Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers

Introduction

where $\nu>0$ and $\sigma,\lambda\neq 0$ are fixed parameters and $\dot{W}(t,x)$ is Gaussian space-time white noise

It is widely studied in physics as a model of randomly growing interfaces. The derivative $u=\partial_{x}h$ should satisfy the stochastic Burgers equation,

Using renormalization group methods physicists have computed the dynamic scaling exponent (, , )

Roughly, this means that one expects non-trivial behavior under the rescaling

For the totally asymmetric exclusion process and the polynuclear growth models, which can be thought of as discretizations of (1.1), it is now known rigorously that in a weak sense,

for an explicit ${\rm v}$ and scaling function $g_{\rm sc}$ related to the Tracy-Widom distribution. Note that these models are in some sense exactly solvable.

(1.1) and (1.2) are ill-posed because the quadratic non-linear term cannot possibly make sense for a typical realization, which, in the case of (1.1) is expected to look, in $x$, locally, like a Brownian motion with variance $\nu^{-1}\sigma^{2}$. Formally applying the Hopf-Cole transformation

to (1.1) leads to the stochastic heat equation

The advantage is that (1.5) is well-posed . We do not attempt to justify the manipulations leading to (1.5). We define $h$ and $u=\partial_{x}h$ through (1.4). These Hopf-Cole solutions are expected to be the physically relevant solutions of (1.1) and (1.2).

$Z$ can be thought of as an asymptotic model of a directed polymer. There is a Feynman-Kac formula

where the expectation is over an independent Brownian motion $b(s)$, $s\geq 0$ starting at $x$, of variance $\nu$, $\beta=\lambda\nu^{-1}\sigma$, and $:\!\!\exp\!\!:$ is the Wick-ordered exponential (see for details). The reason we write (1.6) is to draw attention to the analogy with directed polymers, a typical model being

where $X(m,r)$, $m\in\{1,2,\ldots\}$, $r\in\{\ldots,-1,0,1,\ldots\}$ are independent and identically distributed random variables, and $s_{m}$ is a simple random walk starting at $x$. Assuming reasonable decay on the tails of the $X$’s, it is expected that for any $\beta$,

Little is known rigorously. Bounds analogous to $\chi\in[3/10,1/2]$ are obtained in , , , , and . The closest results with $\chi=1/3$ are those of for certain last passage percolation models, which are obtained in the $\beta\to\infty$ (zero-temperature) limit. Note the contrast with dimensions $d\geq 3$ where the polymer is known to be diffusive for small $\beta$.

2. Mathematical background

We now survey what is known rigorously about (1.1). In terms of well-posedness, the technology at the present time can only handle far smoother noise terms than the white noise. An unusual type of Wick product version of the problem has been introduced . But besides requiring fairly smooth noises, this does not have the scaling expected , and is therefore believed not to be physically relevant.

The idea in , which leads to what appears to be the physically relevant solution, is to smooth out the white noise in space a little, and then use the Hopf-Cole transformation and the tractability of (1.5) to remove the cutoff. As this is done, one finds one has to subtract a large constant from the equation. The resulting Hopf-Cole solution of (1.1) is given explicitly as the logarithm of the well-defined solution of (1.5). We now recall the details.

The following summarizes previous results, mostly from .

Let $h(0,x)$ be a random initial continuous function satisfying (1.11) and independent of the white noise $\dot{W}$.

2. The process $Z^{\kappa}(t,x)=\exp\{-\lambda\nu^{-1}h_{\kappa}(t,x)\}$ is the unique adapted (mild) solution of the Itô equation,

in the sense of distributions. If we start $Z$ with initial data $Z(0,x)=\exp\{B(x)\}$ where $B(x)$ is a two sided Brownian motion independent of $W$ with variance $\nu^{-1}\sigma^{2}$, then $u$ is stationary in both space and time. In this sense, Gaussian white noise with variance $\nu^{-1}\sigma^{2}$ is invariant for (1.2). Stationarity here means stationarity of $\langle\tau_{x}\varphi,u(t)\rangle$ for smooth functions of compact support $\varphi(x)$ where $(\tau_{x}\varphi)(y)=\varphi(y-x)$. The corresponding $h(t,x)$ and $Z(t,x)$ are not stationary in time with these initial data, but the increments $D_{\delta}h(t,x)=h(t,x+\delta)-h(t,x)$ are space and time stationary.

6. Let $h$ be the Hopf-Cole solution of (1.1), as in (1.14). Then

is the Hopf-Cole solution of (1.1) with new coefficients

If we start (1.2) in equilibrium, ie. $u(0)$ is a white noise with variance $\nu^{-1}\sigma^{2}$, then the time reversed process $u(T-t)$, $t\in[0,T)$ is a solution of (1.2) with $\lambda$ replaced by $-\lambda$, and the spatially reversed process $h(t,-x)\stackrel{{\scriptstyle\rm dist}}{{=}}h(t,x)$.

