Celebratio Mathematica

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.

To in­tro­duce the ba­sic idea be­hind ho­mo­gen­iz­a­tion, we first con­sider the (in­homo­gen­eous) Hamilton–Jac­obi (HJ) equa­tion, $$\begin{array}{}\text{(1)}& u_t=H(x,u_x),\end{array}$$ where $H$ is sta­tion­ary and er­god­ic in the first vari­able $x$. More pre­cisely, we have a prob­ab­il­ity space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ with $\mathcal{F}$ a Borel $\sigma$-field on $\mathrm{\Omega }$ and $\mathbb{P}$ a prob­ab­il­ity meas­ure on $(\mathrm{\Omega },\mathcal{F})$, 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 ${\tau }_x:\mathrm{\Omega }\to \mathrm{\Omega }$ so that ${\tau }_x\circ {\tau }_y={\tau }_{x+y}$, and $\mathbb{P}({\tau }_{x}A)=\mathbb{P}(A)$ for every $A\in \mathcal{F}$ and $x,y\in {\mathbb{R}}^d$. We also as­sume that ${\tau }_x$ is er­god­ic; that is, ${\tau }_{x}A=A$ 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 $H_0:\mathrm{\Omega }\times {\mathbb{R}}^d\to \mathbb{R}$ 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 $$\begin{array}{}\text{(2)}& u^{\epsilon }(x,t;\omega )=\epsilon u(\frac{x}{\epsilon },\frac{t}{\epsilon };\omega ).\end{array}$$ We now have $$\begin{array}{}\text{(3)}& u_t^{\epsilon }=H(\frac{x}{\epsilon },u_x^{\epsilon }).\end{array}$$ We note that the right-hand side of $\text{(3)}$ fluc­tu­ates greatly over mac­ro­scop­ic shifts in the po­s­i­tion $x$. The huge fluc­tu­ation in $H$, though, does not ne­ces­sar­ily im­ply cor­res­pond­ingly huge fluc­tu­ations in $u^{\epsilon }$. This is the ho­mo­gen­iz­a­tion phe­nomen­on; that is, we ex­pect $u^{\epsilon }\to \overline{u}$ as $\epsilon \to 0$, with $\overline{u}$ solv­ing a ho­mo­gen­ized HJ equa­tion $$\begin{array}{}\text{(4)}& {\overline{u}}_t=\overline{H}({\overline{u}}_x),\end{array}$$ where $\overline{H}:{\mathbb{R}}^d\to \mathbb{R}$ is the ho­mo­gen­ized Hamilto­ni­an and does not de­pend on $\omega$.

As our second ex­ample, we con­sider the Hamilton–Jac­obi–Bell­mann equa­tion $$\begin{array}{}\text{(5)}& u_t=H(x,u_x)+\frac{1}{2}\mathrm{\Delta }u,\end{array}$$ with $H(x,p)=H(x,p,\omega )$ as be­fore. We define $u^{\epsilon }$ as in $\text{(2)}$, and then $\text{(3)}$ be­comes $$\begin{array}{}\text{(6)}& u_t^{\epsilon }=H(\frac{x}{\epsilon },u_x^{\epsilon })+\frac{\epsilon }{2}\mathrm{\Delta }{u}^{\epsilon }.\end{array}$$ Again, we ex­pect to have $u^{\epsilon }\to \overline{u}$, with $\overline{u}$ sat­is­fy­ing an equa­tion of the form $\text{(4)}$ for a dif­fer­ent ho­mo­gen­ized Hamilto­ni­an $\overline{H}$. In­deed, the ho­mo­gen­iz­a­tions for both $\text{(3)}$ and $\text{(6)}$ 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 $\overline{H}$. In the case of $\text{(6)}$, $\overline{H}$ is giv­en by $$\begin{array}{}\text{(7)}& \overline{H}(p)=\underset{g}{inf}\underset{\omega }{\mathrm{esssup}}[H_0(p+g(\omega ),\omega )+\frac{1}{2}\mathrm{\nabla }\cdot g(\omega )]\end{array}$$ where the es­sen­tial su­prem­um is taken with re­spect to the prob­ab­il­ity meas­ure $\mathbb{P}$, and the in­fim­um is taken over func­tions $g:\mathrm{\Omega }\to {\mathbb{R}}^d$ such that $\mathbb{E}g=0$ and $\mathrm{\nabla }\cdot g=0$ weakly. Here, $\mathrm{\nabla }$ is the gen­er­at­or of the group $\{{\tau }_x\}$; that is, $$\begin{array}{}\text{(8)}& \mathrm{\nabla }f(\omega )\cdot v=\underset{t\to 0}{lim}\frac{1}{t}(f({\tau }_{tv}\omega )-f(\omega ))\end{array}$$ whenev­er the lim­it ex­ists. We ex­pect a sim­il­ar for­mula to hold in the case of $\text{(3)}$, namely, $$\begin{array}{}\text{(9)}& \overline{H}(p)=\underset{g}{inf}\underset{w}{\mathrm{esssup}}{H}_0(p+g(\omega ),\omega ).\end{array}$$ Be­fore we turn to our next mod­el, we make an ob­ser­va­tion re­gard­ing the ho­mo­gen­iz­a­tion of $\text{(6)}$. Note that, if $$\begin{array}{}\text{(10)}& H(x,p,\omega )=\frac{1}{2}|p{|}^2+b(x,\omega )\cdot p+V(x,\omega )\end{array}$$ and $u$ is a solu­tion of $\text{(5)}$, then, by the Hopf–Cole trans­form, the func­tion $w=e^u$ solves $$\begin{array}{}\text{(11)}& w_t=\frac{1}{2}\mathrm{\Delta }w+b(x,\omega )\cdot \mathrm{\nabla }w+V(x,\omega )w(x,\omega ).\end{array}$$ 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 $$\begin{array}{rl}\text{(12)}& dX(t)& =b(X(t),\omega )dt+d\beta (t),X(0)& =x,\end{array}$$ then $$\begin{array}{}\text{(13)}& w(x,t;\omega )=E^{\omega }w(X(t,x;\omega ),0)\exp ({\int }_0^{t}V(X(s,x;\omega ),\omega )ds).\end{array}$$ 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 $u^{\epsilon }$. If $$\begin{array}{}\text{(14)}& u^{\epsilon }(x,0;\omega )=f(x)\end{array}$$ for a de­term­in­ist­ic ini­tial con­di­tion $f$, then $$\begin{array}{rl}\text{(15)}& u^{\epsilon }(x,t;\omega )=\epsilon \log E^{\omega }\exp [& {\epsilon }^{-1}f(\epsilon X(t/\epsilon ,x/\epsilon ;\omega ))& +{\int }_0^{t/\epsilon }V(X(s,x/\epsilon ;\omega ),\omega )ds]\end{array}$$ In par­tic­u­lar, $$\begin{array}{rl}\text{(16)}& u^{\epsilon }(0,1;\omega )=\epsilon \log E^{\omega }\exp [& {\epsilon }^{-1}f(X({\epsilon }^{-1};\omega ))& +{\int }_0^{{\epsilon }^{-1}}V(X(s;\omega ),\omega )ds]\end{array}$$ where $X(s;\omega ):=X(s,0;\omega )$ is the dif­fu­sion start­ing from the ori­gin. On the oth­er hand, since $\overline{H}$ is con­vex (which is evid­ent from $\text{(7)}$) we may use the Hope–Lax–Olein­ik for­mula to write $$\begin{array}{}\text{(17)}& \overline{u}(x,t)=\underset{y}{sup}(f(y)-t\overline{L}(\frac{y-x}{t})),\end{array}$$ where $\overline{L}$ is the con­vex con­jug­ate of $\overline{H}$. In par­tic­u­lar, $$\begin{array}{}\text{(18)}& \underset{\epsilon \to 0}{lim}{u}^{\epsilon }(0,1;\omega )=\overline{u}(0,1)=\underset{y}{sup}(f(y)-\overline{L}(y)).