1/\delta\rfloor}\mathbf{1}\left(B(\delta i)\in K\right).$ (91)

The first is the Hölder norm of the paths, the second measures the regularity
of the occupation measure at a grid of times.

######  Proposition 73.

Let $B,B^{\prime}$ be independent copies of planar Brownian motion or Brownian
bridge on the time interval $[0,1]$, variance $I$ and arbitrary drift or
endpoints. Let $\varepsilon,\delta,\gamma>0$. Let $A$ be the event that

$\|B\|_{\kappa}<\varepsilon/(5\delta^{\kappa})\qquad\mbox{ and that }\qquad
M(B,\varepsilon,\delta)\leq 1/(6\sqrt{\gamma\delta}).$

Then

$E\exp\Big{\\{}\mathbf{1}_{A\times A}\gamma\alpha(B,B^{\prime})\Big{\\}}\leq
E\exp\Big{\\{}c\gamma\delta\sum_{i,i^{\prime}}\mathbf{1}(|B(\delta
i)-B^{\prime}(\delta
i^{\prime})|<4\varepsilon)\Big{\\}}\leq\exp\\{c\gamma\delta^{-1}\\}.$

###### Proof.

Let $S=\\{(0,0),(0,1),(1,0),(1,1)\\}$, let
$\operatorname{box}_{(0,0)}(\varepsilon)$ be the set of boxes with lower left
corner in $2\varepsilon\mathbb{Z}^{2}$, and for $\sigma\in S$ let
$\operatorname{box}_{\sigma}(\varepsilon)$ be the set of translates of boxes
in $\operatorname{box}_{(0,0)}(\varepsilon)$ by $\sigma\varepsilon$. This
partitions $\operatorname{box}(\varepsilon)$ into 4 subsets.

Set
$a^{*}_{i,i^{\prime}}=\alpha(B_{[i,i+1]\delta},B^{\prime}_{[i^{\prime},i^{\prime}+1]\delta}$),
and consider the versions

$\displaystyle a_{i,i^{\prime}}=$ $\displaystyle
a^{*}_{i,i^{\prime}}\mathbf{1}\\{|B(s)-B(i\delta)|\leq\varepsilon/5,s\in[i,i+1]\delta\\}$
$\displaystyle\quad\times\mathbf{1}\\{|B^{\prime}(s)-B^{\prime}(i^{\prime}\delta)|\leq\varepsilon/5,s\in[i^{\prime},i^{\prime}+1]\delta\\}$

For $K\in\operatorname{box}(\varepsilon)$ define the asymmetric quantity

$Y_{K}=\sum_{i,i^{\prime}}\mathbf{1}(B(i\delta)\in K)a_{i,i^{\prime}}$

Then on $A\times A$ we have

$\alpha(B,B^{\prime})=\sum_{\sigma\in
S}\sum_{K\in\operatorname{box}_{\sigma}(\varepsilon)}Y_{K}.$

and so by Cauchy-Schwarz

$E\exp\\{\mathbf{1}_{A\times
A}\gamma\alpha(B,B^{\prime})\\}\leq\prod_{\sigma\in
S}\Big{(}E\exp\big{\\{}4\gamma\sum_{K\in\operatorname{box}_{\sigma}(\varepsilon)}Y_{K}\big{\\}}\Big{)}^{1/4}.$

Now condition on the sigma field $\mathcal{G}_{\delta}$ given by values of
$B,B^{\prime}$ at times in $\delta\mathbb{Z}$.

If $B(\delta i)\in K$ and $B(\delta j)\in K^{\prime}$ for
$K\not=K^{\prime}\in\operatorname{box}_{\sigma}(\varepsilon)$ (with the same
$\sigma$!) then by the modulus of continuity, on the event $A\times A$ the
segments of $B$ on $\delta[i,i+1]$ and $\delta[j,j+1]$ cannot intersect the
same segment of $B^{\prime}$.

Thus, conditionally on $\mathcal{G}_{\delta}$, the collections of mutual
intersection local times
$\mathcal{A}_{K}=\\{a_{i,i^{\prime}}:i,i^{\prime}\in\mathbb{Z},B(i)\in K\\}$
are independent as $K$ changes. Thus

$E\Big{[}\exp\big{\\{}4\gamma\sum_{K\in\operatorname{box}_{\sigma}(\varepsilon)}Y_{K}\big{\\}}|\mathcal{G}_{\delta}\Big{]}=\prod_{K\in\operatorname{box}_{\sigma}(\varepsilon)}E\Big{[}\exp\big{\\{}4\gamma
Y_{K}\big{\\}}|\mathcal{G}_{\delta}\Big{]}$

Now let $L_{K}$ be the number of $i$ so that $B(\delta i)\in K$ and let
$L^{*}_{K}$ be the number of $i^{\prime}$ that $B^{\prime}(\delta i^{\prime})$
is in $K^{*}$, the union of 9 boxes in $\operatorname{box}_{\varepsilon}$ that
intersect $K$. Then $L_{K}L_{K}^{*}$ is an upper bound on the number of
summands in $Y_{K}$. Using Hölder’s inequality again,