Note that only consider the case $\lambda=\nu=1/2$ and $\sigma=1$, but their proofs work in general. 6 is not stated there, but it follows readily from their methods.

The Hopf-Cole solution (1.14) of KPZ (1.1) is obtained as a limit of solutions of (1.12), i.e. after subtraction of a divergent term $C_{\kappa}(0)$. An important open problem is to show that a corresponding version of (1.1) with an appropriate Wick ordered nonlinearity is well-posed. We do not address this issue here. Since (1.14) is expected to be the relevant solution, we simply study it directly.

3. Statement of results.

We can now state our main results about $h$.

Let $h(t,x)$ be the Hopf-Cole solution of (1.1) as in (1.14) with $Z(t,x)$ the solution of (1.5), with initial data $Z(0,x)=\exp\{B(x)\}$ where $B(x)$ is a two sided Brownian motion independent of $W$ with variance $\nu^{-1}\sigma^{2}$. Let $\operatorname{Var}(h(t,x))$ denote the variance of $h(t,x)$. $\operatorname{Var}(h(t,x))$ is a symmetric function of $x$, non-decreasing in $|x|$, and

Furthermore, there exist $c_{0}=c_{0}(\sigma,\nu,\lambda)<\infty$, and $C_{1}=C_{1}(\sigma,\nu,\lambda)<\infty$, $C_{2}=C_{2}(m,\sigma,\nu,\lambda)<\infty$, such that for $t\geq c_{0}$ we have

The dependence of the constants $c_{0}$ and $C$ on $m,\sigma,\nu,\lambda$ is as follows: We can take

Theorem 1.20 tells us that the scaling exponent for the quantities studied here follows the physical prediction, as can be seen by integrating (1.3). This provides considerable support for the notion that this process $h$ is the sought after solution of (1.1), even though is not known presently how to show directly that $h(t,x)=-\lambda^{-1}\nu\log Z(t,x)$ solves (1.1), or indeed, what it means to solve (1.1).

Sometimes we want to indicate the dependence of the solution on the parameters by writing $h(t,x;\lambda,\nu,\sqrt{\nu})$. It is interesting to take $\lambda=1$, $\sigma^{2}=\nu$ because $h(t,x;1,\nu,\sqrt{\nu})=\nu h(\nu^{-3}t,\nu^{-2}x;1,1,1)$ corresponds to the solution of $\partial_{t}h=-(\partial_{x}h)^{2}+\nu\partial_{x}^{2}h+\sqrt{\nu}\dot{W}$ and note that the result implies that there is a $\nu_{0}>0$ and fixed $0<c_{1}\leq c_{2}<\infty$ such that $c_{1}t^{2/3}\leq\operatorname{Var}(h(t,0;1,\nu,\sqrt{\nu}))\leq c_{2}t^{2/3}$ for $t\geq 1$ for all $\nu\leq\nu_{0}$.

The following proposition provides us with our definition of the space-time correlation measure of the stochastic Burgers equation.

$S(t,\cdot)$ is symmetric: $S(t,A)=S(t,-A)$ for Borel sets $A$ where $-A=\{-x:x\in A\}$. The connection with the process $h$ is that

The route to constructing $S(t)$ and proving (1.22) is somewhat circuitous. The measure $S(t)$ is constructed as a weak limit from the rescaled correlations of a particle process. Then we show that, in the sense of distributions,

Finally, after studying solutions of the stochastic heat equation, we can deduce (1.22). Here are the bounds on $S(t)$.

Let $u(t)=\partial_{x}h(t)$ be the distributional derivative of the Hopf-Cole solution (1.14) of (1.1), in equilibrium with initial data white noise with variance $\nu^{-1}\sigma^{2}$ and let $S(t,dx)$ be the space-time correlation measure defined through (1.21). With the same constants as in Theorem 1.20, for $t\geq c_{0}$ we have

The upper bound holds for $1\leq m<3$. The lower bound holds for all $m\geq 1$. In particular, the diffusivity

We turn to proofs. The first issue is to develop the connection with the exclusion process.

Weakly asymmetric simple exclusion

Let $\hat{\eta}=2\eta-1$ and define the height function

where $v_{\varepsilon}=\frac{1}{2}\varepsilon^{-3/2}-\frac{1}{4!}\varepsilon^{-1/2}$ and the closest integer $[x]$ is given by

As $\varepsilon\searrow 0$, the distributions ${\mathcal{P}}_{\varepsilon}$ converge weakly to ${\mathcal{P}}$, the distribution of the Hopf-Cole solution $h$ (1.14) of (1.1) with $\lambda=1/2$, $\nu=1/2$, and $\sigma=1$.

This was proved in using the slightly different height function $\zeta^{\text{BG}}(t,x)$ related to ours by

with $[\cdot]$ defined by (2.3). The rescaled space-time correlation functions are given by

and a discrete absolute value in terms of the closest integer function by

We begin by building on some well-known properties. The assumption that density is $1/2$ is used repeatedly.