\end{array}$$ By a cel­eb­rated lemma of Varadhan, $\text{(18)}$ is equi­val­ent to say­ing that, for al­most all $\omega$, the dif­fu­sion $\hat{X}$ sat­is­fies a large-de­vi­ation prin­ciple with rate func­tion $\overline{L}$. When $b\equiv 0$ and $$\begin{array}{}\text{(19)}& -V(x,\omega )=\sum _{j\in I}{V}_0(x-x_j),\end{array}$$ 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 $\text{(6)}$. Write $P^{\omega }$ for the law of the pro­cess $\hat{X}(\cdot ;\omega )$. What we have in $\text{(18)}$ is an ex­ample of a quenched large-de­vi­ation prin­ciple. We may also con­sider the an­nealed law $$\begin{array}{}\text{(20)}& \overline{P}=\int P^{\omega }\mathbb{P}(d\omega )\end{array}$$ and won­der wheth­er an an­nealed large-de­vi­ation prin­ciple is true for the pro­cess $\hat{X}$. More pre­cisely, wheth­er or not $$\begin{array}{c}\underset{\epsilon \to 0}{lim}\epsilon \log \int E^{\omega }\exp [{\epsilon }^{-1}f(X({\epsilon }^{-1},\omega ))\hfill \\ \text{(21)}& \hfill +{\int }_0^{{\epsilon }^{-1}}V(X(s,\omega ),\omega )ds]\mathbb{P}(d\omega )=\underset{y}{sup}(f(y)-J(y))\end{array}$$ for a suit­able rate func­tion $J$. In terms of $u^{\epsilon }$, this is equi­val­ent to say­ing $$\begin{array}{}\text{(22)}& \underset{\epsilon \to 0}{lim}\epsilon \log \int e^{{\epsilon }^{-1}{u}^{\epsilon }(0,1;\omega )}\mathbb{P}(d\omega )=\underset{y}{sup}(f(y)-J(y)).\end{array}$$ This would fol­low if we can es­tab­lish a large-de­vi­ation prin­ciple for the con­ver­gence of $u^{\epsilon }$ to $\overline{u}$. That is, if we can find a func­tion $K_f(y;x,t)$ such that $$\begin{array}{}\text{(23)}& \underset{\epsilon \to 0}{lim}\epsilon \log \int e^{{\epsilon }^{-1}\lambda u^{\epsilon }(x,t;\omega )}\mathbb{P}(d\omega )=\underset{y}{sup}(\lambda y-K_f(y;x,t)).\end{array}$$ The an­nealed large de­vi­ation $\text{(22)}$ in the case $b\equiv 0$ and $V$ from $\text{(19)}$ can be found in the manuscript [e4], but $\text{(23)}$ re­mains open even when $b=0$.

It is worth men­tion­ing that there is also a vari­ation­al de­scrip­tion for the large-de­vi­ation rate func­tion $\overline{L}$, namely $$\begin{array}{}\text{(24)}& \overline{L}(v)=\underset{a}{inf}\underset{\mu \in {\mathrm{\Gamma }}_{a,v}}{inf}\int L_0(\omega ,a(\omega ))\mu (d\omega ),\end{array}$$ where $L_0(\omega ,v)$ is the con­vex con­jug­ate of $H_0(\omega ,p)$ and ${\mathrm{\Gamma }}_{a,v}$ is the set of in­vari­ant meas­ures for the dif­fu­sions $${\mathcal{A}}_a=a(\omega )\cdot \mathrm{\nabla }+\frac{1}{2}\mathrm{\Delta }\text{with}\int a(\omega )\mu (d\omega )=v.$$ In the case of $\text{(3)}$, the gen­er­at­or ${\mathcal{A}}_a$ takes the form $a\cdot \mathrm{\nabla }$ and, when $H$ is peri­od­ic in $x$ (that is, when $\mathrm{\Omega }$ is the $d$-di­men­sion­al tor­us with $\mathbb{P}$ be­ing the uni­form meas­ure), the for­mula $\text{(24)}$ 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.