$E\Big{[}\exp\big{\\{}4\gamma
Y_{K}\big{\\}}|\mathcal{G}_{\delta}\Big{]}\leq\prod_{i,j:B(\delta i)\in
K}E\Big{[}\exp\big{\\{}4L_{K}L_{K}^{*}\gamma\alpha_{ij}\big{\\}}|\mathcal{G}_{\delta}\Big{]}^{1/L_{K}L_{K}^{*}}$

On the set $A\times A$ we have $4L_{K}L_{K}^{*}\gamma\delta<1$. By Lemma 72,
the conditional expectation is bounded by

$E\Big{[}\exp\big{\\{}4L_{K}L_{K}^{*}\gamma\alpha_{ij}\big{\\}}|\mathcal{G}_{\delta}\Big{]}\leq\exp(cL_{K}L_{K}^{*}\gamma\delta).$

Putting everything together, we get

$E\exp\\{\mathbf{1}_{A\times A}\gamma\alpha(B,B^{\prime})\\}\leq
E\exp\Big{\\{}c\gamma\delta\sum_{K\in\operatorname{box}(\varepsilon)}L_{k}L^{*}_{K}\Big{\\}}$

Where we have

$\sum_{K\in\operatorname{box}(\varepsilon)}L_{K}L^{*}_{K}\leq\sum_{i,i^{\prime}}\mathbf{1}(|B(\delta
i)-B^{\prime}(\delta i^{\prime})|<4\varepsilon).$

The latter can be thought of as the mutual intersection local time of a pair
of Gaussian random walks. ∎

It remains to bound $M(B,\delta,\varepsilon)$. We use a standard argument.

######  Lemma 74.

There is $c,c^{\prime}>0$ so that the following holds. For planar Brownian
motion with arbitrary drift and covariance $I$ on the interval $[0,1]$ and for
all $0<\delta\leq\varepsilon^{2}\leq 1/e^{2}$ and $a\geq 1$ we have

$P(M(B,\delta,\varepsilon)>a\delta^{-1}\varepsilon^{2}(\log\varepsilon)^{2})\leq
c^{\prime}\delta^{-1}\varepsilon^{ca}.$

The same result holds for any planar Brownian bridge with covariance $I$.

###### Proof.

For planar Brownian $B$ motion with identity covariance arbitrary starting
point and drift, the probability that it will not visit a fixed disk $D$ of
radius $\varepsilon$ between times $\varepsilon^{2}$ and $1$ is bounded below
by $c/|\\!\log\varepsilon|$, uniformly over the starting point. Let

$L_{j}=\sum_{i=0}^{j}\mathbf{1}(B(\delta i)\in D),\qquad L_{*}=L_{\lfloor
1/\delta\rfloor}.$

Let $\ell=\lceil\varepsilon^{2}/\delta\rceil$ and $J_{k}$ be the smallest $j$
so that $L_{j}=\ell k$. By the strong Markov property of $B$ applied at time
$\delta J_{k}$, we have

$P(L_{*}\geq\ell(k+1)|L_{*}\geq\ell k)\leq c/|\\!\log\varepsilon|.$

This inequality implies that $L_{*}$ is dominated by $\ell$ times a geometric
random variable $G$ with success probability $c/|\\!\log\varepsilon|$.

By this and the strong Markov property, applied at time $i\delta$, the number
$M_{i}$ of total returns up to time $1$ to a box containing $B(i\delta)$ is
dominated by $\ell G$.

Hence by the union bound for all $x\geq 0$,

$P(\max_{i}M_{i}>x)\leq\lceil\delta^{-1}\rceil P(\ell
G>x)=\lceil\delta^{-1}\rceil(1-c/|\\!\log\varepsilon|)^{\lfloor
x/\ell\rfloor}.$

Setting $x=a\varepsilon^{2}\delta^{-1}(\log\varepsilon)^{2}$, the tail bound
for the geometric random variable completes the proof for Brownian motion.

For Brownian bridge $R$ we prove the result for the time intervals $[0,1/2]$
and $[1/2,1]$ separately. There, it follows from the Brownian motion case and
absolute continuity, Proposition 65. ∎

######  Corollary 75.

Let $R$ be planar Brownian bridge run until an arbitrary time with arbitrary
endpoints, and let $R^{\prime}$ be an independent copy. For every $\gamma\geq
0$, $p<1$, there is an event $A$ so that

$PA>p,\qquad\mbox{ and }\qquad E\exp\\{\mathbf{1}_{A\times
A}\gamma\alpha(R,R^{\prime})\\}<\infty.$

###### Proof.

By scaling, it suffices to show this for bridges run until time 1. Fix
$1/4<\kappa^{\prime}<\kappa<1/2$. As $\delta\to 0$, set
$\varepsilon=\delta^{\kappa^{\prime}}$. Then by Lemma 74, and Levy’s modulus
of continuity,

$P(M\leq a\delta^{2\kappa^{\prime}-1}(\log\delta)^{2})\to
1,\qquad\mbox{and}\qquad
P(\|B\|_{\kappa}\leq\varepsilon/(5\delta^{\kappa}))\to 1,$

and for the intersection $A_{\delta}$ of these events $PA_{\delta}\to 1$.
Choosing $\delta$ small enough so that $PA_{\delta}>p$ and
$\gamma<c\delta^{1-4\kappa^{\prime}}/(\log\delta)^{4}$, the conditions of
Proposition 73 are satisfied and therefore we indeed have

$E\exp\\{\mathbf{1}_{A_{\delta}\times
A_{\delta}}\gamma\alpha(R,R^{\prime})\\}<\infty.\qed$

###  6.3 Uniform exponential moments for large drift

We need a bound on the exponential moments which improves with the distance of
the starting points. First we show such a bound for short times.

