# Celebratio Mathematica

### Large deviation and homogenization

#### by Fraydoun Rezakhanlou

The main theme of this ex­pos­it­ory art­icle is to re­view some of Raghu Varadhan and his col­lab­or­at­ors’ con­tri­bu­tions to the ques­tion of ho­mo­gen­iz­a­tion for the fol­low­ing stochast­ic mod­els:

• sta­tion­ary Hamilton–Jac­obi (HJ) and Hami­lon–Jac­obi–Bell­man (HJB) equa­tion;
• ran­dom walk in ran­dom en­vir­on­ment (RWRE);
• simple ex­clu­sion pro­cess (SEP).

All the above mod­els share sim­il­ar scal­ing be­ha­vi­ors and, in some sense, rep­res­ent evolving height func­tions which are gov­erned by loc­al and ran­dom growth rules. In fact, the law of a RWRE sat­is­fies an equa­tion which re­sembles a dis­crete HJB equa­tion, and the growth rates of the particle cur­rents in SEP are de­scribed by a non­lin­ear func­tion of the height dif­fer­ences. Re­view­ing Raghu Varadhan’s fun­da­ment­al con­tri­bu­tions sheds light on some uni­ver­sal be­ha­vi­or of stochast­ic growth mod­els.

#### The Hamilton–Jacobi and Hamilon–Jacobi–Bellman equations

To in­tro­duce the ba­sic idea be­hind ho­mo­gen­iz­a­tion, we first con­sider the (in­homo­gen­eous) Hamilton–Jac­obi (HJ) equa­tion, $$\label{eq1.1} u_t = H(x,u_x),$$ where $H$ is sta­tion­ary and er­god­ic in the first vari­able $x$. More pre­cisely, we have a prob­ab­il­ity space $(\Omega,\mathcal{F},\mathbb{P})$ with $\mathcal{F}$ a Borel $\sigma$-field on $\Omega$ and $\mathbb{P}$ a prob­ab­il­ity meas­ure on $(\Omega,\mathcal{F})$, which is in­vari­ant with re­spect to a fam­ily of trans­la­tion op­er­at­ors; that is, for every $x \in \mathbb{R}^d$, there ex­ists a meas­ur­able func­tion $\tau_x: \Omega \to \Omega$ so that $\tau_x \circ \tau_y = \tau_{x+y}$, and $\mathbb{P}(\tau_xA) = \mathbb{P}(A)$ for every $A \in \mathcal{F}$ and $x,y \in \mathbb{R}^d$. We also as­sume that $\tau_x$ is er­god­ic; that is, $\tau_xA = A$ for all $x \in \mathbb{R}^d$ im­plies that either $\mathbb{P}(A) = 1$ or 0.

Now, $H(x,p,{\omega}) = H_0(\tau_x{\omega},p)$ where $H_0: \Omega \times \mathbb{R}^d \to \mathbb{R}$ is a meas­ur­able func­tion. We think of $(x,t,u)$ as the mi­cro­scop­ic co­ordin­ates, with the graph of $u(\,\cdot\,,t)$ rep­res­ent­ing a ran­dom in­ter­face. To switch to mac­ro­scop­ic co­ordin­ates, we set $$\label{eq1.2} u^{\varepsilon}(x,t;{\omega}) = \varepsilon u\Bigl( \frac {x}{\varepsilon},\frac {t}{\varepsilon};{\omega}\Bigr).$$ We now have $$\label{eq1.3} u_t^{\varepsilon} = H\Bigl( \frac {x}{\varepsilon}, u_x^{\varepsilon}\Bigr).$$ We note that the right-hand side of \eqref{eq1.3} fluc­tu­ates greatly over mac­ro­scop­ic shifts in the po­s­i­tion $x$. The huge fluc­tu­ation in $H$, though, does not ne­ces­sar­ily im­ply cor­res­pond­ingly huge fluc­tu­ations in $u^{\varepsilon}$. This is the ho­mo­gen­iz­a­tion phe­nomen­on; that is, we ex­pect $u^{\varepsilon} \to {\bar u}$ as $\varepsilon \to 0$, with ${\bar u}$ solv­ing a ho­mo­gen­ized HJ equa­tion $$\label{eq1.4} {\bar u}_t = {\bar H}({\bar u}_x),$$ where ${\bar H}: \mathbb{R}^d \to \mathbb{R}$ is the ho­mo­gen­ized Hamilto­ni­an and does not de­pend on $\omega$.