As our second class of ex­amples, we con­sider a dis­crete ver­sion of the dif­fu­sion $\text{(12)}$. 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 $\mathrm{\Omega }={\mathcal{P}}^{{\mathbb{Z}}^d}$, and $\omega \in \mathrm{\Omega }$ is writ­ten as $$\omega =(p_a:a\in {\mathbb{Z}}^d).$$ Giv­en $\omega \in \mathrm{\Omega }$, 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, $$P^{\omega }(X(n+1)=y|X(n)=x)=p_x(y-x).$$ Giv­en a func­tion $g:{\mathbb{Z}}^d\to \mathbb{R}$, we write $$T_{n}g(x)=E^{\omega }g(X(n,x;\omega )),$$ so that $$T_{1}g(x)=\sum _{y\in {\mathbb{Z}}^d}g(y)p_x(y-x).$$ To com­pare with $\text{(11)}$ in the case $V\equiv 0$, we also write $$w(x,n)=T_{n}g(x)$$ for a giv­en ini­tial $g$. This trivi­ally solves $$w(x,n+1)=(T_{1}w(\cdot ,n))(x).$$ To com­pare with $\text{(5)}$, we set $u=\log w$ so that $$u(x,n+1)-u(x,n)=(Au(\cdot ,n))(x)$$ 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 $$\overline{u}(x,t)=\underset{\epsilon \to 0}{lim}\epsilon u([\frac{x}{\epsilon }],[\frac{t}{\epsilon }];\omega ),$$ 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, $\overline{u}$ solves $\text{(4)}$ provided that $\underset{\epsilon \to 0}{lim}{u}^{\epsilon }(x,0)=\overline{u}(x,0)=f(x)$ ex­ists ini­tially. The func­tion $\overline{L}$ (the con­vex con­jug­ate of $\overline{H}$) is the quenched large-de­vi­ation rate func­tion for $X(n;\omega )$. More pre­cisely, for any bounded con­tinu­ous $f:{\mathbb{R}}^d\to \mathbb{R}$, $$\underset{n\to \mathrm{\infty }}{lim}{n}^{-1}\log E{e}^{f(n^{-1}X(n,0;\omega ))}=\underset{y}{sup}(f(y)-\overline{L}(y)).$$ 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 $\text{(7)}$ is the fol­low­ing for­mula of Rosen­bluth [e12]: $$\begin{array}{}\text{(25)}& \overline{H}(p)=\underset{g}{inf}\underset{\omega }{\mathrm{esssup}}\sum _y{p}_0(y)e^{p\cdot z+g(\omega ,z)}\end{array}$$ with in­fim­um over func­tions $(g(\cdot ,z):\mathrm{\Omega }\to \mathbb{R}:z\in {\mathbb{Z}}^d)$ 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 $\overline{P}=\int P^{\omega }\mathbb{P}(d\omega )$ 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 $$N_{x,z}(n)=\mathrm{\#}\{i\in \{0,1,2,\dots ,n\}:X(i)=x,Z(i)=z\}.$$ We cer­tainly have $$\begin{array}{rl}& P(X(1;\omega )=x_1,\dots ,X(n,\omega )=x_n)=\prod _{z,x\in {\mathbb{Z}}^d}(p_x(z){)}^{N_{x,z}(n)},\\ & \overline{P}(X(1)=x_1,\dots ,X(n)=x_n)=\prod _{x\in {\mathbb{Z}}^d}\int \prod _z(p(z){)}^{N_{x,z}(n)}\beta (dp),\end{array}$$ where now $$N_{x,z}(n)=\mathrm{\#}\{i\in \{0,1,\dots ,n-1\}:x_i=x,x_{i+1}-x_i=z\}.