######  Lemma 76.

Let $B,B^{\prime}$ be independent planar Brownian motions started at
$(x,y),(x^{\prime},y^{\prime})\in\mathbb{R}^{2}$ with identity covariance
matrix and a shared drift $(0,N),N\geq 0$ and run until time $t$. Let
$0<\gamma,t$ and $\gamma t\leq\pi/4$. Then, there is a constant $c$ such that

$Ee^{\gamma\alpha(B,B^{\prime})}\leq 1+\frac{c\gamma
t}{1+N\sqrt{t}}e^{-\frac{|y-y^{\prime}|^{2}}{8t}}\leq\exp\big{\\{}\tfrac{c\gamma
t}{1+N\sqrt{t}}e^{-\frac{|y-y^{\prime}|^{2}}{8t}}\big{\\}}.$ (92)

and

$E\big{[}\alpha(B,B^{\prime})^{2}\big{]}\leq\frac{ct^{2}}{1+N^{2}t}e^{-\frac{|y-y^{\prime}|^{2}}{8t}}.$
(93)

###### Proof.

By symmetry, we may assume $(x^{\prime},y^{\prime})=(0,0)$ and $y\geq 0$. By
Lemma 70, we may assume $t=1$. Let $\tau,\tau^{\prime}$ be the first times the
second coordinate of $B$, respectively $B^{\prime}$, equals $y/2$. By
symmetry, $\tau^{\prime}$ is an independent copy of $\tau$ and

$P(\tau\leq 1)=P(\max_{t\in[0,1]}B_{2}(t)\geq y/2)=P(|B_{2}(1)|\geq y/2)\leq
e^{-{y}^{2}/8}$ (94)

Let $A_{0}=[0,\tau)$, and $A_{1}=[\tau,1]$. Define $A_{i}^{\prime}$
analogously. By bilinearity (Lemma 58),

$\alpha(B,B^{\prime})=\sum_{i,j\in\\{0,1\\}}\alpha(B_{A_{i}},B^{\prime}_{A^{\prime}_{j}})\leq\alpha(B_{A_{1}},B^{\prime})+\alpha(B,B^{\prime}_{A^{\prime}_{1}}),$

since $\alpha(B_{A_{0}},B^{\prime}_{A^{\prime}_{0}})=0$, and $\alpha$ is
nonnegative. Cauchy-Schwarz gives

$Ee^{\gamma\alpha(B,B^{\prime})}\leq
E[e^{2\gamma\alpha(B_{A_{1}},B^{\prime})}]^{1/2}E[e^{2\gamma\alpha(B,B^{\prime}_{A^{\prime}_{1}})}]^{1/2}=Ee^{2\gamma\alpha(B_{A_{1}},B^{\prime})},$

since the two terms are equal, by symmetry. By conditioning on the
$\sigma$-field at time $\tau$ and using Lemma 72,

$E[e^{2\gamma\alpha(B_{A_{1}},B^{\prime})}-1]\leq P(\tau\leq
1)\frac{c\gamma}{1+N}.$

By the same argument,

$E\alpha^{2}(B_{A_{1}},B^{\prime})\leq P(\tau\leq 1)\frac{c}{1+N^{2}}.$

Together with (94) these give (92) and (93). ∎

The following lemma will be needed in the proof of Proposition 78.

######  Lemma 77.

Let $B$ be standard one-dimensional Brownian motion. For every
$\gamma,\lambda>0$, and integer $k\geq\gamma^{2}$, for the sum over integer
times,

$E\exp\left\\{\frac{\gamma}{\sqrt{k}}\sum_{i=0}^{k-1}e^{-\lambda
B^{2}(i)}\right\\}\leq 2\exp\\{4\pi\gamma^{2}e^{\lambda}/\lambda\\}.$

###### Proof.

For $y\in[0,1]$, we have $e^{-y}\leq 1-y/2$, so for $a\in[0,1]$,

$\displaystyle
E\exp\left\\{-a\int_{0}^{1}e^{-\lambda(B(t)-x)^{2}}dt\right\\}\leq
1-\frac{a}{2}E\int_{0}^{1}e^{-\lambda(B(t)-x)^{2}}dt.$

By Jensen’s inequality applied to both the expectation and time integral, this
is at most

$\displaystyle
1-\frac{a}{2}\exp\left\\{-\lambda\int_{0}^{1}E(B(t)-x)^{2}\,dt\right\\}=1-2a_{1}e^{-\lambda
x^{2}}\leq\exp\\{-2a_{1}e^{-\lambda x^{2}}\\},\qquad
a_{1}=\frac{ae^{-\lambda/2}}{4}.$