As our second ex­ample, we con­sider the Hamilton–Jac­obi–Bell­mann equa­tion $$\label{eq1.5} u_t = H(x,u_x) + \tfrac {1}{2} \mathop{\Delta} u,$$ with $H(x,p) = H(x,p,{\omega})$ as be­fore. We define $u^{\varepsilon}$ as in \eqref{eq1.2}, and then \eqref{eq1.3} be­comes $$\label{eq1.6} u_t^{\varepsilon} = H\Bigl( \frac {x}{\varepsilon},u_x^{\varepsilon}\Bigr) + \frac {\varepsilon}{2} \mathop{\Delta} u^{\varepsilon}.$$ Again, we ex­pect to have $u^{\varepsilon} \to {\bar u}$, with ${\bar u}$ sat­is­fy­ing an equa­tion of the form \eqref{eq1.4} for a dif­fer­ent ho­mo­gen­ized Hamilto­ni­an ${\bar H}$. In­deed, the ho­mo­gen­iz­a­tions for both \eqref{eq1.3} and \eqref{eq1.6} have been achieved by Sou­gan­id­is [e4], Reza­khan­lou and Tarv­er [e6], Lions and Sou­gan­id­is [e11], and Kosy­gina, Reza­khan­lou and Varadhan [5], provided that $H(x,p)$ is con­vex in $p$ and sat­is­fies suit­able tech­nic­al as­sump­tions on which we do not elab­or­ate here. (See also Kosy­gina and Varadhan [8] when $H$ is al­lowed to de­pend on the time vari­able.) Not­ably, [5] ob­tains a vari­ation­al for­mula for ${\bar H}$. In the case of \eqref{eq1.6}, ${\bar H}$ is giv­en by $$\label{eq1.7} {\bar H}(p) = \inf_g \mathop{\mathrm{esssup}}_\omega \Bigl[H_0\bigl(p+ g({\omega}),{\omega}\bigr) + \tfrac {1}{2} \nabla\cdot g(\omega)\Bigr]$$ where the es­sen­tial su­prem­um is taken with re­spect to the prob­ab­il­ity meas­ure $\mathbb{P}$, and the in­fim­um is taken over func­tions $g: \Omega \to \mathbb{R}^d$ such that $\mathbb{E} \,g=0$ and $\nabla\cdot g=0$ weakly. Here, $\nabla$ is the gen­er­at­or of the group $\{\tau_x\}$; that is, $$\label{eq1.8} \nabla f({\omega}) \cdot v = \lim_{t \to 0} \frac {1}{t}\bigl(f(\tau_{tv}{\omega}) - f({\omega})\bigr)$$ whenev­er the lim­it ex­ists. We ex­pect a sim­il­ar for­mula to hold in the case of \eqref{eq1.3}, namely, $$\label{eq1.9} {\bar H}(p) = \inf_g \mathop{\mathrm{esssup}}_w\, H_0\bigl(p+g({\omega}),{\omega}\bigr).$$ Be­fore we turn to our next mod­el, we make an ob­ser­va­tion re­gard­ing the ho­mo­gen­iz­a­tion of \eqref{eq1.6}. Note that, if $$\label{eq1.10} H(x,p,{\omega}) = \tfrac {1}{2} |p|^2 + b(x,{\omega}) \cdot p + V(x,{\omega})$$ and $u$ is a solu­tion of \eqref{eq1.5}, then, by the Hopf–Cole trans­form, the func­tion $w = e^u$ solves $$\label{eq1.11} w_t = \tfrac {1}{2} \mathop{\Delta} w + b(x,{\omega}) \cdot \nabla w + V(x,{\omega})\,w(x,{\omega}).$$ By the Feyn­mann–Kac for­mula, there is a prob­ab­il­ist­ic rep­res­ent­a­tion for $w$ us­ing a dif­fu­sion with a drift $b$. More pre­cisely, if $X(t,x;{\omega})$ de­notes the solu­tion to \begin{align} \label{eq1.12} dX(t) & = b(X(t),{\omega})\,dt + d\beta(t), \\ X(0) & = x, \nonumber \end{align} then $$\label{eq1.13} w(x,t;{\omega}) = E^\omega w(X(t,x;{\omega}),0)\exp\Bigl(\int_0^t V\bigl(X(s,x;{\omega}),{\omega}\bigr)\,ds\Bigr).$$ Here, $\beta$ is a stand­ard Browni­an mo­tion, and $E^\omega$ de­notes the ex­pec­ted value for the pro­cess $X(t)$. The func­tion $V$ is the po­ten­tial and, if $V \le 0$, then $-V$ may be in­ter­preted as a killing rate for the dif­fu­sion $X$. With this in­ter­pret­a­tion, $w(x,t;{\omega})$ is the ex­pec­ted value of $w({\hat X}(t),0)$, with ${\hat X}$ de­not­ing the dif­fu­sion with the killing. We now would like to use our prob­ab­il­ist­ic rep­res­ent­a­tion to re­write $u^{\varepsilon}$. If $$\label{eq1.14} u^{\varepsilon}(x,0;{\omega}) = f(x)$$ for a de­term­in­ist­ic ini­tial con­di­tion $f$, then \begin{align} \label{eq1.15} u^{\varepsilon}(x,t;{\omega}) = \varepsilon \log E^\omega \exp\Bigl[ & \varepsilon^{-1}f(\varepsilon X(t/\varepsilon,x/\varepsilon;{\omega})) \\& + \int_0^{t/\varepsilon}V(X(s,x/\varepsilon;{\omega}),\omega)\,ds \Bigr] \nonumber \end{align} In par­tic­u­lar, \begin{align} \label{eq1.16} u^{\varepsilon}(0,1;{\omega}) = \varepsilon \log E^\omega \exp\Bigl[ & \varepsilon^{-1}f(X(\varepsilon^{-1};{\omega})) \\& + \int_0^{\varepsilon^{-1}} V(X(s;{\omega}),\omega)\,ds \Bigr] \nonumber \end{align} where $X(s;{\omega}) := X(s,0;{\omega})$ is the dif­fu­sion start­ing from the ori­gin. On the oth­er hand, since ${\bar H}$ is con­vex (which is evid­ent from \eqref{eq1.7}) we may use the Hope–Lax–Olein­ik for­mula to write $$\label{eq1.17} {\bar u}(x,t) = \sup_y \Bigl( f(y) - t {\bar L} \Bigl( \frac {y-x}{t}\Bigr)\Bigr),$$ where ${\bar L}$ is the con­vex con­jug­ate of ${\bar H}$. In par­tic­u­lar, $$\label{eq1.18} \lim_{\varepsilon \to 0} u^{\varepsilon}(0,1;{\omega}) = {\bar u}(0,1) = \sup_y \bigl(f(y)-{\bar L}(y)\bigr).$$ By a cel­eb­rated lemma of Varadhan, \eqref{eq1.18} is equi­val­ent to say­ing that, for al­most all $\omega$, the dif­fu­sion ${\hat X}$ sat­is­fies a large-de­vi­ation prin­ciple with rate func­tion ${\bar L}$. When $b \equiv 0$ and $$\label{eq1.19} -V(x,{\omega}) = \sum_{j \in I} V_0(x-x_j),$$ with $\omega = \{x_j: j \in I\}$ be­ing a Pois­son point pro­cess and $V_0$ a con­tinu­ous func­tion of com­pact sup­port, the large-de­vi­ation prin­ciple for ${\hat X}$ was earli­er es­tab­lished by Szn­it­man [e2].