$$ Evid­ently, $\overline{P}$ 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 $$\begin{array}{rl}{W}_n& =(0-X(n),X(1)-X(n),\dots ,X(n-1)-X(n),X(n)-X(n))\\ & =(s_{-n},\dots ,s_{-1},s_0=0)\end{array}$$ 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 $\overline{P}$, the se­quence $W_1,W_2,\dots$ is a Markov chain with the fol­low­ing rule: $$\begin{array}{rl}\text{(26)}& \overline{P}(W_{n+1}=T_z{W}_n\mid W_n)& =\frac{\overline{P}(T_z{W}_n)}{\overline{P}(W_n)}& =\frac{{\int }_{\mathcal{P}}p(z)\prod _{a}p(a{)}^{N_{0,a}}\beta (dp)}{{\int }_{\mathcal{P}}\prod _{a}p(a{)}^{N_{0,a}}\beta (dp)},\end{array}$$ 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, $T_z{W}_n$ 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 $\text{(26)}$, where $W=W_n\in \bigcup _{m=0}^{\mathrm{\infty }}{\mathbf{W}}_m$ 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 $W\in {\mathbf{W}}_{\mathrm{\infty }}$ with $N_{0,a}<\mathrm{\infty }$ for every $a\in D$. If we let ${\mathbf{W}}_{\mathrm{\infty }}^{\mathrm{tr}}$ de­note the set of tran­si­ent walks, then the ex­pres­sion $q(W,z)=\mathbf{q}(W,T_{z}W)$ giv­en by $\text{(26)}$ defines the trans­ition prob­ab­il­ity for a Markov chain in ${\mathbf{W}}_{\mathrm{\infty }}^{\mathrm{tr}}$. 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 $$\mathcal{I}(\mu )={\int }_{{\mathbf{W}}_{\mathrm{\infty }}^{\mathrm{tr}}}{\mathbf{q}}_{\mu }(W,z)\log \frac{{\mathbf{q}}_{\mu }(W,z)}{\mathbf{q}(W,z)}\mu (dW),$$ where $\mu$ is any $T$-in­vari­ant meas­ure on ${\mathbf{W}}_{\mathrm{\infty }}^{\mathrm{tr}}$, 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 $$H(v)=inf\{I(\mu ):\int z_0\mu (dW)=v\},$$ 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].

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 $t\in [{\tau }_1^i+\cdots +{\tau }_k^i,{\tau }_1^i+\cdots +{\tau }_{k+1}^i)$ with ${\tau }_j^i$ in­de­pend­ent mean-1 ex­po­nen­tial ran­dom vari­ables. Giv­en a real­iz­a­tion of $\omega$ and an ini­tial height func­tion $$h^0\in \mathrm{\Gamma }=\{h:\mathbb{Z}\to \mathbb{Z}\mid 0\le h(i+1)-h(i)\le 1\},$$ we con­struct $h(i,t)=h(i,t;\omega )$ such that $h(\cdot ,t;\omega )\in \mathrm{\Gamma }$ for all $t$. More pre­cisely, at each Pois­son time $t={\tau }_1^i+\cdots +{\tau }_k^i$, the height $h(i,t)$ in­creases by one unit provided that the res­ult­ing height func­tion $h^i$ be­longs to $\mathrm{\Gamma }$; 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 $h\in \mathrm{\Gamma }$ is non­decreas­ing, we may define its in­verse $x\in {\mathrm{\Gamma }}^{\prime }$, where $${\mathrm{\Gamma }}^{\prime }=\{x:\mathbb{Z}\to \mathbb{Z}\mid x(h+1)>x(h)\}.