This implies that

$E[\exp\\{-a\int_{i}^{i+1}e^{-\lambda
B^{2}(t)}dt\\}\mid\mathcal{F}_{i}]\leq\exp\\{-2a_{1}e^{-\lambda B^{2}(i)}\\}$

and so

$M_{j}=\exp\left\\{2a_{1}\sum_{i=0}^{j-1}e^{-\lambda
B^{2}(i)}-a\int_{0}^{j}e^{-\lambda B^{2}(t)}\,dt\right\\}$

is a supermartingale. By Cauchy-Schwarz

$E\exp\left\\{a_{1}\sum_{i=0}^{k-1}e^{-\lambda
B^{2}(i)}\right\\}\leq(EM_{k})^{1/2}\left(E\exp\left\\{a\int_{0}^{k}e^{-\lambda
B^{2}(t)}\,dt\right\\}\right)^{1/2}$

with $EM_{k}\leq 1$. Setting $a_{1}=\gamma/\sqrt{k}$, $a=4\gamma
e^{\lambda/2}/\sqrt{k}$, $a^{\prime}=4\gamma\sqrt{\pi/(\lambda
k)}e^{\lambda/2}$,

$E\exp\left\\{a\int_{0}^{k}e^{-\lambda
B^{2}(t)}\,dt\right\\}=E\exp\left\\{a^{\prime}\int\sqrt{\lambda/\pi}e^{-\lambda
x^{2}}\ell(k,x)dx\right\\}$

where $\ell(k,x)=\int_{0}^{k}\delta_{0}(B(t)-x)dt$ is the local time up to
time $k$ at $x$. Since $\int\sqrt{\lambda/\pi}e^{-\lambda x^{2}}dx=1$, we have
by Jensen’s inequality,

$E\exp\left\\{a^{\prime}\int\sqrt{\lambda/\pi}e^{-\lambda
x^{2}}\ell(k,x)dx\right\\}\leq E\int\sqrt{\lambda/\pi}e^{-\lambda
x^{2}}\exp\\{a^{\prime}\,\ell(k,x)\\}dx.$

By stochastic domination,

$E\exp\\{a^{\prime}\,\ell(k,x)\\}\leq
E\exp\\{a^{\prime}\,\ell(k,0)\\}=E\exp\\{4\gamma\sqrt{\pi/\lambda}e^{\lambda/2}|B_{1}|\\}\leq
2\exp\\{8\pi\gamma^{2}e^{\lambda}/\lambda\\}$

since $\ell(k,0)$ has the same distribution as $|B(k)|$ and $\sqrt{k}|B(1)|$
and using $Ee^{|X|}\leq Ee^{X}+Ee^{-X}$. ∎

######  Proposition 78.

Let $B,B^{\prime}$ be two independent standard planar Brownian motions with
drift $(0,N)$ started at some arbitrary $z,z^{\prime}\in\mathbb{R}^{2}$ on the
time interval $[0,1]$. There is a $c<\infty$ such that for all $N,r\geq 1$,
and $0\leq\gamma\leq\tfrac{\pi}{8r}$,

$E\big{[}e^{\gamma N\alpha(B,B^{\prime})}\mathbf{1}_{A_{r}^{2}}\big{]}\leq
e^{c\gamma^{2}},\qquad A_{r}:=\Big{\\{}\sup_{t\in[0,1]}|B_{2}(t)-Nt|\leq
r/2\Big{\\}}.$

###### Proof.

Consider overlapping time intervals

$I_{j}=[j\tfrac{r}{N},(j+1)\tfrac{r}{N}),\qquad
J_{j}=[(j-\tfrac{1}{2})\tfrac{r}{N},(j+\tfrac{1}{2})\tfrac{r}{N}).$

On $A_{r}$, the restrictions $B_{I_{j}}$ and $B^{\prime}_{I_{i}}$ of the paths
to those intervals can intersect only when $|i-j|\leq 1$. Even if $j=i+1$, the
intersection can only happen at a time in $J_{j}$. This implies that on
$A_{r}$,

$\alpha(B,B^{\prime})\leq\alpha_{1}+\alpha_{2},\qquad\alpha_{1}=\sum_{j=0}^{k-1}\alpha(B_{I_{j}},B^{\prime}_{I_{j}}),\qquad\alpha_{2}=\sum_{j=1}^{k-1}\alpha(B_{J_{j}},B^{\prime}_{J_{j}}).$

where $k=\lceil N/r\rceil$. Let $\gamma^{\prime}=2N\gamma$, so that by
assumption $\gamma^{\prime}r/N\leq\pi/4$. Let $Y_{j}$ denote the first
coordinate of $B_{jr/N}-B^{\prime}_{jr/N}$. By Lemma 72,

$E\Big{[}\exp\\{\gamma^{\prime}\alpha(B_{I_{i}},B^{\prime}_{I_{i}})\\}\,\big{|}\,\mathcal{F}_{ir/N}\Big{]}\leq\exp\\{de^{-\frac{1}{8t}Y_{i}^{2}}\\},\qquad
d=\tfrac{c\gamma^{\prime}r/N}{1+N\sqrt{r/N}}\leq\tfrac{c^{\prime}\gamma\sqrt{r}}{\sqrt{N}},$

so the process

$M_{j}=\exp\Big{\\{}\sum_{i=0}^{j-1}\gamma^{\prime}\alpha(B_{I_{i}},B^{\prime}_{I_{i}})-de^{-\frac{1}{8t}Y_{i}^{2}}\Big{\\}}$

is a supermartingale. By Cauchy-Schwarz,

$(Ee^{2\gamma N\alpha_{1}})^{2}\leq
EM_{k}\;E\exp\sum_{i=0}^{k-1}\tfrac{c\gamma\sqrt{r}}{\sqrt{N}}e^{-\frac{m}{8r}Y_{i}^{2}}\leq
2\exp(c\gamma^{2}),$ (95)

where the last inequality uses $EM_{k}\leq 1$ and Lemma 77. The same bound
holds for $\alpha_{2}$, and we conclude by Cauchy-Schwarz. ∎