To see that $S_{\varepsilon}(t,x)$ is a probability density, use the well-known connection with the second class particle:

Here $\mathbf{P}^{1/2}_{\varepsilon}$ is the coupling measure of two ASEP’s that start with one discrepancy at the origin and Bernoulli(1/2) occupations elsewhere, and $\mathbf{x}(\cdot)$ is the position of the second class particle. This setting is discussed in Section 4. A proof of (2.10) can be found for example in . Symmetry of $S_{\varepsilon}(t,x)$ can be seen from the definition (2.5) and the fact that

defines an ASEP $\widetilde{\eta}$ equal in distribution to $\eta$. See for the explicit computation that proves 2.

We now work towards 3. Let $N(t,x)$ denote the current across the bond between site $x$ and $x+1$ up to time $t$. We start by checking that

With $\widetilde{\eta}$ as in (2.11), an $\widetilde{\eta}$-particle jump from $x$ to $y$ is the same as an $\eta$-hole jump from $-x$ to $-y$. Hence $\widetilde{N}(t,0)=N(t,-1)$. By the distributional equality $\widetilde{\eta}\overset{d}{=}\eta$, \operatorname{Cov}\Bigl{[}N(t,0),\sum_{y=-x+1}^{x}\eta(t,y)\Bigr{]}=\operatorname{Cov}\Bigl{[}\widetilde{N}(t,0),\sum_{y=-x+1}^{x}\widetilde{\eta}(t,y)\Bigr{]}=\operatorname{Cov}\Bigl{[}N(t,-1),-\,\sum_{y=-x}^{x-1}\eta(t,y)\Bigr{]}=-\,\operatorname{Cov}\Bigl{[}N(t,0),\sum_{y=-x+1}^{x}\eta(t,y)\Bigr{]} and (2.12) is verified.

The right-hand side of (2.13) is symmetric in $x$ by invariance under spatial translations.

Next, note that by the finite range of ASEP, for fixed $\varepsilon>0$ and $t\geq 0$ there exist $C_{1}<\infty$, $C_{2}>0$ such that

This proves (3). To prove (4), start with the observation

Then by (2) and by integration by parts (that is, by shifting the integration variable),

where $B_{\varepsilon,t,m}(N)$ are sums of integrals of $\lvert x\pm\varepsilon\rvert_{\varepsilon}^{m}[\operatorname{Var}(h_{\varepsilon}(t,x))-|x|_{\varepsilon}]$ over intervals of length $\varepsilon$ around $\pm N$. By 3 these are exponentially small in $N$ as $\varepsilon$, $t$ and $m$ are fixed. Taking $N\to\infty$ gives 4.

The key technical estimate which will be proved in Section 4 is

With the same constants as in Theorem 1.20, for all $0<\varepsilon<1/4$, $1\leq m<3$, and $t\geq c_{0}$,

1. For $0<\varepsilon<1/4$ and $t\geq c_{0}$,

2. For each $t>0$, the family of probability measures $\{S_{\varepsilon}(t,x)dx\}_{0<\varepsilon<1/4}$ is tight.

Part 1 follows from Theorem 2.17 and case $m=1$ of 4 of Proposition 2.2 because $\Delta_{\varepsilon}(|x|_{\varepsilon})=\varepsilon^{-1}$ for $x\in[-\varepsilon/2,\varepsilon/2)$ and vanishes elsewhere.

For $t\geq c_{0}$ tightness of $\{S_{\varepsilon}(t,x)dx\}_{0<\varepsilon<1/4}$ follows from the upper bound in (2.17). For $0<t<c_{0}$ recall the second class particle connection (2.10). Proposition 4 in proves that the second moment

is monotone nondecreasing in $t$. Thus the large-$t$ bound gives the tightness for all $t>0$.

Proofs of the main results

As a preliminary point we discuss the regularity of $\operatorname{Var}(h(t,x))$. The control comes from the weak limit $h_{\varepsilon}\to h$. We have

As mentioned in the proof of Corollary 2.4, this last quantity is nondecreasing in $t$. Consequently by the upper bound in (2.17) and the i.i.d. mean zero spatial increments of $h_{\varepsilon}(t,x)$ [see (2.1)] we conclude that $\operatorname{Var}(h_{\varepsilon}(x,t))$ is locally bounded as a function of $(t,x)$, uniformly in $\varepsilon>0$. By the weak limit $\operatorname{Var}(h(t,x))\leq\varliminf_{\varepsilon\to 0}\operatorname{Var}(h_{\varepsilon}(t,x))$ and so $\operatorname{Var}(h(x,t))$ is locally bounded. The $x$-symmetry of $\operatorname{Var}(h(t,x))$ follows from part 6 of Proposition 1.1, or from the weak limit $h_{\varepsilon}\to h$ and the distributional symmetry of $\zeta_{\varepsilon}(t,\,\cdot)$. For any fixed $x_{0}$, $h(t,x)-h(t,x_{0})$ is a Brownian motion in $x$ and hence the continuity of $x\mapsto\operatorname{Var}(h(t,x))$. By studying the stochastic heat equation we prove in the Appendix that

Let $S(t,dx)$ denote a weak limit point of $S_{\varepsilon}(t,x)dx$ as $\varepsilon\searrow 0$. Taking the limit in (3.2), the last expression becomes the right-hand side of (1.21).

