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{equation} \label{eq1.1} u_t = H(x,u_x), \end{equation} 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 \begin{equation} \label{eq1.2} u^{\varepsilon}(x,t;{\omega}) = \varepsilon u\Bigl( \frac {x}{\varepsilon},\frac {t}{\varepsilon};{\omega}\Bigr). \end{equation} We now have \begin{equation} \label{eq1.3} u_t^{\varepsilon} = H\Bigl( \frac {x}{\varepsilon}, u_x^{\varepsilon}\Bigr). \end{equation} 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 \begin{equation} \label{eq1.4} {\bar u}_t = {\bar H}({\bar u}_x), \end{equation} 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 \begin{equation} \label{eq1.5} u_t = H(x,u_x) + \tfrac {1}{2} \mathop{\Delta} u, \end{equation} 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 \begin{equation} \label{eq1.6} u_t^{\varepsilon} = H\Bigl( \frac {x}{\varepsilon},u_x^{\varepsilon}\Bigr) + \frac {\varepsilon}{2} \mathop{\Delta} u^{\varepsilon}. \end{equation} 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 \begin{equation} \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] \end{equation} 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, \begin{equation} \label{eq1.8} \nabla f({\omega}) \cdot v = \lim_{t \to 0} \frac {1}{t}\bigl(f(\tau_{tv}{\omega}) - f({\omega})\bigr) \end{equation} whenev­er the lim­it ex­ists. We ex­pect a sim­il­ar for­mula to hold in the case of \eqref{eq1.3}, namely, \begin{equation} \label{eq1.9} {\bar H}(p) = \inf_g \mathop{\mathrm{esssup}}_w\, H_0\bigl(p+g({\omega}),{\omega}\bigr). \end{equation} Be­fore we turn to our next mod­el, we make an ob­ser­va­tion re­gard­ing the ho­mo­gen­iz­a­tion of \eqref{eq1.6}. Note that, if \begin{equation} \label{eq1.10} H(x,p,{\omega}) = \tfrac {1}{2} |p|^2 + b(x,{\omega}) \cdot p + V(x,{\omega}) \end{equation} and \( u \) is a solu­tion of \eqref{eq1.5}, then, by the Hopf–Cole trans­form, the func­tion \( w = e^u \) solves \begin{equation} \label{eq1.11} w_t = \tfrac {1}{2} \mathop{\Delta} w + b(x,{\omega}) \cdot \nabla w + V(x,{\omega})\,w(x,{\omega}). \end{equation} 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 \begin{equation} \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). \end{equation} 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 \begin{equation} \label{eq1.14} u^{\varepsilon}(x,0;{\omega}) = f(x) \end{equation} 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 \begin{equation} \label{eq1.17} {\bar u}(x,t) = \sup_y \Bigl( f(y) - t {\bar L} \Bigl( \frac {y-x}{t}\Bigr)\Bigr), \end{equation} where \( {\bar L} \) is the con­vex con­jug­ate of \( {\bar H} \). In par­tic­u­lar, \begin{equation} \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). \end{equation} 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 \begin{equation} \label{eq1.19} -V(x,{\omega}) = \sum_{j \in I} V_0(x-x_j), \end{equation} 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 \begin{equation} \label{eq1.20} {\bar P} = \int P^{\omega} \mathbb{P}(d{\omega}) \end{equation} 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 \begin{equation} \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). \end{equation} 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 \begin{equation} \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). \end{equation} 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 \begin{equation} \label{eq1.24} {\bar L}(v) = \inf_a \inf_{\mu \in \Gamma_{a,v}} \int L_0({\omega},a({\omega}))\,\mu(d{\omega}), \end{equation} 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.

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]: \begin{equation} \label{eq1.25} {\bar H}(p) = \inf_g \mathop{\mathrm{esssup}}_{\omega} \sum_y p_0(y) \,e^{p \cdot z + g({\omega},z)} \end{equation} 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].

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, \begin{equation} \label{eq1.31} \lim_{\varepsilon \to 0} \varepsilon \log \mathbb{P}(u^{\varepsilon}(x,t) \le u) = -\infty \end{equation} 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 \begin{equation} \label{eq1.32} W_t = K(W_x,W_u). \end{equation} 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 \begin{equation} \label{eq1.34} W(x,0,t) = bt\Bigl(I_2 \Bigl( \frac {-x}{bt} \Bigr)\Bigr)^+ \end{equation} 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 \begin{equation} \label{eq1.36} {\bar \rho}_t = ({\bar \rho}(1-{\bar \rho}))_x \end{equation} 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, \begin{equation} \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)} \end{equation} 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 \begin{equation} \label{eq1.38} \varphi({\bar \rho})_t + q({\bar \rho})_x \le 0 \end{equation} 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.