{0}t\geq\Cr{constHn}$. Combining this with the previous two display,
inequality (3.41) follows with
$\Cr{randomindexapproxtopamsppedv0}:=(\mathcal{N}\vee\Cr{constHn})/v_{0}$ and
$\Cr{constapproxtopamwithspeedv0}$ suitable. ∎

We introduce the so-called _breakpoint inverse_

$T_{x}^{u_{0},a}:=\inf\big{\\{}t\geq 0:u^{u_{0}}(t,x)\geq a\big{\\}},\quad
x\in\mathbb{R},\ a\in[0,\infty),\ u_{0}\in\mathcal{I}_{\text{PAM}},$ (3.42)

and abbreviate

$T_{x}^{(a)}:=T_{x}^{\mathds{1}_{(-\infty,0]},a}.$ (3.43)

Next, we state an important approximation result for $T_{x}^{u_{0},a}$, $x\geq
0$, in terms of the centered logarithmic moment generating functions.

######  Lemma 3.14.

There exists a constant $\Cl{constapproxofTn}<\infty$ and a $\mathbb{P}$-a.s.
finite random variable
$\Cl[constant]{constprobapproxofTn}=\Cr{constprobapproxofTn}(a,u_{0})$,
$a>0,u_{0}\in\mathcal{I}_{\text{PAM}}$, such that for all $x\geq 1,$

$\Big{|}T_{x}^{u_{0},a}-\frac{1}{v_{0}L(\overline{\eta}(v_{0}))}\sum_{i=1}^{x}L_{i}^{\zeta}(\overline{\eta}(v_{0}))\Big{|}\leq\Cr{constprobapproxofTn}+\Cr{constapproxofTn}\ln
x.$ (3.44)

Additionally, for each $u_{0}\in\mathcal{I}_{\text{PAM}}$ and $a>0,$

$\lim_{x\to\infty}\frac{T_{x}^{u_{0},a}}{x}=\frac{1}{v_{0}}\quad\mathbb{P}\text{-a.s}.$
(3.45)

###### Proof.

We set $t=x/v_{0}$ and let

$h_{t}:=\frac{1}{v_{0}L(\overline{\eta}(v_{0}))}\sum_{i=1}^{v_{0}t}\big{(}L(\overline{\eta}(v_{0})-L_{i}^{\zeta}(\overline{\eta}(v_{0}))\big{)}.$

We first note that due to Lemma A.2, the family
$(L(\overline{\eta}(v_{0}))-L_{i}^{\zeta}(\overline{\eta}(v_{0})))_{i\in\mathbb{Z}}$
satisfies the assumptions of Corollary A.5 with all $m_{i}$ equal to some
large enough finite constant and thus $\sum_{n}\mathbb{P}(|h_{n}|\geq
C\sqrt{n\ln n})<\infty$ for some $C>1$ large enough. The first Borel-Cantelli
lemma then readily supplies us with $|h_{n}|<C\sqrt{n\ln n}$ $\mathbb{P}$-a.s.
for all $n$ large enough. To control non-integer $t$, we recall

$\displaystyle h_{t}-h_{\lfloor t\rfloor}$
$\displaystyle=\frac{1}{v_{0}L(\overline{\eta}(v_{0}))}\big{(}v_{0}(t-\lfloor
t\rfloor)L(\overline{\eta}(v_{0}))-\ln
E_{v_{0}t}\big{[}e^{\int_{0}^{H_{v_{0}\lfloor
t\rfloor}}(\zeta(B_{s})+\overline{\eta}(v_{0}))\mathop{}\\!\mathrm{d}s}\big{]}\big{)}$

and hence $\mathbb{P}$-a.s. that $|h_{t}-h_{\lfloor t\rfloor}|\leq
1+\frac{\sqrt{2(\texttt{es}-\texttt{ei}-\overline{\eta}(v_{0}))}}{|L(\overline{\eta}(v_{0}))|}$
by (BDD) and [4, (2.0.1), p. 204], thus giving