(The constant term on the right-hand side of definition (2.2) vanishes since we are integrating the height function against a derivative.) This is a sum of independent mean zero random variables, and

which is bounded uniformly in $\varepsilon$.

Note first of all that from part 6 of Proposition 1.1, it suffices to prove all results with $\lambda=1/2$, $\nu=1/2$, $\sigma=1$.

The upper bounds of (1.19) and (1.24) follow from the weak convergence and from the upper bounds in (2.17) and in 1 of Corollary 2.4.

Let $1\leq m<3$. For the upper bound of (1.20) we collect these ingredients: Inequality $\operatorname{Var}(h_{\varepsilon}(t,x))-|x|_{\varepsilon}\geq 0$ from 3 of Proposition 2.2, the fact that under the weak limit

and for $x\neq 0$, $\Delta_{\varepsilon}(\lvert x\rvert_{\varepsilon}^{m})\to m(m-1)\lvert x\rvert^{m-2}/2$. Combine the upper bound in (2.17) with identity (2.9), let $\varepsilon\searrow 0$ in (2.9) and use Fatou’s Lemma.

To prove the lower bound of (1.24), let $t>c_{0}$ be fixed and choose a non-negative smooth function $f(x)$ with compact support such that $f(x)\geq|x|^{m}$ for $|x|\leq At^{2/3}$. We have

Choose $\delta>0$ such that $m+\delta<3$. By Chebyshev’s inequality and Theorem 2.17,

Since this is true for all such $f$, we conclude that the lower bound of (1.24) holds.

We are unable to do this by direct approximation due to lack of control of moments of $h_{\varepsilon}(t,x)$ higher than $2$. By direct calculation $E[h_{\varepsilon}(t,x)]=t/4!$ and we can take the $\varepsilon\searrow 0$ limit by uniform integrability that follows from the boundedness of $\operatorname{Var}(h_{\varepsilon}(t,x))$ argued in the beginning of this section. Consequently

Since increments are mean zero and stationary in space (part 5 of Proposition 1.1), the latter is equal to

Define the “tent function” $\varphi_{\delta}(x)=(\delta^{-1}-\delta^{-2}|x|)1_{|x|\leq\delta}$. We have shown that

with $\ast$ denoting convolution. Let $\delta\searrow 0$. Since $\operatorname{Var}(h(t,x))$ is locally bounded we can take the limit on the left. In the limit we obtain

This completes the proof of Proposition 1.6.

We now complete the proof of Proposition 1.6 and Theorem 1.20 with the following

This proposition implies the remaining parts of Theorem 1.20 because symmetry implies

From this follow $\operatorname{Var}(h(t,x))-\lvert x\rvert\geq 0$ and the identities

Then we can apply the bounds from (1.24).

Suppose first that $v^{\prime\prime}/2$ is a continuous probability density. Then

satisfies $g^{\prime\prime}=v^{\prime\prime}$ and thereby

for constants $a$, $b$. From symmetry deduce $a=1$. Taking $x=0$ identifies $b=v(0)-\tfrac{1}{2}\int_{\infty}^{\infty}\lvert z\rvert v^{\prime\prime}(z)\,dz.$ Now (3.6) holds for smooth $v$. Take a symmetric compactly supported smooth approximate identity $\{\phi_{\delta}\}_{\delta>0}$, apply (3.6) to $\phi_{\delta}*v$ and let $\delta\searrow 0$.

Continuing the proof of Proposition 3.5, apply (3.6) to $v(x)=\operatorname{Var}(h(t,x))$ to get

From the Appendix we get $\operatorname{Var}(h(t,x))-\lvert x\rvert\to 0$. Combining this with above gives first

Second class particle estimate

In this section we prove the key estimate for the moment of a second class particle. The context is the asymmetric simple exclusion process (ASEP) jumping to the right with rate $p=1/2$ and to the left with rate $q=1/2+\varepsilon^{1/2}$. Throughout $\varepsilon\in(0,1/4)$, with the real interest being the limit $\varepsilon\searrow 0$. Probabilities associated to this process are denoted by $P_{\varepsilon}^{\rho}$ when the process is stationary with Bernoulli $\rho$ occupations. The macroscopic flux function is $H_{\varepsilon}(\rho)=-\varepsilon^{1/2}\rho(1-\rho)$ and the characteristic speed $V^{\rho}_{\varepsilon}=H_{\varepsilon}^{\prime}(\rho)=-\varepsilon^{1/2}(1-2\rho)$.