In words, the large-de­vi­ation prin­ciple for the dif­fu­sion ${\hat X}(\,\cdot\,;{\omega})$ is equi­val­ent to ho­mo­gen­iz­a­tion for the equa­tion \eqref{eq1.6}. Write $P^{\omega}$ for the law of the pro­cess ${\hat X}(\,\cdot\,;{\omega})$. What we have in \eqref{eq1.18} is an ex­ample of a quenched large-de­vi­ation prin­ciple. We may also con­sider the an­nealed law $$\label{eq1.20} {\bar P} = \int P^{\omega} \mathbb{P}(d{\omega})$$ and won­der wheth­er an an­nealed large-de­vi­ation prin­ciple is true for the pro­cess ${\hat X}$. More pre­cisely, wheth­er or not \begin{multline} \label{eq1.21} \quad \lim_{\varepsilon \to 0} \varepsilon \log \int E^\omega \exp\Bigl[ \varepsilon^{-1} f(X(\varepsilon^{-1},{\omega})) \\ + \int_0^{\varepsilon^{-1}} V(X(s,{\omega}),\omega)\,ds \Bigr]\, \mathbb{P}(d{\omega}) = \sup_y \bigl(f(y)-J(y)\bigr) \end{multline} for a suit­able rate func­tion $J$. In terms of $u^{\varepsilon}$, this is equi­val­ent to say­ing $$\label{eq1.22} \lim_{\varepsilon \to 0} \varepsilon \log \int e^{\varepsilon^{-1}u^{\varepsilon}(0,1;{\omega})} \mathbb{P}(d{\omega}) = \sup_y \bigl(f(y)-J(y)\bigr).$$ This would fol­low if we can es­tab­lish a large-de­vi­ation prin­ciple for the con­ver­gence of $u^{\varepsilon}$ to ${\bar u}$. That is, if we can find a func­tion $K_f(y;x,t)$ such that $$\label{eq1.23} \lim_{\varepsilon \to 0} \varepsilon \log \int e^{\varepsilon^{-1}\lambda u^{\varepsilon}(x,t;{\omega}) } \mathbb{P}(d{\omega}) = \sup_y \bigl(\lambda y-K_f(y;x,t)\bigr).$$ The an­nealed large de­vi­ation \eqref{eq1.22} in the case $b \equiv 0$ and $V$ from \eqref{eq1.19} can be found in the manuscript [e4], but \eqref{eq1.23} re­mains open even when $b=0$.

It is worth men­tion­ing that there is also a vari­ation­al de­scrip­tion for the large-de­vi­ation rate func­tion ${\bar L}$, namely $$\label{eq1.24} {\bar L}(v) = \inf_a \inf_{\mu \in \Gamma_{a,v}} \int L_0({\omega},a({\omega}))\,\mu(d{\omega}),$$ where $L_0({\omega},v)$ is the con­vex con­jug­ate of $H_0({\omega},p)$ and $\Gamma_{a,v}$ is the set of in­vari­ant meas­ures for the dif­fu­sions $\mathcal{A}_a =a({\omega}) \cdot \nabla + \tfrac {1}{2} \mathop{\Delta} \quad\text{with}\quad \int a({\omega})\,\mu(d\omega) = v .$ In the case of \eqref{eq1.3}, the gen­er­at­or $\mathcal{A}_a$ takes the form $a \cdot \nabla$ and, when $H$ is peri­od­ic in $x$ (that is, when $\Omega$ is the $d$-di­men­sion­al tor­us with $\mathbb{P}$ be­ing the uni­form meas­ure), the for­mula \eqref{eq1.24} is equi­val­ent to a for­mula of Math­er for the av­er­aged Lag­rangi­an and our ho­mo­gen­iz­a­tion is closely re­lated to the weak KAM the­ory. See Fathi and Maderna [e13] and Evans and Gomez [e8] for more de­tails.