$$ 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 $\mathbf{1}\mathbf{1}(x(h,t)-x(h-1,t)>1)$. In words, $x(h)$ de­creases by one unit with rate 1, provided that the res­ult­ing con­fig­ur­a­tion $x_h$ is still in ${\mathrm{\Gamma }}^{\prime }$. 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 ${\omega }^{\prime }=(p_h^{\prime }(t):h\in \mathbb{Z})$ 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 $$\begin{array}{rl}& u^{\epsilon }(x,t;\omega )=\epsilon h([\frac{x}{\epsilon }],\frac{t}{\epsilon };\omega ),\\ & x^{\epsilon }(u,t;\omega )=\epsilon x([\frac{u}{\epsilon }],\frac{t}{\epsilon };\omega ),\end{array}$$ and as a ho­mo­gen­iz­a­tion we ex­pect to have $u^{\epsilon }\to \overline{u}$, $x^{\epsilon }\to \overline{x}$, with $\overline{u}$ and $\overline{x}$ de­term­in­ist­ic solu­tions to Hamilton–Jac­obi equa­tions $$\begin{array}{}\text{(27)}& & {\overline{u}}_t={\overline{H}}_1({\overline{u}}_x)={\overline{u}}_x(1-{\overline{u}}_x),\text{(28)}& & {\overline{x}}_t={\overline{H}}_2({\overline{x}}_u)=\frac{1}{{\overline{x}}_u}-1.\end{array}$$ (See [e1].) As for the large de­vi­ations, we will be in­ter­ested in $$\begin{array}{rl}\text{(29)}& \underset{\epsilon \to 0}{lim}\epsilon \log \mathbb{P}(u^{\epsilon }(x,t)\ge u)& =\underset{\epsilon \to 0}{lim}\epsilon \log \mathbb{P}(x^{\epsilon }(u,t)\ge x)& =:-W(x,u,t).\end{array}$$ Evid­ently, $W(x,u,t)=0$ if $u\le \overline{u}(x,t)$ or $x\le \overline{x}(u,t)$. However, we have that $W(x,u,t)>0$ whenev­er $u>\overline{u}(x,t)$ or $x>\overline{x}(u,t)$. As it turns out, $$\begin{array}{}\text{(30)}& \underset{\epsilon \to 0}{lim}\epsilon \log \mathbb{P}(u^{\epsilon }(x,t)\le u)=-\mathrm{\infty }\end{array}$$ for $u<\overline{u}(x,t)$ be­cause, for such a num­ber $u$, $${lim inf}_{\epsilon \to 0}{\epsilon }^2\log \mathbb{P}(u^{\epsilon }(x,t)\le u)>0,$$ as was demon­strated by Jensen and Varadhan [1] (see also [e7] and [3]). Quot­ing from [1], the state­ment $\text{(29)}$ 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 $O({\epsilon }^{-1})$ by simply slow­ing down $x(h_0,t)$. This can be achieved for an en­tropy price of or­der $O({\epsilon }^{-1})$. However, for $x^{\epsilon }(u,t)\le \overline{x}(u,t)-\delta$, with $\delta >0$, we need to speed up $O({\epsilon }^{-1})$-many particles for a time in­ter­val of or­der $O({\epsilon }^{-1})$. This re­quires an en­tropy price of or­der $O({\epsilon }^{-2})$.