Let $\mathbf{P}^{\rho}_{\varepsilon}$ denote the probability measure of the basic coupling of two processes $\zeta^{-}(t)\leq\zeta(t)$ with this initial configuration: $\zeta^{-}(0,0)=0<1=\zeta(0,0)$, and for $x\neq 0$, $\zeta^{-}(0,x)=\zeta(0,x)$ have mean $\rho$ and they are independent across the sites $x$. Let ${\mathbf{x}}(t)$ denote the position of the discrepancy between $\zeta^{-}(t)$ and $\zeta(t)$, in other words, the position of the second class particle started at the origin. The mean speed of the second class particle is the characteristic speed (Corollary 2.5 in or Theorem 2.1 in ):

From (2.10), Theorem 2.17 is equivalent to the case $\rho=1/2$ of the following theorem.

With the same constants as in Theorem 1.20, for all $0<\varepsilon<1/4$, $1\leq m<3$, and $t\geq c_{0}\varepsilon^{-2}$,

The remainder of the section proves Theorem 4.11, with separate subsections for the upper and lower bound.

Let $B\in(0,\infty)$. There exists $C\in(0,\infty)$ and $c_{1}(B)\in(0,\infty)$ such that the following bounds hold for all $0<\rho<1$, $u\geq 1$, $0<\varepsilon<1/4$, and $t\geq c_{1}(B)\varepsilon^{-1/2}$.

(i) For $B\varepsilon^{1/3}t^{2/3}\leq u\leq 20t/3$,

First we obtain the bounds for $\mathbf{P}^{\rho}_{\varepsilon}({\mathbf{x}}(t)\leq V^{\rho}_{\varepsilon}t-u)$. By an adjustment of the constant $C$ we can assume that $u$ is a positive integer. Fix a density $0<\rho<1$ and let $\lambda\in(0,\rho)$. Consider a basic coupling of three ASEP’s $\zeta\geq\zeta^{-}\geq\eta$ with this initial configuration:

(a) Initially $\{\zeta(0,x):x\neq 0\}$ are i.i.d. Bernoulli($\rho$) and $\zeta(0,0)=1$.

(b) Initially $\zeta^{-}(0,x)=\zeta(0,x)-\delta_{0}(x)$.

(c) Initially $\{\eta(0,x):x\neq 0\}$ are i.i.d. Bernoulli($\lambda$) and $\eta(0,0)=0$. The coupling of the initial occupations is such that $\zeta(0,x)\geq\eta(0,x)$ for all $x\neq 0$.

Recall that basic coupling means that the processes share common Poisson clocks.

Let these second class particles preserve their labels in the dynamics and stay ordered. Thus the $\zeta(t)$ configuration consists of first class particles (the $\eta(t)$ process) and second class particles (the $X_{j}(t)$’s). Let $\mathbf{P}_{\varepsilon}$ denote the joint probability distribution of these coupled processes. The marginal distribution of $(\zeta,\zeta^{-},{\mathbf{x}})$ under $\mathbf{P}_{\varepsilon}$ is the same as under $\mathbf{P}_{\varepsilon}^{\rho}$.

Basic coupling preserves $\mathbf{x}(t)\in\{X_{j}(t)\}$. Define the label $m(t)$ by $\mathbf{x}(t)=X_{m(t)}(t)$ with initial value $m(0)=0$. The label $m(t)$ performs a walk on the labels of the $\{X_{j}\}$ with rates $p$ to the left and $q$ to the right, but jumps permitted only when $X_{j}$ particles are adjacent. Through a comparison with a reversible walk, Lemma 5.2 in gives the bound

To get the first step of the estimation, note that if $\mathbf{x}(t)\leq V^{\rho}_{\varepsilon}t-u$ and $m(t)>-k$, then $X_{-k}(t)<\lfloor{V^{\rho}_{\varepsilon}t}\rfloor-u$. Then among the $\zeta-\eta$ particles only $X_{-k+1},\dotsc,X_{0}$ could have crossed from the left side of $1/2$ to the right side of $\lfloor{V^{\rho}_{\varepsilon}t}\rfloor-u+1/2$ during time $(0,t]$. Thereby $J^{\zeta-\eta}_{\lfloor{V^{\rho}_{\varepsilon}t}\rfloor-u}(t)\leq k$, and by an appeal to (4.14) we have

Case 1. $B\varepsilon^{1/3}t^{2/3}\leq u\leq 5\rho\varepsilon^{1/2}t$. Note that $20t/3>5\rho\varepsilon^{1/2}t$. Choose

By assuming $t\geq C(B)\varepsilon^{-1/2}$ we guarantee that $u\geq 1$ and

In the next inequality below the $-3$ in the definition (4.17) of $k$ absorbs $c_{1}$ from line (4.16). Let $\overline{X}=X-EX$ denote a centered random variable. Continuing with the probability from line (4.15):

$C$ is a constant that can change from line to line but is independent of all parameters.