#### The random walk in a random environment

As our second class of ex­amples, we con­sider a dis­crete ver­sion of the dif­fu­sion \eqref{eq1.12}. This is simply a ran­dom walk in a ran­dom en­vir­on­ment (RWRE). To this end, let us write $\mathcal{P}$ for the space of prob­ab­il­ity dens­it­ies on the $d$-di­men­sion­al lat­tice $\mathbb{Z}^d$; that is, $p \in \mathcal{P}$ if $p: \mathbb{Z}^d \to [0,1]$ with $\sum_z p(z) = 1$. We set $\Omega = \mathcal{P}^{\mathbb{Z}^d}$, and $\omega \in \Omega$ is writ­ten as $\omega = (p_a: a \in \mathbb{Z}^d).$ Giv­en $\omega \in \Omega$, we write $X(n,a;{\omega})$ to de­note a ran­dom walk at time $n$ with start­ing point $a \in \mathbb{Z}^d$ and trans­ition prob­ab­il­it­ies $p_a,a \in \mathbb{Z}^d$. More pre­cisely, $P^\omega\bigl(X(n+1) = y \mathrel{\big|} X(n) = x\bigr) = p_x(y-x).$ Giv­en a func­tion $g: \mathbb{Z}^d \to \mathbb{R}$, we write $T_ng(x) = E^\omega g(X(n,x;{\omega})),$ so that $T_1g(x) = \sum_{y \in \mathbb{Z}^d} g(y)\,p_x(y-x).$ To com­pare with \eqref{eq1.11} in the case $V \equiv 0$, we also write $w(x,n) = T_ng(x)$ for a giv­en ini­tial $g$. This trivi­ally solves $w(x,n+1) = \bigl(T_1 w(\,\cdot\,,n)\bigr)(x).$ To com­pare with \eqref{eq1.5}, we set $u = \log w$ so that $u(x,n+1)-u(x,n) = \bigl(Au(\,\cdot\,,n)\bigr)(x)$ where $Ag(x) = \log T_1 e^g(x)-g(x) = \log \sum_z e^{g(x+z)-g(x)} p_x(z).$ Now, ho­mo­gen­iz­a­tion means that we are in­ter­ested in ${\bar u}(x,t) = \lim_{\varepsilon \to 0} \varepsilon u\Bigl( \Bigl[ \frac {x}{\varepsilon} \Bigr],\Bigl[ \frac {t}{\varepsilon}\Bigr];{\omega}\Bigr),$ provided that ${\omega}$ is dis­trib­uted ac­cord­ing to an er­god­ic sta­tion­ary prob­ab­il­ity meas­ure $\mathbb{P}$, where $\tau_x\omega = (p_{y+x}: y \in \mathbb{Z}^d).$ (Here, $[a]$ de­notes the in­teger part of $a$.)

Again, ${\bar u}$ solves \eqref{eq1.4} provided that $\lim_{\varepsilon \to 0} {u^\varepsilon}(x,0)={\bar u}(x,0) = f(x)$ ex­ists ini­tially. The func­tion ${\bar L}$ (the con­vex con­jug­ate of ${\bar H}$) is the quenched large-de­vi­ation rate func­tion for $X(n;{\omega})$. More pre­cisely, for any bounded con­tinu­ous $f: \mathbb{R}^d \to \mathbb{R}$, $\lim_{n \to \infty} n^{-1} \log Ee^{f(n^{-1}X(n,0;{\omega}))} = \sup_y\bigl(f(y)-{\bar L}(y)\bigr).$ This has been es­tab­lished un­der an el­lipt­i­city con­di­tion on $p_x$ by Varadhan [3]. See Bolthausen and Szn­it­man [e9] for a sur­vey on earli­er res­ults. The ana­log of \eqref{eq1.7} is the fol­low­ing for­mula of Rosen­bluth [e12]: $$\label{eq1.25} {\bar H}(p) = \inf_g \mathop{\mathrm{esssup}}_{\omega} \sum_y p_0(y) \,e^{p \cdot z + g({\omega},z)}$$ with in­fim­um over func­tions $\bigl(g(\,\cdot\,,z): \Omega \to \mathbb{R}:z\in\mathbb{Z}^d\bigr)$ such that $\mathbb{E} g(\,\cdot\,,z)=0$ and $g$ is a “closed 1-form”. By the lat­ter we mean that, for every loop $x_0,x_1,\dots,x_{k-1},x_k=x_0$, we have that $\sum_{r=0}^{k-1}g(\tau_{x_r}\omega,x_{k+1}-x_k)=0$.

We now turn to the an­nealed large de­vi­ations for a RWRE. For this, we need to se­lect a tract­able law for the en­vir­on­ment. Pick a prob­ab­il­ity meas­ure $\beta$ on $\mathcal{P}$ and set $\mathbb{P}$ to be the product of $\beta$ to ob­tain a law on $\mathcal{P}^{\mathbb{Z}^d}$. The an­nealed meas­ure ${\bar P} = \int P^{\omega} \mathbb{P}(d{\omega})$ has a simple de­scrip­tion. For this, we write $Z(n) = X(n+1) - X(n)$ for the jump the walk per­forms at time $n$. We also define $N_{x,z}(n) = \#\bigl\{i\in \{0,1,2,\dots,n\}: X(i) = x, Z(i) = z\bigr\}.$ We cer­tainly have \begin{align*} & P\bigl(X(1;{\omega}) = x_1,\dots,X(n,{\omega}) = x_n\bigr) = \prod_{z,x \in \mathbb{Z}^d} (p_x(z))^{N_{x,z}(n)}, \\& {\bar P}\bigl(X(1) = x_1,\dots,X(n) = x_n\bigr) = \prod_{x \in \mathbb{Z}^d} \int \prod_z(p(z))^{N_{x,z}(n)}\beta(dp), \end{align*} where now $N_{x,z}(n) = \#\bigl\{i \in \{0,1,\dots,n-1\}: x_i = x, x_{i+1} - x_i = z\bigr\}.$ Evid­ently, ${\bar P}$ is the law of a non-Markovi­an walk in $\mathbb{Z}^d$. Varadhan [3] es­tab­lished the an­nealed large-de­vi­ations prin­ciple un­der a suit­able el­lipt­i­city con­di­tion on $\beta$. The meth­od re­lies on the fact that the en­vir­on­ment seen from the walk­er is a Markov pro­cess for which Don­sker–Varadhan the­ory may ap­ply if we have enough con­trol on the trans­ition prob­ab­il­it­ies.