$|h_{t}|<c_{1}\sqrt{t\ln t}\ \mathbb{P}\text{-a.s.\ for all $t$ large
enough.}$ (3.46)

To show the desired inequality, we note that (3.44) is equivalent to

$\big{|}T_{v_{0}t}^{u_{0},a}-(t-h_{t})\big{|}\leq\Cr{constprobapproxofTn}+\Cr{constapproxofTn}\ln(v_{0}t).$
(3.47)

For proving the latter, observe that it is sufficient to show that we can
choose $\Cr{constapproxofTn}>0$ as well as a $\mathbb{P}$-a.s. finite random
variable $\mathcal{T},$ such that

$\displaystyle\begin{split}u^{\delta^{\prime}\mathds{1}_{[-\delta^{\prime},0]}}(t-h_{t}+\Cr{constapproxofTn}\ln
t,v_{0}t)&\geq a\quad\text{ and }\quad
u^{\mathds{1}_{(-\infty,0]}}(t-h_{t}-\Cr{constapproxofTn}\ln
t,v_{0}t)<\frac{a}{2C^{\prime}},\,\forall t\geq\mathcal{T}.\end{split}$ (3.48)

with $\delta^{\prime},C^{\prime}$ from (PAM-INI). Indeed, due to (PAM-INI),
the first inequality in (3.48) implies $T_{v_{0}t}^{u_{0},a}\leq
T_{v_{0}t}^{\delta^{\prime}\mathds{1}_{[-\delta^{\prime},0]},a}\leq
t-h_{t}+\Cr{constapproxofTn}\ln t$ for $t\geq\mathcal{T}$. To use the second
inequality, first note that $T_{v_{0}t}^{u_{0},a}\geq
T_{v_{0}t}^{C^{\prime}\mathds{1}_{(-\infty,0]},a}=T_{v_{0}t}^{\mathds{1}_{(-\infty,0]},a/C^{\prime}}$.
Then, using
$u^{\mathds{1}_{(-\infty,0]}}(s,x)=\texttt{E}_{x}^{\xi}\big{[}N^{\leq}(s,0)\big{]}$
and Lemma A.6, (3.48) implies
$T_{v_{0}t}^{\mathds{1}_{(-\infty,0]},a/C^{\prime}}\geq
t-h_{t}-\Cr{constapproxofTn}\ln t$ for all $t\geq\mathcal{T}$. For
$t<\mathcal{T}$ on the other hand, we can use that $\mathbb{P}$-a.s., the
family $T_{v_{0}t}^{u_{0},a},$ $t<\mathcal{T},$ as well as the
$L_{i}^{\zeta}(\overline{\eta}(v_{0}))$ are uniformly bounded, allowing us to
upper bound the remaining cases of (3.47) by some finite random variable
$\Cr{constprobapproxofTn}$.

Thus, in order to show (3.48), note that for $\alpha\in\mathbb{R}$ and
uniformly in $u_{0}\in\mathcal{I}_{\text{PAM}}$ we have that

$\displaystyle\ln u^{u_{0}}(t-h_{t}+\alpha\ln t,v_{0}t)$
$\displaystyle=\ln\Big{(}\frac{u^{u_{0}}(t-h_{t}+\alpha\ln
t,v_{0}t)}{u^{u_{0}}(t,v_{0}t)}\Big{)}+\sum_{i=1}^{v_{0}t}\big{(}L_{i}^{\zeta}(\overline{\eta}(v_{0}))-L(\overline{\eta}(v_{0}))\big{)}+a_{t}$
$\displaystyle=(-h_{t}+\alpha\ln
t)(\texttt{es}-\overline{\eta}(v_{0}))+v_{0}L(\overline{\eta}(v_{0}))h_{t}+b(\alpha,t)=\alpha(\texttt{es}-\overline{\eta}(v_{0}))\ln
t+b(\alpha,t),$

for some error terms $a_{t}$ and $b(\alpha,t)$ fulfilling
$|a_{t}|\leq\Cr{constapproxtopamwithspeedv0}\ln t$ and