An important shortcoming of Proposition 78 is that the range $[0,\pi/(8r)]$ of
exponents $\gamma$ for which it gives a bound gets worse with the truncation
parameter $r$. The remedy is provided by cutting off paths with large Hölder
norm, 91.

######  Proposition 79.

Let $B,B^{\prime}$ be two independent standard planar Brownian motions with
drift $(0,N)$ on the time interval $[0,1]$. Alternatively, let $B,B^{\prime}$
be Browniaan bridges from $(0,0)$ to $N$. Let $0<\kappa<1/2$. Then for all and
$\gamma>0$, $r>1$, and $N\geq cr((\gamma r)^{3}+1)$ we have

$E\left[e^{\gamma N\alpha(B,B^{\prime})}\mathbf{1}_{H^{2}}\right]\leq
2e^{b\gamma^{2}},\qquad H=\\{\|B\|_{\kappa}\leq r/2\\}.$

###### Proof.

We start with the Brownian motion case. Let $A_{r,t}$ be the event that
$\sup_{s\in[0,t]}|B_{2}(s)-Ns|\leq r$. Let

$\mathfrak{e}(t,N,r,\gamma)=\sup E\big{[}e^{\gamma
N\alpha(B,B^{\prime})}\mathbf{1}_{A_{r,t}^{2}}\big{]},$

where the sup is over all starting points for $B,B^{\prime}$, which still have
a shared drift $(0,N)$.

Consider the time intervals $I_{j}=[\tfrac{j}{m},\tfrac{j+1}{m}]$,
$J_{j}=[(j-\tfrac{1}{2})\tfrac{1}{m},(j+\tfrac{1}{2})\tfrac{1}{m}]$. When
$1/m>r/N$ and $H^{2}$ holds, then $B_{I_{j}}$ and $B^{\prime}_{I_{i}}$ have
disjoint support unless $|i-j|\leq 1$. Moreover, if $j=i+1$ they can only have
intersecting support within the time interval $J_{j}$. This implies that

$\alpha(B,B^{\prime})\leq\alpha_{1}+\alpha_{2},\qquad\alpha_{1}=\sum_{j=0}^{m-1}\alpha(B_{I_{j}},B_{I_{j}}),\qquad\alpha_{2}=\sum_{j=1}^{m-1}\alpha(B_{J_{j}},B^{\prime}_{J_{j}}).$

Let $G_{i}$ be the event that $\sup_{s,t\in I_{i}}|B(s)-B(t)-(0,N)(s-t)|\leq
r/m^{\kappa}$, and abbreviate

$\mathfrak{e}=\mathfrak{e}(1/m,N,rm^{-\kappa},\gamma)=\mathfrak{e}(1,N/m^{1/2},rm^{1/2-\kappa},\gamma/m^{1/2}),$

where the equality is by scaling, Lemma 70. Then
$R_{j}=\mathfrak{e}^{-j}e^{N\gamma\sum_{i=0}^{j-1}\mathbf{1}_{G_{i}}\alpha(B_{I_{i}},B^{\prime}_{I_{i}})}$
is a supermartingale, and since on $H^{2}$ all $G_{i}$ occur, we have

$E[e^{\gamma
N\alpha_{1}}\mathbf{1}_{H^{2}}]\leq\mathfrak{e}^{m}ER_{m}\leq\mathfrak{e}^{m}.$

Similarly, $E[e^{\gamma
N\alpha_{2}}\mathbf{1}_{H^{2}}]\leq\mathfrak{e}^{m-1}$. By Cauchy-Schwarz and
Proposition 78 we get

$E[e^{\gamma
N\alpha(B,B^{\prime})}\mathbf{1}_{H^{2}}]\leq\mathfrak{e}(1,N/m^{1/2},rm^{1/2-\kappa},2\gamma/m^{1/2})^{m}\leq
e^{4b^{\prime}\gamma^{2}}$

as long as $2\gamma rm^{-\kappa}\leq\pi/8$. The motion case follows with
$m=\lceil(16\gamma r/\pi)^{1/\kappa}\rceil$.

For the bridge, on $\mathbf{1}_{H^{2}}$, for $N$ large enough,
$\alpha(B_{[0,1/3]},B^{\prime}_{[2/3,1]})=\alpha(B_{[2/3,1]},B^{\prime}_{[0,1/3]})=0$
and then since $\alpha$ is bilinear and non-negative we have

$\alpha(R,R^{\prime})\leq\alpha(R_{0\to 2/3},R^{\prime}_{0\to
2/3})+\alpha(R_{1/3\to 1},R^{\prime}_{1/3\to 1}).$ (96)

The two quantities have the same distribution by symmetry. The first two-
thirds of the bridge is absolutely continuous with bounded derivative with
respect to the first two-thirds of the motion, Proposition 65. Cauchy-Schwarz
concludes the proof. ∎

##  7 Proofs of the main theorems

In this section we prove Theorems 2, 3, 4 and 6. Theorem 5 is a special case
of Theorem 6. The hard work has been done, but the pieces need to be
assembled.