If we set \begin{align*} W_n & = \bigl(0 - X(n),X(1) - X(n),\dots,X(n-1) - X(n),X(n) - X(n)\bigr) \\ & = (s_{-n},\dots,s_{-1},s_0 = 0) \end{align*} for the chain seen from the loc­a­tion $X(n)$, then we ob­tain a walk of length $n$ that ends at 0. The space of such walks is de­noted by $\mathbf{W}_n$. Un­der the law ${\bar P}$, the se­quence $W_1,W_2,\dots$ is a Markov chain with the fol­low­ing rule: \begin{align} \label{eq1.26} {\bar P}(W_{n+1} = T_zW_n \mid W_n) & = \frac {{\bar P}(T_zW_n)}{{\bar P}(W_n)} \\& = \frac {\int_{\mathcal{P}} p(z) \prod_a p(a)^{N_{0,a}}\beta(dp)}{\int_{\mathcal{P}} \prod_a p(a)^{N_{0,a}}\beta(dp)}, \nonumber \end{align} where $N_{0,a} = N_{0,a}(W_n)$ is the num­ber of jumps of size $a$ from 0 for the walk $W_n$. Here, $T_zW_n$ de­notes a walk of size $n+1$ which is formed by trans­lat­ing the walk $W_n$ by $-z$ so that it ends at $-z$ in­stead of 0, and then mak­ing the new jump of size $z$ so that it ends at 0.

We wish to es­tab­lish a large-de­vi­ation prin­ciple for the Markov chain with trans­ition prob­ab­il­ity $q(W,z)$ giv­en by \eqref{eq1.26}, where $W = W_n \in \bigcup_{m=0}^{\infty} \mathbf{W}_m$ and $z$ is the jump size. We as­sume that, with prob­ab­il­ity one, the sup­port of $p_0(\,\cdot\,)$ is con­tained in the set $D=\{z: |z| \le C_0\}$. Nat­ur­ally, $q$ ex­tends to those in­fin­ite walks $W \in \mathbf{W}_{\infty}$ with $N_{0,a} < \infty$ for every $a \in D$. If we let $\mathbf{W}_{\infty}^{\mathrm{tr}}$ de­note the set of tran­si­ent walks, then the ex­pres­sion $q(W,z) = \mathbf{q}(W,T_zW)$ giv­en by \eqref{eq1.26} defines the trans­ition prob­ab­il­ity for a Markov chain in $\mathbf{W}_{\infty}^{\mathrm{tr}}$. Don­sker–Varadhan the­ory sug­gests that the em­pir­ic­al meas­ure $\frac {1}{n} \sum_{m=0}^{n-1} \delta_{W_m}$ sat­is­fies a large-de­vi­ation prin­ciple with a rate func­tion $\mathcal{I}(\mu) = \int_{\mathbf{W}_{\infty}^{\mathrm{tr}}} \mathbf{q}_{\mu}(W,z) \log \frac {\mathbf{q}_{\mu}(W,z)}{\mathbf{q}(W,z)} \mu(dW),$ where $\mu$ is any $T$-in­vari­ant meas­ure on $\mathbf{W}_{\infty}^{\mathrm{tr}}$, and $\mathbf{q}_{\mu}(W,z)$ is the con­di­tion­al prob­ab­il­ity of a jump of size $z$, giv­en the past his­tory. We then use the con­trac­tion prin­ciple to come up with a can­did­ate for the large-de­vi­ation rate func­tion $H(v) = \inf\Bigl\{I(\mu): \int z_0\mu(dW) = v\Bigr\},$ where $(z_j: j \in \mathbb{Z})$ de­notes the jumps of a walk $W$. Sev­er­al tech­nic­al dif­fi­culties arise as one tries to ap­ply Don­sker–Varadhan the­ory, be­cause of the non-com­pact­ness of the state space and the fact that the trans­ition prob­ab­il­it­ies are not con­tinu­ous. These is­sues are handled mas­ter­fully in [3].

#### The simple exclusion process

We now turn to our fi­nal mod­el. This time, our en­vir­on­ment $\omega = (p_i(t): i \in \mathbb{Z})$ is a col­lec­tion of in­de­pend­ent Pois­son clocks. More pre­cisely, $p_i$ with $i \in \mathbb{Z}$ are in­de­pend­ent, and each $p_i$ is a Pois­son pro­cess of rate 1; $p_i(t) = k$ for $t \in [\tau_1^i + \dots + \tau_k^i,\tau_1^i + \dots + \tau_{k+1}^i)$ with $\tau_j^i$ in­de­pend­ent mean-1 ex­po­nen­tial ran­dom vari­ables. Giv­en a real­iz­a­tion of $\omega$ and an ini­tial height func­tion $h^0 \in \Gamma = \bigl\{h: \mathbb{Z} \to \mathbb{Z} \mid 0 \le h(i+1) - h(i) \le 1\bigr\},$ we con­struct $h(i,t) = h(i,t;{\omega})$ such that $h(\,\cdot\,,t;{\omega}) \in \Gamma$ for all $t$. More pre­cisely, at each Pois­son time $t = \tau_1^i + \dots + \tau_k^i$, the height $h(i,t)$ in­creases by one unit provided that the res­ult­ing height func­tion $h^i$ be­longs to $\Gamma$; oth­er­wise, the in­crease is sup­pressed.