$|b(\alpha,t)|\leq\Cr{constapproxtopamwithspeedv0}\ln
t+\Cr{const_{t}ime_{p}ert_{g}enau}+\Cr{const_{t}ime_{p}ert_{g}enau}\cdot|\alpha\ln
t-h_{t}|\cdot\Big{(}\sqrt{\frac{\ln t}{t}}+\frac{|\alpha\ln
t-h_{t}|}{t}\Big{)}$

for all $t$ large enough. Indeed, the first equality is due to Lemma 3.13, the
second due to the time perturbation Lemma 3.11, the last one due to the
identity $\texttt{es}-\overline{\eta}(v_{0})=v_{0}L(\overline{\eta}(v_{0})).$
Then due to $|h_{t}|\leq c_{1}\sqrt{t\ln t}$ for large $t$ (cf. (3.46)),
choosing
$\Cr{constapproxofTn}=\alpha>\frac{2\Cr{const_{t}ime_{p}ert_{g}enau}\cdot
C^{2}}{\texttt{es}-\overline{\eta}(v_{0})}$ the latter term tends to infinity,
supplying us with (3.48).

To complete the proof, equation (3.45) is a direct consequence of (3.44) and
(2.9). ∎

Recall the definition $\overline{m}^{u_{0},a}=\overline{m}^{\xi,u_{0},a}$ from
(1.6).

######  Corollary 3.15.

For all $u_{0}\in\mathcal{I}_{\text{PAM}}$ and $a>0$ we have

$\frac{\overline{m}^{u_{0},a}(t)}{t}\displaystyle\mathop{\longrightarrow}_{t\rightarrow\infty}v_{0}\quad\mathbb{P}\text{-a.s.}$
(3.49)

###### Proof.

For an upper bound, we have
$\limsup_{t\to\infty}\frac{\overline{m}^{u_{0},a}(t)}{t}=\limsup_{t\to\infty}\frac{\overline{m}^{u_{0},a}(t)}{T_{\overline{m}^{u_{0},a}(t)}^{u_{0},a}}\frac{T_{\overline{m}^{u_{0},a}(t)}^{u_{0},a}}{t}\leq
v_{0}$, where the last inequality is due to
$T_{\overline{m}^{u_{0},a}(t)}^{u_{0},a}\leq t$ and (3.45). To get a lower
bound, we can use the properties of the Lyapunov exponent from Proposition
A.3, giving $\liminf_{t\to\infty}\frac{\overline{m}^{u_{0},a}(t)}{t}\geq v$
for all $v\in(0,v_{0})$, and we can conclude. ∎

######  Lemma 3.16.

For every $a>0$ there exists a constant
$\Cl{constantlemmaT_{m}teqt}=\Cr{constantlemmaT_{m}teqt}(a)>0$ and a
$\mathbb{P}$-a.s. finite random variable
$\Cl[index_{p}rob_{c}ont_{t}ime]{timelemmaT_{m}teqt}=\Cr{timelemmaT_{m}teqt}(a)$
such that for all $u_{0}\in\mathcal{I}_{\text{PAM}}$ and
$t\geq\Cr{timelemmaT_{m}teqt},$

$t-\Cr{constantlemmaT_{m}teqt}\leq T^{u_{0},a}_{\overline{m}^{u_{0},a}(t)}\leq
t.$ (3.50)

###### Proof.

By definition, the inequality $T^{u_{0},a}_{\overline{m}^{u_{0},a}(t)}\leq t$
follows directly. To show $t-\Cr{constantlemmaT_{m}teqt}\leq
T^{u_{0},a}_{\overline{m}^{u_{0},a}(t)}$, recall that due to (3.49) we can use
time perturbation. Indeed, by defining
$\Cr{constantlemmaT_{m}teqt}:=\Cr{const_{t}ime_{p}erturbation}\ln\big{(}\Cr{const_{t}ime_{p}erturbation}\cdot
3C^{\prime}/a\big{)}$ with $C^{\prime}$ from (PAM-INI) and
$\Cr{const_{t}ime_{p}erturbation}$ from Lemma 3.11 b), for all $t$ large
enough