###  7.1 Proof of Theorems 2 and 3

Explicit solution and uniqueness. Definition 1 is what we rigorously mean by a
solution of the Wick-ordered heat equation. Proposition 26 shows that the
projections $E_{P}[u(x,t)|\mathcal{F}_{n}]$ satisfy a finitary version of the
equation. These projections are martingales.

Proposition 27 shows that the finitary version of the Wick-ordered heat
equation has a unique solution $u_{n}(x,t)=p(x,t)Z_{n}(x,t)$ with

$Z_{n}(x,t)=E_{Q}\exp\bigg{\\{}\sum_{j=1}^{n}m_{j}\xi_{j}-\tfrac{1}{2}\sum_{j=1}^{n}m_{j}^{2}\bigg{\\}},\quad
m_{j}=\int_{0}^{t}{\rm e}_{j}(B(s))\,ds.$ (97)

Proposition 28 shows that if the martingale $Z_{n}(x,t)$ is uniformly
integrable for all $x,t$, then its limit is the unique solution of the Wick-
ordered SHE. So we fix $x,t$.

In Section 4, we identify the martingale $Z_{n}$ as a randomized shift.
Proposition 39.2 gives criteria for $Z_{n}$ to converge in $L^{1}$ in terms of
the intersection exponential $\mathfrak{e}(m)$ (see (61)). Here

$\mathfrak{e}(m)=\lim_{n\to\infty}E_{Q^{\otimes
2}}e^{\alpha_{n}},\qquad\alpha_{n}=\sum_{i=1}^{n}m_{i}m_{i}^{\prime}.$

is defined in terms of two independent copies of $m$ in (97). Proposition 66
shows that $\alpha_{n}$ has a limit $\alpha\geq 0$ in $L^{k}$ for every $k>0$,
the mutual intersection local time of two independent copies of $B$. Lemma 44
shows that for $\alpha\geq 0$ we have $\mathfrak{e}(m)\leq Ee^{\alpha}$.

By Proposition 39.2, to show martingale convergence, it suffices to show that
for every $p<1$ there is set of paths $A$ with $QA>p$ so that
$\mathfrak{e}(\mathbf{1}_{A}m)\leq\infty$.

More concretely, for Brownian bridges, we need to show that there is a set of
paths $A$ with probability arbitrarily close to 1 so that the mutual
intersection local time satisfies $E_{Q^{\otimes 2}}\exp\\{\mathbf{1}_{A\times
A}\,\alpha\\}<\infty$.

For the set $A$, we control two properties of the paths: a Hölder norm and the
maximal mass of a discretized version of the occupation measure, see (91).

Proposition 73 shows that under such control, the exponential moments of the
intersection local time exist, and Lemma 74 bounds the maximal mass. Using
these results, Corollary 75 provides the set $A$ of paths with $QA>p$ so that
$E_{Q^{\otimes 2}}\exp\\{\mathbf{1}_{A\times A}e^{\alpha}\\}<\infty$. Thus we
have found the solution of the Wick-ordered heat equation.

Convergence in $L^{2}(P)$. For $t<t_{c}$ we have $Ee^{\alpha}<\infty$. By
Lemma 44, we have $\mathfrak{e}(m)<\infty$, and Proposition 39 implies that
$Z_{n}\to Z$ in $L^{2}(P)$.

Basis independence. The solution we have found is the partition function of a
randomized shift with $\alpha$ given by the mutual intersection local time.
Proposition 56 shows that when the mutual intersection local time exists, it
is unique. In particular, it does not depend on the basis ${\rm e}_{j}$. The
partition function $Z$ is basis-independent by Remark 47. Uniqueness in this
type of construction is due to Shamov (2016) (Theorem 17 and Corollary 18),
which show that the underlying extended Gaussian covariance kernel uniquely
determines the partition function $Z$. For us, the kernel is on path space
$\Omega$, and is given by the mutual intersection local time $\alpha$. Theorem
2 follows.

Polymer measure. The randomized shift representation has been established
along the way in Proposition 39.2. It gives rise to a unique polymer measure
by Theorem 50. This completes the proof of Theorem 3. Proposition 54 implies
the claimed polymer convergence. ∎

######  Remark 80.

This proof extends to the case when the initial condition $\varsigma$ is a
general probability measure.

###  7.2 Proof of Theorem 4

Explicit solution and uniqueness. By Proposition 31, the conditional
expectation $z_{n}=E_{Q}[z|\mathcal{F}_{n}]$ of the equation has the unique
solution given by

$z_{n}(x,t)=Z_{n}(y,0;x,t)q(x-y,t)$
$Z_{n}=Z_{n}(0,y;t,x)=E_{Q}\exp\left(\sum_{j=1}^{n}m_{j}\xi_{j}-\frac{1}{2}m_{j}^{2}\right),\qquad
m_{j}=\int_{0}^{t}e_{j}(b(s),s)ds.$

and $b(s)$ is a one-dimensional Brownian bridge between space-time points
$(0,0)$ to $(x,t)$.