The pro­cess $h(\,\cdot\,,t)$ is a Markov pro­cess with the rule $h \to h^i$ with rate $\eta(i+1)\,(1-\eta(i))$, where $\eta(i) = h(i) - h(i-1)$. The pro­cess $(\eta(i,t;{\omega}): i \in \mathbb{Z})$ is also Markovi­an, with the in­ter­pret­a­tion that $\eta(i,t) = 1$ if the site $i$ is oc­cu­pied by a particle, and $\eta(i,t) = 0$ if the site $i$ is va­cant. Now, the growth $h \to h^i$ is equi­val­ent to jump­ing a particle from site $i+1$ to $i$, provided that the site $i$ is va­cant. Since $h \in \Gamma$ is non­decreas­ing, we may define its in­verse $x\in \Gamma^{\prime}$, where $\Gamma^{\prime} = \bigl\{x: \mathbb{Z} \to \mathbb{Z} \mid x(h+1) > x(h)\bigr\}.$ Since $h$ in­creases at a site $i+1$ if the site $i$ is oc­cu­pied by a particle, we may re­gard $x(h)$ as the po­s­i­tion of a particle of la­bel $h$. Equi­val­ently, we may in­ter­pret $h(i)$ as the la­bel of a particle at an oc­cu­pied site $i$.

The pro­cess $x(h,t;{\omega})$ is also a Markov pro­cess with the rule $x(h,t) \to x(h,t) - 1$ with rate $\mathbf{1}\!\!\mathbf{1} \bigl(x(h,t)-x(h-1,t) > 1\bigr)$. In words, $x(h)$ de­creases by one unit with rate 1, provided that the res­ult­ing con­fig­ur­a­tion $x_h$ is still in $\Gamma^{\prime}$. For the con­struc­tion of $x(h,t;{\omega})$ we may use the clocks ${\omega}$ or, equi­val­ently, we may use clocks that are as­signed to sites $h \in \mathbb{Z}$. More pre­cisely, if ${\omega}^{\prime} = (p^{\prime}_h(t): h \in \mathbb{Z})$ is a col­lec­tion of in­de­pend­ent Pois­son pro­cesses of rate 1, then we de­crease $x(h)$ by one unit when the clock $p_h^{\prime}$ rings. The pro­cesses $x(h,t;{\omega})$ and $x(h,t;{\omega}^{\prime})$ have the same dis­tri­bu­tion. If we define $\zeta(h,t) = x(h,t) - x(h-1,t) - 1$, then $\zeta(h,t)$ rep­res­ents the gap between the $h$-th and $(h-1)$-th particles in the ex­clu­sion pro­cess. The pro­cess $(\zeta(h,t): h \in \mathbb{Z})$ is the cel­eb­rated zero-range pro­cess and can be re­garded as the oc­cu­pa­tion num­ber at site $h$. The $\zeta$-pro­cess is also Markovi­an, where a $\zeta$-particle at site $h$ jumps to site $h+1$ with rate $\mathbf{1}\!\!\mathbf{1} (\zeta(h) > 0)$.

As in the pre­vi­ous sec­tions, we set \begin{align*} & u^{\varepsilon}(x,t;{\omega}) = \varepsilon h\Bigl( \Bigl[ \frac {x}{\varepsilon} \Bigr], \frac {t}{\varepsilon};{\omega}\Bigr), \\ & x^{\varepsilon}(u,t;{\omega}) = \varepsilon x\Bigl( \Bigl[\frac {u}{\varepsilon}\Bigr], \frac {t}{\varepsilon};{\omega}\Bigr), \end{align*} and as a ho­mo­gen­iz­a­tion we ex­pect to have $u^{\varepsilon} \to {\bar u}$, $x^{\varepsilon} \to {\bar x}$, with ${\bar u}$ and ${\bar x}$ de­term­in­ist­ic solu­tions to Hamilton–Jac­obi equa­tions \begin{align} \label{eq1.28} & {\bar u}_t = {\bar H}_1({\bar u}_x) = {\bar u}_x(1-{\bar u}_x), \\ \label{eq1.29} & {\bar x}_t = {\bar H}_2({\bar x}_u) = \frac {1}{{\bar x}_u} - 1. \end{align} (See [e1].) As for the large de­vi­ations, we will be in­ter­ested in \begin{align} \label{eq1.30} \lim_{\varepsilon \to 0} \varepsilon \log \mathbb{P}(u^{\varepsilon}(x,t) \ge u) & = \lim_{\varepsilon \to 0} \varepsilon \log \mathbb{P}(x^{\varepsilon}(u,t) \ge x) \\& =: -W(x,u,t). \nonumber \end{align} Evid­ently, $W(x,u,t) = 0$ if $u \le {\bar u}(x,t)$ or $x \le {\bar x}(u,t)$. However, we have that $W(x,u,t) > 0$ whenev­er $u > {\bar u}(x,t)$ or $x > {\bar x}(u,t)$. As it turns out, $$\label{eq1.31} \lim_{\varepsilon \to 0} \varepsilon \log \mathbb{P}(u^{\varepsilon}(x,t) \le u) = -\infty$$ for $u < {\bar u}(x,t)$ be­cause, for such a num­ber $u$, ${\liminf}_{\varepsilon\to 0} \varepsilon^2 \log \mathbb{P}(u^{\varepsilon}(x,t) \le u) > 0,$ as was demon­strated by Jensen and Varadhan [1] (see also [e7] and [3]). Quot­ing from [1], the state­ment \eqref{eq1.30} has to do with the fact that one may slow down $x(h,t)$ for $h \le h_0$ in a time in­ter­val of or­der $O(\varepsilon^{-1})$ by simply slow­ing down $x(h_0,t)$. This can be achieved for an en­tropy price of or­der $O(\varepsilon^{-1})$. However, for $x^{\varepsilon}(u,t) \le {\bar x}(u,t) - \delta$, with $\delta > 0$, we need to speed up $O(\varepsilon^{-1})$-many particles for a time in­ter­val of or­der $O(\varepsilon^{-1})$. This re­quires an en­tropy price of or­der $O(\varepsilon^{-2})$.