$\displaystyle
u^{\mathds{1}_{(-\infty,0]}}(t-\Cr{constantlemmaT_{m}teqt},\overline{m}^{u_{0},a}(t))$
$\displaystyle\leq\Cr{const_{t}ime_{p}erturbation}e^{-\Cr{constantlemmaT_{m}teqt}/\Cr{const_{t}ime_{p}erturbation}}u^{u_{0}}(t,\overline{m}^{u_{0},a}(t))<\frac{a}{2C^{\prime}}$

and thus, recalling
$u^{\mathds{1}_{(-\infty,0]}}(s,x)=\texttt{E}_{x}^{\xi}\big{[}N^{\leq}(s,0)\big{]}$
and Lemma A.6, we get the lower bound
$T^{u_{0},a}_{\overline{m}^{u_{0},a}(t)}\geq
T^{C^{\prime}\mathds{1}_{(-\infty,0]},a}_{\overline{m}^{u_{0},a}(t)}=T^{\mathds{1}_{(-\infty,0]},a/C^{\prime}}_{\overline{m}^{u_{0},a}(t)}\geq
t-\Cr{constantlemmaT_{m}teqt}$ for all $t$ large enough and we can conclude. ∎

Recall definition (3.43) for $T_{x}^{(a)}$.

######  Corollary 3.17.

There exists $\overline{K}\in(1,\infty)$ such that $\mathbb{P}$-a.s., for all
$a>0$ and for all $x$ large enough

$\sup_{|y|\leq 1}T^{(a)}_{x+y}-\overline{K}\leq T^{(a)}_{x}\leq\inf_{|y|\leq
1}T^{(a)}_{x+y}+\overline{K}.$ (3.51)

###### Proof.

We set
$\overline{K}:=1+\Cr{const_{t}ime_{p}erturbation}\big{(}\ln(2\Cr{const_{t}ime_{p}erturbation}\Cr{const_{s}pace_{p}erturbation})+\Cr{const_{s}pace_{p}erturbation}\big{)}$.
Then due to (3.45), $\mathbb{P}$-a.s. for all $x$ large enough we have

$\frac{x+y^{\prime}}{T^{(a)}_{x+y^{\prime\prime}}\pm\overline{K}}\in
V\quad\forall\ y^{\prime},y^{\prime\prime}\in[-1,1].$

This allows us to apply the inequalities (3.33) and (3.40) for
$u_{0}=\mathds{1}_{(-\infty,0]}$. Indeed, for all $|y|\leq 1,$

$\displaystyle u(T^{(a)}_{x+y}-\overline{K},x)$
$\displaystyle\leq\Cr{const_{s}pace_{p}erturbation}e^{\Cr{const_{s}pace_{p}erturbation}}u(T^{(a)}_{x+y}-\overline{K},x+y)\leq\Cr{const_{s}pace_{p}erturbation}e^{\Cr{const_{s}pace_{p}erturbation}}\Cr{const_{t}ime_{p}erturbation}e^{-(\overline{K}-1)/\Cr{const_{t}ime_{p}erturbation}}u(T^{(a)}_{x+y}-1,x+y)<\frac{a}{2},$

where the first inequality is due to (3.40), the second one due to (3.33) and
the last one uses $u(T^{(a)}_{x+y}-1,x+y)<a$. By Lemma A.6 we get $u(t,x)<a$
for all $t\leq T^{(a)}_{x+y}-\overline{K}$ and thus the left-hand side in
(3.51). Analogously, first applying (3.40) and then (3.33), we have