We develop bounds on the variances above, first for $J^{\zeta}$. Utilize the coupling with a stationary density $\rho$ process. Then apply the basic identity

that links the variance of the current with the second class particle. (This is proved in Corollary 2.4 in and in Theorem 2.1 in .) We find

For the second variance on line (4.19) we begin in the same way:

Here we switched to a stationary density $\lambda$ process and introduced a second class particle ${\mathbf{x}}^{\lambda}$ in this process. In order to get the same bound as on line (4.21) we wish to switch from ${\mathbf{x}}^{\lambda}(t)$ to the second class particle ${\mathbf{x}}(t)$ in the density-$\rho$ process. To this end we utilize a coupling developed in Section 3 of . Because the density $\rho$ process has higher particle density than the density $\lambda$ process, the second class particle in density $\lambda$ moves on average faster in the direction of the drift. Theorem 3.1 of allows us to couple ${\mathbf{x}}^{\lambda}$ and ${\mathbf{x}}$ so that ${\mathbf{x}}(t)\geq{\mathbf{x}}^{\lambda}(t)$ with probability 1. Thus continuing from line (4.22),

Now \mathbf{E}_{\varepsilon}\bigl{[}{\mathbf{x}}(t)-{\mathbf{x}}^{\lambda}(t)\bigr{]}=(V^{\rho}_{\varepsilon}-V^{\lambda}_{\varepsilon})t=2\varepsilon^{1/2}t(\rho-\lambda) and from the choice (4.17) of $\lambda$, $2\varepsilon^{1/2}t(\rho-\lambda)\leq u$, hence

Insert bounds (4.21) and (4.23) into (4.19) to get

Insert (4.18) and (4.25) into line (4.15) to get

and we have verified (4.12) for $\mathbf{P}^{\rho}_{\varepsilon}({\mathbf{x}}(t)\leq V^{\rho}_{\varepsilon}t-u)$ for Case 1.

Case 2. $u\geq 5\rho\varepsilon^{1/2}t$. Let $Z_{t}$ be a nearest-neighbor random walk with rates $p=1/2$ to the right and $q=1/2+\varepsilon^{1/2}$ to the left. We have the stochastic domination $Z_{t}\leq{\mathbf{x}}(t)$ because no matter what the environment next to ${\mathbf{x}}(t)$, it has a weaker left drift than $Z_{t}$. Then, since $V^{\rho}_{\varepsilon}=-\varepsilon^{1/2}(1-2\rho)$, $2\rho\varepsilon^{1/2}t\leq 2u/5$, and $\varepsilon<1/4$,

For $\alpha\in(0,1]$, utilizing $({e^{\alpha}+e^{-\alpha}})/2\leq 1+\alpha^{2}$ and $e^{-\alpha}\geq 1-\alpha$,

We can estimate P\{Z_{t}\leq-\varepsilon^{1/2}t-\tfrac{3}{5}u\}\leq\exp\bigl{(}-\tfrac{3}{5}\alpha u+2\alpha^{2}t\bigr{)} and choose $\alpha=1\wedge\frac{3u}{20t}$ to obtain

Combining (4.26) and (4.28) gives Lemma 4.13 for $\mathbf{P}^{\rho}_{\varepsilon}({\mathbf{x}}(t)\leq V^{\rho}_{\varepsilon}t-u)$.

The corresponding upper tail bound $\mathbf{P}^{\rho}_{\varepsilon}({\mathbf{x}}(t)-V^{\rho}_{\varepsilon}t\geq u)$ is obtained from that for $\mathbf{P}^{\rho}_{\varepsilon}({\mathbf{x}}(t)-V^{\rho}_{\varepsilon}t\leq-u)$ by a particle-hole interchange followed by a reflection of the lattice. For details we refer to Lemma 5.3 in . This completes the proof of Lemma 4.13.

Integrate Lemma 4.13 to get the bound (4.11) on the moments of the second class particle. First for $m=1$.

$C_{1}(B)$ is a new constant that depends on $B$. Set $B=C^{1/3}$ to obtain

We can fix a constant $c_{0}$ large enough so that, for a new constant $C$,

Restrict to $t$ that satisfy this requirement and substitute (4.29) into Lemma 4.13. Then upon using $u\geq B\varepsilon^{1/3}t^{2/3}$ and redefining $C$ once more, we have for $B\varepsilon^{1/3}t^{2/3}\leq u\leq 20t/3$:

Now take $1<m<3$ and use (4.30) together with Lemma 4.13

This gives $\mathbf{E}_{\varepsilon}^{\rho}\lvert{\mathbf{x}}(t)-V^{\rho}_{\varepsilon}t\rvert^{m}\leq\frac{C}{3-m}\varepsilon^{m/3}t^{2m/3}$ provided $t\geq c_{0}\varepsilon^{-2}$ for a large enough $c_{0}$.