As was ob­served by Sep­päläin­en [e5], both the $h$ and $x$ pro­cesses en­joy a strong mono­ton­icity prop­erty. More pre­cisely, if we write $x(h,t;{\omega}) = T_t^{\omega}x^0(h)$ for the $x$-pro­cess start­ing from the ini­tial con­fig­ur­a­tion $x^0 \in \Gamma^{\prime}$, then $T_t^{\omega}(\sup_{\alpha} x_{\alpha}^0) = \sup_{\alpha} T_t^{\omega}(x_{\alpha}^0)$. In words, if the ini­tial height $x^0 = \sup_{\alpha} x_{\alpha}^0$ is the su­prem­um of a fam­ily of height func­tions $x_{\alpha}^0$, then it suf­fices to evolve each $x_{\alpha}^0$ sep­ar­ately for a giv­en real­iz­a­tion of $\omega$, and take the su­prem­um af­ter­wards. From this, it is not hard to show that such a strong mono­ton­icity must be val­id for $W$ and this, in turn, im­plies that $W$ solves a HJ equa­tion of the form $$\label{eq1.32} W_t = K(W_x,W_u).$$ Here, the ini­tial data $W(x,u,0)$ is the large-de­vi­ation rate func­tion at ini­tial time. Of course we as­sume that there is a large-de­vi­ation rate func­tion ini­tially, and would like to de­rive a large-de­vi­ation prin­ciple at later times. In the case of the ex­clu­sion or zero-range pro­cess, it is not hard to guess what $K$ is, be­cause, when the pro­cess is at equi­lib­ri­um, the height func­tion at a giv­en site has a simple de­scrip­tion. To con­struct the equi­lib­ri­um meas­ures for the $x$-pro­cess, we pick a num­ber $b \in (0,1)$ and define a ran­dom ini­tial-height func­tion $x(\,\cdot\,,0)$ by the re­quire­ment that $x(\,\cdot\,,0) = 0$ and that $(x(h+1,0) - x(h,0)-1: h \in \mathbb{Z})$ are in­de­pend­ent geo­met­ric ran­dom vari­ables of para­met­er $b$. That is, $x(h+1,0) - x(h,0) = k+1$ with prob­ab­il­ity $(1-b)b^k$. Let us write $\mathbb{P}^b$ for the law of the cor­res­pond­ing pro­cess $x(h,t;{\omega})$. Us­ing Cramer’s large-de­vi­ation the­or­em, we can read­ily cal­cu­late that, for $u$ pos­it­ive, \begin{align} \label{eq1.33} W(x,u,0) & = -\lim_{\varepsilon\to 0} \varepsilon \log \mathbb{P}^b(x^{\varepsilon}(u,0) \ge x) \\& = u\Bigl( I_1\Bigl( \frac {x}{u} + 1,b \Bigr)\Bigr)^+ \nonumber \end{align} where $I_1(r,b) = r \log[ r/(b(1+r))] - \log[(1-b)(1+r)]$. As is well known (see for ex­ample Chapter VIII, Co­rol­lary 4.9 of Lig­gett [e10]), $-x(0,t)$ is a Pois­son pro­cess which de­creases one unit with rate $b$. Again Cramer’s the­or­em yields $$\label{eq1.34} W(x,0,t) = bt\Bigl(I_2 \Bigl( \frac {-x}{bt} \Bigr)\Bigr)^+$$ where $I_2(r) = r \log r - r + 1$. The ex­pres­sions \eqref{eq1.32}\eqref{eq1.34} provide us with enough in­form­a­tion to fig­ure out what $K$ is. We refer to [e3] for a large-de­vi­ation prin­ciple of the form \eqref{eq1.30} for a re­lated particle sys­tem known as Ham­mers­ley’s mod­el.

Al­tern­at­ively, we may study the large de­vi­ation of the particle dens­it­ies. For this pur­pose, we define the em­pir­ic­al meas­ure by \begin{align*} \pi^{\varepsilon}(t,dx) & = \pi^{\varepsilon}(t,dx;{\omega}) \\& = \varepsilon \sum_i \delta_{\varepsilon i}(dx)\,\eta(i,t/\varepsilon;{\omega}). \end{align*} We re­gard $\pi^{\varepsilon}$ as an ele­ment of the Skoro­hod space $X = \mathcal{D}([0,T];M)$, where $M$ is the space of loc­ally bounded meas­ures. The law $\omega\mapsto \pi^{\varepsilon}(t,dx;{\omega})$ in­duces a prob­ab­il­ity meas­ure $\mathcal{P}^{\varepsilon}$ on $X$.