$\displaystyle u(T^{(a)}_{x+y}+\overline{K},x)$
$\displaystyle\geq\Cr{const_{s}pace_{p}erturbation}^{-1}e^{-\Cr{const_{s}pace_{p}erturbation}}u(T^{(a)}_{x+y}+\overline{K},x+y)\geq\Cr{const_{s}pace_{p}erturbation}^{-1}e^{-\Cr{const_{s}pace_{p}erturbation}}\Cr{const_{t}ime_{p}erturbation}^{-1}e^{\overline{K}/\Cr{const_{t}ime_{p}erturbation}}u(T^{(a)}_{x+y},x+y)\geq
a$

for all $|y|\leq 1$, giving the right-hand side of (3.51). ∎

######  Corollary 3.18.

Let $\overline{m}^{a}(t)=\overline{m}^{\xi,\mathds{1}_{(-\infty,0]},a}(t)$,
$a>0$, be defined in (1.6). Then for all $0<\varepsilon\leq M$ there exists
$C=C(\varepsilon,M)$ such that $\mathbb{P}$-a.s. for all $t$ large enough

$0\leq\overline{m}^{\varepsilon}(t)-\overline{m}^{M}(t)\leq C.$

###### Proof.

The first inequality is clear. By Corollary 3.15, we can use the second
inequality from Lemma 3.12 b) and get the claim by defining
$C:=\Cr{const_{s}pace_{p}erturbation}\ln\big{(}\Cr{const_{s}pace_{p}erturbation}\cdot
M/\varepsilon\big{)}$ with $\Cr{const_{s}pace_{p}erturbation}$ from Lemma 3.12
b) to get $u^{\mathds{1}_{(-\infty,0]}}(t,\overline{m}^{\varepsilon}(t)-C)\geq
M$ and thus $\overline{m}^{M}(t)\geq\overline{m}^{\varepsilon}(t)-C$ for all
$t$ large enough. ∎

###  3.6 Proof of Theorem 1.3

Using the preparatory results from the previous sections, it is now possible
to obtain an invariance principle for the front of the solution to (PAM).
Roughly speaking, up to some error which can be controlled by the results from
the previous sections, we have $\overline{m}(t)\approx\ln u(t,v_{0}t)$ and can
then use the invariance principle from Theorem 1.2 to conclude.

###### Proof of Theorem 1.3.

Let $u_{0}\in\mathcal{I}_{\text{PAM}}$, $a>0$ and abbreviate $u=u^{u_{0}}$ and
$\overline{m}:=\overline{m}^{u_{0},a}$. We first assume
$\sigma_{v_{0}}^{2}>0$. Then we have to show that the sequence of processes

$[0,\infty)\ni
t\mapsto\frac{\overline{m}(nt)-v_{0}nt}{\sqrt{n\widetilde{\sigma}_{v_{0}}^{2}}},\quad
n\in\mathbb{N},$ (3.52)

where $\widetilde{\sigma}_{v_{0}}^{2}>0$ is given in (3.57) below, converges
in $\mathbb{P}$-distribution to standard Brownian motion in the Skorohod space
$D([0,\infty))$. Notice that $[0,\infty)\ni t\mapsto\overline{m}(t)$ might not
be càdlàg _only_ in $0$. To avoid this issue, above convergence is defined as
convergence of the sequence of processes in (3.52) where we set
$\overline{m}(t)\equiv 0$ for $t$ such that $\overline{m}(t)\leq 0$, making it
càdlàg.

Due to the limiting behavior and the continuity of $x\mapsto u(t,x)$ for
$t>0$, the value $r(t):=\overline{m}(t)-v_{0}t$ is the largest solution to

$\ln u(t,v_{0}t+r(t))=-\ln 2.$

We define

$\mathcal{L}(t,h):=\ln\frac{u(t,v_{0}t+h)}{u(t,v_{0}t)},\quad t>0,\
h\in\mathbb{R},$ $U(t):=-\ln u(t,v_{0}t)-\ln 2=\mathcal{L}(t,r(t)),\quad t\geq
0,$

let

$\delta\in\big{(}0,|L(\overline{\eta}(v_{0}))|\big{)}$

and $\varepsilon(t)$ be a positive function such that $\varepsilon(t)\to 0$
and $\varepsilon(t)t^{1/2}\to\infty$. Then by Lemma 3.12, there is
$C(\varepsilon)>0$ such that for $t$ large enough and all $h\in\mathbb{R}$
fulfilling $C(\varepsilon)\ln t\leq|h|\leq\varepsilon(t)t$ and
$v_{0}+\frac{h}{t}\in V$ we have