2. Proof of the lower bound of Theorem 4.11

By Jensen’s inequality it suffices to prove the lower bound for $m=1$. Let $C_{UB}$ denote the constant in the upper bound statement that we just proved. We can also assume $c_{0}\geq 1$. Fix a constant $b>0$ and set

Fix a density $\rho\in(0,1)$ and define an auxiliary density $\lambda=\rho-bt^{-1/3}\varepsilon^{-1/6}$. Define positive integers

Construct a basic coupling of three processes $\eta\leq\eta^{+}\leq\zeta$ with the following initial state:

(a) Initially $\eta$ has i.i.d. Bernoulli($\lambda$) occupations $\{\eta(0,x):x\neq n\}$ and $\eta(0,n)=0$.

(c) Initially $\zeta$ has independent occupation variables, coupled with $\eta(0)$ as follows:

(c.1) $\zeta(0,x)=\eta(0,x)$ for $0\leq x<n$ and $\zeta(0,n)=1$.

(c.2) For $x>n$ and $x<0$ variables $\zeta(0,x)$ are i.i.d. Bernoulli($\rho$) and $\zeta(0,x)\geq\eta(0,x)$.

Let again the random label $m(t)$ satisfy ${\mathbf{x}}^{(n)}(t)=X_{m(t)}(t)$, with initial value $m(0)=0$. In basic coupling $m(\cdot)$ jumps to the left with rate $q$ and to the right with rate $p$, but only when there is an $X$ particle adjacent to $X_{m(\cdot)}$. As in the proof of the upper bound, Lemma 5.2 in gives the bound

By the upper bound already proved and by the choice of $a_{1}$,

This gives a lower bound for the complementary event,

The reasoning behind the second inequality above is as follows: ${\mathbf{x}}^{(n)}(t)>\lfloor{V^{\rho}t}\rfloor$ and $m(t)<k$ imply $X_{k}(t)>\lfloor{V^{\rho}t}\rfloor$ and consequently $-k\leq J^{\zeta-\eta}_{\lfloor{V^{\rho}t}\rfloor}(t)=J^{\zeta}_{\lfloor{V^{\rho}t}\rfloor}(t)-J^{\eta}_{\lfloor{V^{\rho}t}\rfloor}(t)$.

Put $k=\lfloor{a_{2}t^{1/3}\varepsilon^{1/6}}\rfloor-2$. Observe from (4.33) that $\mathbf{P}\{m(t)\geq k\}\leq e^{-2}<1/4$ follows from $a_{2}t^{1/3}\varepsilon^{1/6}\geq 2\varepsilon^{-1/2}+3$, which is guaranteed by $t\geq c_{0}\varepsilon^{-2}$ and the definition of $a_{2}$. Hence

Consider line (4.36). The $\eta$ process can be coupled with a stationary $P^{\lambda}$-process with at most one discrepancy. The mean current in the stationary process is

After Chebyshev above we applied the basic identity (4.20) for which we introduced a second class particle ${\mathbf{x}}(t)$ in a density $\lambda$ system under the measure $\mathbf{P}^{\lambda}$. Then we replaced $\lfloor{V^{\rho}t}\rfloor$ with $V^{\lambda}t$ and applied the upper bound and properties of $a_{2}$.

Put this last bound back into line (4.36) to get

Let $\gamma$ denote the distribution of the initial $\zeta(0)$ configuration described by (a)–(c) in the beginning of this section. As before $\nu^{\rho}$ is the density $\rho$ i.i.d. Bernoulli measure. The Radon-Nikodym derivative is

Here condition $t\geq c_{0}\varepsilon^{-2}$ implies a bound $c_{2}(\rho)<\infty$ independent of $t$ and $\varepsilon$. From (4.38) and Schwarz’s inequality

Continue from line (4.40), recalling (4.31):

This completes the proof of the lower bound and thereby the proof of Theorem 4.11.

Appendix: Properties of the solution

Define the current of $u=\partial_{x}h$ across $x$ up to time $t$ by

and the mass of $u$ in the interval $[0,x]$ at time $t$ by

${\rm Var}(h(t,x))-|x|=\operatorname{Cov}(N(t,0),N(t,x))$.

Since $h(0,0)=0$ we have $h(t,x)=M(t,x)+N(t,0)$. From the invariance of white noise ($5$ of Prop 1.1), ${\rm Var}(M(t,x))=|x|$. Hence

To see this, note that we always have the conservation law

But $\operatorname{Cov}(M(t,x),(M(t,x)-M(0,x)))=\frac{1}{2}{\rm Var}(M(t,x)-M(0,x))$. By the conservation law again $\frac{1}{2}{\rm Var}(M(t,x)-M(0,x))=\frac{1}{2}{\rm Var}(N(t,x)-N(t,0))$. Finally, by the translation invariance, $\frac{1}{2}{\rm Var}(N(t,x)-N(t,0))={\rm Var}(N(t,0))-\operatorname{Cov}(N(t,x),N(t,0))$. This gives (5.3).