The hy­dro­dynam­ic lim­it for the ex­clu­sion pro­cess means that $\mathcal{P}^{\varepsilon} \to \mathcal{P}$ where $\mathcal{P}$ is con­cen­trated on the single en­tropy solu­tion of $$\label{eq1.36} {\bar \rho}_t = ({\bar \rho}(1-{\bar \rho}))_x$$ for a giv­en ini­tial data ${\bar \rho}(x,0) = {\bar \rho}^0(x)$. The func­tion $\hat \rho$ is re­lated to the mac­ro­scop­ic height func­tion $\bar u$ by ${\bar \rho} = {\bar u}_x$. In [1], a large-de­vi­ation prin­ciple has been es­tab­lished for the con­ver­gence of $\mathcal{P}^{\varepsilon}$. Roughly, $$\label{eq1.37} \mathcal{P}^{\varepsilon}\bigl(\pi^{\varepsilon}(t,dx) \mbox{ is near } \mu(t,dx) \bigr)\approx e^{-\varepsilon^{-1}\mathcal{I}(\mu)}$$ with the fol­low­ing rate func­tion $\mathcal{I}$: First, $\mathcal{I}(\mu) = +\infty$ un­less $\mu(t,dx) = m(x,t)\,dx$ and $m$ is a weak solu­tion of \eqref{eq1.36}. However, when $0 < \mathcal{I}(m) < \infty$, then $m$ is a non-en­trop­ic solu­tion of \eqref{eq1.36}. In fact $\mathcal{I}(\mu) = \mathcal{I}_0(\mu) + \mathcal{I}_{\mathrm{dyn}}(\mu)$, where $\mathcal{I}_0(\mu)$ is the large-de­vi­ation rate func­tion com­ing from the ini­tial de­vi­ation and de­pends only on our choice of ini­tial con­fig­ur­a­tions, and $\mathcal{I}_{\mathrm{dyn}}(\mu)$ is the con­tri­bu­tion com­ing from dy­nam­ics and quant­it­at­ively meas­ures how the en­tropy con­di­tion is vi­ol­ated. By “en­tropy con­di­tion” we mean that, for a pair $(\varphi,q)$ with $\varphi$ con­vex and $\varphi^{\prime}{\bar H}^{\prime}_1 = q^{\prime}$ for $\bar H_1(p) = p(1-p)$, we have $$\label{eq1.38} \varphi({\bar \rho})_t + q({\bar \rho})_x \le 0$$ in the weak sense. The left-hand side is a neg­at­ive dis­tri­bu­tion, which can only be a neg­at­ive meas­ure. As our dis­cus­sions around \eqref{eq1.32} and \eqref{eq1.33} in­dic­ate, the in­vari­ant meas­ures play an es­sen­tial role in de­term­in­ing the large-de­vi­ations rate func­tion. As it turns out, the rel­ev­ant $\varphi$ to choose is simply the large-de­vi­ation rate func­tion for the in­vari­ant meas­ure, which is giv­en by $\varphi(m) = m \log m + (1-m) \log (1-m) + \log 2.$ Here, for the in­vari­ant meas­ure we choose a Bernoulli meas­ure $\nu$ un­der which $(\eta(i): i \in \mathbb{Z})$ are in­de­pend­ent and $\nu(\eta(i)=1) = 1/2$. To meas­ure the fail­ure of the en­tropy solu­tion, we take a weak solu­tion $m$ for which the cor­res­pond­ing $\varphi(m)_t + q(p)_x = \gamma = \gamma^+ - \gamma^-$ is a meas­ure, with $\gamma^+$ and $\gamma^-$ rep­res­ent­ing the pos­it­ive and neg­at­ive part of $\gamma$. We now have $\mathcal{I}_{\mathrm{dyn}}(\mu) = \gamma^+(\mathbb{R} \times [0,T]).$

It is cus­tom­ary in equi­lib­ri­um stat­ist­ic­al mech­an­ics to rep­res­ent a state as a prob­ab­il­ity meas­ure with dens­ity $(1/Z) e^{-\beta H}$, with $H$ some type of en­ergy and $Z$ the nor­mal­iz­ing con­stant. In non-equi­lib­ri­um stat­ist­ic­al mech­an­ics, a large-de­vi­ation prin­ciple of the form \eqref{eq1.37} of­fers an ana­log­ous ex­pres­sion, with $\mathcal{I}(\mu)$ play­ing the role of “ef­fect­ive” en­ergy (or, rather, po­ten­tial). What we learn from [1] is that, after the en­tropy solu­tion, the most fre­quently vis­ited con­fig­ur­a­tions are those as­so­ci­ated with non-en­trop­ic solu­tions, and the en­trop­ic price for such vis­its is meas­ured by the amount the in­equal­ity \eqref{eq1.38} fails. Even though the en­tropy solu­tions for scal­ar con­ser­va­tion laws are rather well un­der­stood, our un­der­stand­ing of non-en­trop­ic solu­tions is rather poor, per­haps be­cause we had no reas­on to pay at­ten­tion to them be­fore. The re­mark­able work [1] urges us to look more deeply in­to non-en­trop­ic solu­tions for gain­ing in­sight in­to the way the mi­cro­scop­ic dens­it­ies de­vi­ate from the solu­tion to the mac­ro­scop­ic equa­tions.

### Works

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