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 $x^0\in {\mathrm{\Gamma }}^{\prime }$, then $T_t^{\omega }(\underset{\alpha }{sup}{x}_{\alpha }^0)=\underset{\alpha }{sup}{T}_t^{\omega }(x_{\alpha }^0)$. In words, if the ini­tial height $x^0=\underset{\alpha }{sup}{x}_{\alpha }^0$ 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 $$\begin{array}{}\text{(31)}& W_t=K(W_x,W_u).\end{array}$$ 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, $$\begin{array}{rl}\text{(32)}& W(x,u,0)& =-\underset{\epsilon \to 0}{lim}\epsilon \log {\mathbb{P}}^b(x^{\epsilon }(u,0)\ge x)& =u(I_1(\frac{x}{u}+1,b){)}^{+}\end{array}$$ 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 $$\begin{array}{}\text{(33)}& W(x,0,t)=bt(I_2(\frac{-x}{bt}){)}^{+}\end{array}$$ where $I_2(r)=r\log r-r+1$. The ex­pres­sions $\text{(31)}$–$\text{(33)}$ provide us with enough in­form­a­tion to fig­ure out what $K$ is. We refer to [e3] for a large-de­vi­ation prin­ciple of the form $\text{(29)}$ 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 $$\begin{array}{rl}{\pi }^{\epsilon }(t,dx)& ={\pi }^{\epsilon }(t,dx;\omega )\\ & =\epsilon \sum _i{\delta }_{\epsilon i}(dx)\eta (i,t/\epsilon ;\omega ).\end{array}$$ We re­gard ${\pi }^{\epsilon }$ as an ele­ment of the Skoro­hod space $X=\mathcal{D}([0,T];M)$, where $M$ is the space of loc­ally bounded meas­ures. The law $\omega \mapsto {\pi }^{\epsilon }(t,dx;\omega )$ in­duces a prob­ab­il­ity meas­ure ${\mathcal{P}}^{\epsilon }$ on $X$.

The hy­dro­dynam­ic lim­it for the ex­clu­sion pro­cess means that ${\mathcal{P}}^{\epsilon }\to \mathcal{P}$ where $\mathcal{P}$ is con­cen­trated on the single en­tropy solu­tion of $$\begin{array}{}\text{(34)}& {\overline{\rho }}_t=(\overline{\rho }(1-\overline{\rho }){)}_x\end{array}$$ for a giv­en ini­tial data $\overline{\rho }(x,0)={\overline{\rho }}^0(x)$. The func­tion $\hat{\rho }$ is re­lated to the mac­ro­scop­ic height func­tion $\overline{u}$ by $\overline{\rho }={\overline{u}}_x$. In [1], a large-de­vi­ation prin­ciple has been es­tab­lished for the con­ver­gence of ${\mathcal{P}}^{\epsilon }$. Roughly, $$\begin{array}{}\text{(35)}& {\mathcal{P}}^{\epsilon }({\pi }^{\epsilon }(t,dx)\text{ is near }\mu (t,dx))\approx e^{-{\epsilon }^{-1}\mathcal{I}(\mu )}\end{array}$$ with the fol­low­ing rate func­tion $\mathcal{I}$: First, $\mathcal{I}(\mu )=+\mathrm{\infty }$ un­less $\mu (t,dx)=m(x,t)dx$ and $m$ is a weak solu­tion of $\text{(34)}$. However, when $0<\mathcal{I}(m)<\mathrm{\infty }$, then $m$ is a non-en­trop­ic solu­tion of $\text{(34)}$. 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 $(\phi ,q)$ with $\phi$ con­vex and ${\phi }^{\prime }{\overline{H}}_1^{\prime }=q^{\prime }$ for ${\overline{H}}_1(p)=p(1-p)$, we have $$\begin{array}{}\text{(36)}& \phi (\overline{\rho }{)}_t+q(\overline{\rho }{)}_x\le 0\end{array}$$ in the weak sense. The left-hand side is a neg­at­ive dis­tri­bu­tion, which can only be a neg­at­ive meas­ure. As our dis­cus­sions around $\text{(31)}$ and $\text{(32)}$ in­dic­ate, the in­vari­ant meas­ures play an es­sen­tial role in de­term­in­ing the large-de­vi­ations rate func­tion. As it turns out, the rel­ev­ant $\phi$ to choose is simply the large-de­vi­ation rate func­tion for the in­vari­ant meas­ure, which is giv­en by $$\phi (m)=m\log m+(1-m)\log (1-m)+\log 2.$$ 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 $$\phi (m{)}_t+q(p{)}_x=\gamma ={\gamma }^{+}-{\gamma }^{-}$$ 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 $\text{(35)}$ 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 $\text{(36)}$ 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.