We need to show that this is an $L^{2}(P)$ martingale, that is, it has a
bounded $L^{2}$-norm. By Proposition 33 we have

$E_{P}Z_{n}^{2}=E_{Q}e^{\alpha_{n}},\qquad\alpha_{n}=\sum_{i=1}^{n}m_{j}m_{j}^{\prime}.$

By Proposition 68, $\alpha_{n}$ has a limit in every $L^{k}({Q^{\otimes 2}})$.
The limit $\alpha(b,b^{\prime})$ is the mutual intersection local time of two
independent copies of $(b(s),s)$ up to time $t$, and $E_{{Q^{\otimes
2}}}\cosh\alpha<\infty.$ By Lemma 44, we

$E_{P}[Z_{n}^{2}]=E_{{Q^{\otimes 2}}}\,e^{\alpha_{n}}\leq E_{{Q^{\otimes
2}}}\,\cosh{\alpha}.$

so $Z_{n}\to Z$ in $L^{2}(P)$. With this construction, taking conditional
expectations given $\mathcal{F}_{n}$ as in Proposition 30, we see that for
every $n$ the $\mathcal{F}_{n}$-conditional expectations of the two sides of
the equation (52) of Definition 29 agree. So the limit $z$ is indeed a
solution. We have thus found the unique solution in an explicit form.

Agreement with Itô. The Skorokhod integral formulation in Definition 29 of the
one-dimensional Wick-ordered stochastic heat equation coincides with the mild
Itô formulation. This is because by Remark 24, the two integrals are the same.
Since there is a unique solution, the solution in Alberts et al. (2014) agrees
with ours.

Polymer measure and CDRP. Associated to the partition function $Z$ there is a
random polymer measure $M$ on $\Omega$ defined by the formula (13), see
Section 4.6. We claim that it coincides with the continuum directed random
polymer constructed in Alberts et al. (2014), Definition 4.1. Their
construction is a random continuous function on $[0,1]$, determined by its
finite dimensional distributions given the noise $\xi$. Since $\Omega$ is the
space of continuous functions, it suffices to show that their finite
dimensional distributions agree with ours. This is the content of the next
lemma. Let $z(x,s;y,t)=q(y-x,t-s)Z(x,s;y,t)$ denote the solution of the SHE
started at time $s$ from the initial condition $\delta_{x}$.

######  Lemma 81.

Let $M$ be the unnormalized polymer measure on paths $b$ from ${\rm p}$ to
${\rm q}$, let $k\geq 1$ and set $(x_{0},s_{0})=\rm p$, $(x_{k+1},s_{k+1})=\rm
q$. Pick $s_{1}<\dots<s_{k}$ in $(s_{0},s_{k+1})$. Let $\varphi$ be a bounded
nonnegative test function on $\mathbb{R}^{k}$. Then

$E_{M}\varphi(b(s_{1}),\dots,b(s_{k}))=\int\,\varphi(x_{1},\ldots,x_{k})\prod_{i=0}^{k}z(x_{i},s_{i};x_{i+1},s_{i+1})\,dx_{1}\dots
dx_{k}\qquad P-\mbox{a.s.}$

###### Proof.

To keep the notation manageable, let

$\varphi=\varphi(b(s_{1}),\ldots,b(s_{k})),\qquad
Z_{i}=Z(b(s_{i}),s_{i};b(s_{i+1}),s_{i+1}).$

By expressing the integral as a $Q$-expectation and accounting for the heat
equation $q$ factors between $z$ and $Z$, the claim translates to

$E_{M}\varphi=E_{Q}\big{[}\,\varphi\,Z_{0}\cdots\,Z_{k}\,\big{]}\qquad
P-\mbox{a.s.}$ (98)

Let $F_{i}$ be a bounded nonnegative test function on $(\Xi,\mathcal{F},P)$
depending only on the noise at times in between $s_{i}$ and $s_{i+1}$. Let
$F=F_{0}\cdots F_{k}$. Let $\mathcal{S}\subset\mathcal{G}$ denote the
$\sigma$-field generated by $b(s_{1}),\ldots,b(s_{k})$. By definition of the
polymer measure, Fubini, and since $\varphi$ is $\mathcal{S}$-measurable, we
have

$\displaystyle E_{P}E_{M}[\varphi F]$ $\displaystyle=E_{P}E_{Q}[\varphi
F(\xi+m)]=E_{P}E_{Q}E_{Q}[\varphi F(\xi+m)|\mathcal{S}]$
$\displaystyle=E_{Q}\Big{[}\varphi\,E_{P}E_{Q}[F(\xi+m)|\mathcal{S}]\Big{]}.$
(99)

Given $\mathcal{S}$ and $\xi$ fixed, by the Markov property of $b$ the random
variables $F_{0}(\xi+m),\ldots,F_{k}(\xi+m)$ are independent over $Q$, so

$E_{Q}[F(\xi+m)|\mathcal{S}]=\prod_{i=0}^{k}E_{Q}[F_{i}(\xi+m)|\mathcal{S}].$

Because the noises between times $s_{i}$ and $s_{i+1}$ are $P$-independent as
$i$ varies, we have