From (5.3) we can rewrite the right hand side of (5.2) as

The proof is completed by noting that the second term vanishes by symmetry. To see it, note that ${\rm Var}(h(t,-x))={\rm Var}(h(t,x))$ and by translation invariance $\operatorname{Cov}(N(t,-x),N(t,0))=\operatorname{Cov}(N(t,x),N(t,0))$. Hence $\operatorname{Cov}(M(t,x),N(t,0)+N(t,x))=\operatorname{Cov}(M(t,-x),N(t,0)+N(t,-x))$. But translating by $x$ gives $\operatorname{Cov}(M(t,-x),N(t,0)+N(t,-x))=\operatorname{Cov}(-M(t,x),N(t,x)+N(t,0))$.

$\lim_{|x|\to\infty}\operatorname{Cov}(N(t,0),N(t,x))=0.$

The two propositions combine to prove (5.1).

Let $Z_{i}(t,x)$, $i=1,2$, be the solutions of (1.5) with $W_{i}$, $i=1,2$ and initial data $Z_{i}(0,x)=\exp\{B_{i}(x)\}$, where $(W_{1},dB_{1})$ and $(W_{2},dB_{2})$ are the same on $(-\infty,R)$ and independent on $(R,\infty)$. Then there is a finite $C$ such that for $R\geq|x|+2t$,

Let $p(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-x^{2}/2t}$ be the heat kernel. We can write

First we obtain a preliminary bound on $E[Z^{2}_{i}(t,x)]$. By Schwarz’s inequality it is bounded above by (dropping the $i$ for clarity),

Call $g(t,x)=E[Z^{2}_{i}(t,x)]$ and let $P_{t}$ denote the heat semigroup. $g(0,x)=e^{2|x|}$ and have shown that

Iterating once we see that this is bounded above by

The last term can be simplified by noting that $P_{t-s}P_{s-u}=P_{s-u}$, applying Fubini’s theorem, and using $\int_{0}^{s}{\scriptstyle\frac{du}{\sqrt{(t-s)(s-u)}}}=\pi$. The result is

If we let $\bar{g}(0)=g(0)$ and $\bar{g}(t)$ satisfy (5.9) with equality instead of inequality, then $\bar{g}-g$ satisfies $(\bar{g}-g)(t)\geq\int_{0}^{t}P_{t-s}(\bar{g}-g)(s)ds$ with $(\bar{g}-g)(0)=0$. By the maximum principle for the heat equation, $g\leq\bar{g}$. $\bar{g}(t,x)$ is readily computed with the result that for some finite $C$,

By (5.6) again we have $f(t,x):=E[(Z_{1}(t,x)-Z_{2}(t,x))^{2}]$ is bounded above by twice

Explicit computation gives that (5.11) is equal to

By Schwarz’s inequality, $(\ref{p1})\leq 2\int p(t,x-y)1_{\{y\geq R\}}e^{2y}dy$. Another explicit computation gives that (5.12) is equal to

Hence $f(t)$ satisfies the same equation as $g(t)$ in (5.9) except that this time $f(0,x)\leq 1_{\{y\geq R\}}e^{2y}$. The same argument now shows that there is a finite $C$ such that $f(t,x)\leq Ce^{-R+C(t+|x|)}$ for $R\geq|x|+2t$.

Let us use the notation $\bar{N}(t,x)$ for the normalized current $N(t,x)-E[N(t,x)]$. First of all note that by (1.19) and $5$ of Proposition 1.1, $E[\bar{N}^{2}(t,x)]\leq C(t)$ and does not depend on $x$. Now let $(W_{1},dB_{1})$, $(W_{2},dB_{2})$, $(W_{3},dB_{3})$ be coupled so that $(W_{1},dB_{1})$ and $(W_{2},dB_{2})$ are the same on $(-\infty,x/2)$ and independent on $(x/2,\infty)$, $(W_{2},dB_{2})$ and $(W_{3},dB_{3})$ are the same on $(x/2,\infty)$ and independent on $(-\infty,x/2)$, and $(W_{1},dB_{1})$ and $(W_{3},dB_{3})$ are independent. Let $\bar{N}_{1}$, $\bar{N}_{2}$, $\bar{N}_{3}$ be the currents corresponding to the three different pairs. Of course $\operatorname{Cov}(N(t,0),N(t,x))=E[\bar{N}_{1}(t,0)\bar{N}_{1}(t,x)]$. By Schwarz’s inequality

By independence, $E[\bar{N}_{1}(t,0)\bar{N}_{3}(t,x)]=0$. By symmetry,

For each $L>0$, let $\log_{L}z=\log z$ for $z\geq L^{-1}$ and $\log_{L}z=-\log L$ for $0<z<L^{-1}$. We have

Because $\log_{L}$ is Lipschitz with constant $L$ we have

By Lemma 5.3, for each fixed $L$, this vanishes as $|x|\to\infty$. On the other hand, since $E[(\log Z_{1}(t,0))^{2}]<\infty$, by the dominated convergence theorem,

References