$\displaystyle-\big{(}|L(\overline{\eta}(v_{0}))|+\delta\big{)}h\leq\mathcal{L}(t,h)\leq-\big{(}|L(\overline{\eta}(v_{0}))|-\delta\big{)}h.$
(3.53)

Now, multiplying (3.44) by $v_{0}$, replacing $x$ by $\overline{m}(t)$ in
(3.44) and recalling that $t-\Cr{constantlemmaT_{m}teqt}\leq
T_{\overline{m}(t)}\leq t$ by Lemma 3.16, we get

$\Big{|}(\overline{m}(t)-v_{0}t)-\frac{1}{L(\overline{\eta}(v_{0}))}\sum_{i=1}^{\overline{m}(t)}\big{(}L_{i}^{\zeta}(\overline{\eta}(v_{0}))-L(\overline{\eta}(v_{0}))\big{)}\Big{|}\leq
v_{0}\Cr{constprobapproxofTn}+\Cr{constantlemmaT_{m}teqt}+v_{0}\Cr{constapproxofTn}\ln(\overline{m}(t))$
(3.54)

for all $t$ large enough. Next, recall that $\frac{\overline{m}(t)}{t}\to
v_{0}$ by Corollary 3.15 and that the standardized sum
$\frac{1}{\sqrt{n}}\sum_{i=1}^{nt}\big{(}L_{i}^{\zeta}(\overline{\eta}(v_{0}))-L(\overline{\eta}(v_{0}))\big{)}$
converges in distribution to a non-degenerate Gaussian random variable by
Lemma 3.2. As a consequence, in combination with (3.54), we infer that
$|r(t)|=|\overline{m}(t)-v_{0}t|\in[C(\varepsilon)\ln t,\varepsilon(t)t]$ with
probability tending to $1$ as $t$ tends to infinity. This and (3.53) implies

$r(t)\in\Big{[}\frac{U(t)}{|L(\overline{\eta}(v_{0}))|\mp\delta},\frac{U(t)}{|L(\overline{\eta}(v_{0}))|\pm\delta}\Big{]},$
(3.55)

with probability tending to $1$ as $t$ tends to $\infty$, where the upper sign
is chosen if $U(t)>0$ and the lower sign if $U(t)<0$. If
$\sigma_{v_{0}}^{2}>0$, due to $\Lambda(v_{0})=0$ and Theorem 1.2, the
sequence of processes

$[0,\infty)\ni t\mapsto\frac{1}{\sqrt{nv_{0}\sigma_{v_{0}}^{2}}}U(nt),\quad
n\in\mathbb{N},$ (3.56)

converges in $\mathbb{P}$-distribution to standard Brownian motion. Because
(3.55) holds for all $\delta>0$ small enough, Theorem 1.3 is a direct
consequence of the convergence in distribution of (3.56) by choosing

$\displaystyle\widetilde{\sigma}_{v_{0}}$
$\displaystyle:=\frac{\sqrt{\sigma_{v_{0}}^{2}v_{0}}}{|L(\overline{\eta}(v_{0}))|}.$
(3.57)

where $\sigma_{v_{0}}^{2}$ is defined in (3.1). This gives the second part of
Theorem 1.3. If $\sigma_{v_{0}}^{2}=0$, we can proceed analogously and the
first part of Theorem 1.3 follows from the first part of Theorem 1.2 and
(3.55). ∎

##  4 Log-distance of the fronts of the solutions to PAM and F-KPP

We finally prove our last main result, Theorem 1.4. In Sections 4.1 and 4.2,
we will assume that $u_{0}=w_{0}=\mathds{1}_{(-\infty,0]}$. Indeed, using a
comparison argument in the proof of Theorem 1.4, it will turn out that this is
actually sufficient for or purposes. It should also be mentioned here that the
tools we employ are inherently probabilistic. As a consequence, and for
notational convenience, we will mostly formulate the respective results in
terms of the BBMRE in what follows below; the correspondence to the results in
PDE terms is immediate from (2.2) and Remark 2.2.