$E_{P}E_{Q}[F(\xi+m)|\mathcal{S}]=\prod_{i=0}^{k}E_{P}E_{Q}[F_{i}(\xi+m)|\mathcal{S}]=\prod_{i=0}^{k}E_{P}[Z_{i}F_{i}],$

by the shift representation of the partition functions $Z_{i}$. Since the
factors are independent over $P$, substituting this into (99) we get

$E_{P}E_{M}[\varphi
F]=E_{Q}E_{P}[\varphi\prod_{i=0}^{k}Z_{i}F_{i}]=E_{P}E_{Q}[\varphi Z_{0}\cdots
Z_{k}F].$

As this holds for any product test function $F$, the claim (98) follows. ∎

Next, we prove the main convergence theorem.

###  7.3 Proof of Theorem 6, tightness and polymer limits

We continue the proof started in the introduction.

Tightness. We need to justify the interchange of limits in $n$ and $N$. For
the rest of the proof, we will assume that all scaling parameters are
standard, namely $\nu=\rho=\beta=s=1$, $x=t=0$ and only $y$ is free, since the
general argument is identical by scaling, see Lemma 70.

A condition for the interchange of limits is established in Proposition 46.2.
It is in terms of the self-intersection local times

$\alpha_{N}=\lim_{n\to\infty}\sum_{i=1}^{n}m_{N,i}m^{\prime}_{N,i},\qquad\alpha=\lim_{n\to\infty}\sum_{i=1}^{n}m_{i}m^{\prime}_{i}.$

defined on $\Omega^{2}$, with prime denoting independent copies. See Section
5.1 for more explanation about the $\alpha$s.

One condition is that the limits defining $\alpha_{N}$ and $\alpha$ should
exist in $L^{2}$; this is established in Propositions 66 and 68. Next, we need
$\alpha_{N}\to\alpha$ in law, which, by Lemma 45, follows from
$E\alpha_{N}^{2}\to E\alpha^{2}$. This is established by an explicit
computation in Proposition 71. Finally we need a tightness condition: there is
$\gamma>1$ so that given $p<1$ we have a set of paths $A_{n}$ for every $n$,
so that $QA_{n}>p$ and

$\limsup_{N\to\infty}\mathfrak{e}(\gamma\mathbf{1}_{A}\mathbf{m}_{N})<\infty.$

Since $\alpha_{N}$ are nonnegative, by Lemma 44 it suffices to show
$\limsup_{N}E[{e^{\gamma\alpha_{N}}}\mathbf{1}_{A\times A}]<\infty$. We take
$A$ to be a set of paths with bounded Hölder-$\kappa$ norm (91) for
$\kappa\in(0,1/2)$. The $L^{1}(P)$ convergence claim then follows from
Proposition 79.

Polymer convergence. For every $N$, the formula (9) gives a $P$-random measure
$M_{N}$ on $Q$. These measures are modifications of the law of the pair
$(b_{\rm sp},b_{\rm ti})$. Similarly, the limiting measure $M$ modifies the
law of this pair, but in a way that only depends on the first process.

Proposition 54 implies that $M_{n}\to M$ in probability with respect to weak
convergence of measures. In other words,

$(b_{\rm sp},b_{\rm ti})\mbox{ under }M_{N}\;\;\stackrel{{\scriptstyle\rm
law}}{{\Longrightarrow}}\;\;(b_{\rm sp},b_{\rm ti})\mbox{ under }M.$

By Theorem 4, under $M$, the process $b_{\rm sp}$ is the continuum directed
random polymer (CDRP), and $b_{\rm ti}$ is an independent Brownian bridge.
This implies the following laws converge in probability

$\displaystyle\Big{(}(N^{-1/2}$ $\displaystyle B_{N}(N\rho
r)_{1},N^{-3/2}B_{N}(N\rho r)_{2}),\;r\in[0,s]\Big{)}\mbox{ under $M_{N}$}$
$\displaystyle\;\;\stackrel{{\scriptstyle\rm
law}}{{\Longrightarrow}}\;\;\Big{(}(b_{\rm
sp}(t+r),t+r),\;r\in[0,s]\Big{)}\mbox{ under $M$}.$

This completes the proof of Theorem 6.

######  Remark 82.

The proof also shows that the fluctuations $b_{\rm ti}$ in the limiting time-
direction converge to an unweighted Brownian bridge independent of the
limiting polymer. Also, since $Z_{N}\to Z$ and $Z_{N},Z>0$ a.s. by the 0-1
law, Lemma 37, the normalized polymer measures converge as well:
$M_{N}/Z_{N}\to M/Z$.

Acknowledgments.  JQ and BV are partially supported by NSERC discovery grants.
AR is partially supported by Fondecyt 1220396.

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Jeremy Quastel, Department of Mathematics, University of Toronto, Canada,
<EMAIL_ADDRESS>

Alejandro Ramírez, Departamento de Matemáticas, Pontificia Universidad
Católica de Chile<EMAIL_ADDRESS>

Bálint Virág, Departments of Mathematics and Statistics, University of
Toronto, Canada,
<EMAIL_ADDRESS>