Celebratio Mathematica

Giv­en a second-or­der (pos­sibly de­gen­er­ate) el­lipt­ic dif­fer­en­tial op­er­at­or $$\begin{array}{}\text{(1)}& L=\frac{1}{2}\sum _{i,j=1}^N{a}_{ij}(x)\partial x_i{\partial }_{x_j}+\sum _{i=1}^N{b}_i(x){\partial }_{x_i}\end{array}$$ act­ing on $C^2(\mathbb{R};\mathbb{C})$ and a Borel prob­ab­il­ity meas­ure $\mu$ on ${\mathbb{R}}^N$, what does it mean for a Borel prob­ab­il­ity meas­ure ${\mathbb{P}}_{\mu }$ on $C([0,\mathrm{\infty });{\mathbb{R}}^N)$ to be a dif­fu­sion pro­cess de­term­ined by $L$ with ini­tial value $\mu$? When Varadhan and I asked ourselves this ques­tion in the mid-1960s, we found no an­swer which sat­is­fied us.

It was not that there were no an­swers. In­deed, already in the 1930s Kolmogorov [e1] had mapped out a path which star­ted from $L$ and ended at a meas­ure ${\mathbb{P}}_{\mu }$. Namely, he began by solv­ing what is now called Kolmogorov’s for­ward equa­tion $$\begin{array}{}\text{(2)}& {\partial }_{t}p(t,x,\cdot )=L^{\ast }p(t,x,\cdot )\text{with }p(0,x,\cdot )={\delta }_x,\end{array}$$ where $L^{\ast }$ is the form­al ad­joint of $L$ giv­en by $$L^{\ast }\psi (y)=\frac{1}{2}\sum _{i,j=1}^N{\partial }_{y_i}{\partial }_{y_j}(a_{ij}(y)\psi (y))-\sum _{i=1}^N{\partial }_{y_i}(b_i(y)\psi (y)).$$ He then showed that, un­der suit­able reg­u­lar­ity and non-de­gen­er­acy con­di­tions on the coef­fi­cients $a$ and $b$, there is one and only solu­tion to $\text{(2)}$ which is the dens­ity of a prob­ab­il­ity meas­ure on ${\mathbb{R}}^N$. Fur­ther he showed that $p(t,x,y)$ is a con­tinu­ous func­tion of $(t,x,y)\in (0,\mathrm{\infty })\times {\mathbb{R}}^N\times {\mathbb{R}}^N$ and that it sat­is­fies the Chap­man–Kolmogorov equa­tion $$\begin{array}{}\text{(3)}& p(s+t,x,y)=\int p(t,\xi ,y)p(s,x,\xi )d\xi .\end{array}$$ Know­ing $\text{(3)}$, he could check that, for any $\mu \in \mathbf{M}({\mathbb{R}}^N)$, the fam­ily of prob­ab­il­ity meas­ures giv­en by $$\begin{array}{c}{P}_{\mu }((t_1,\dots ,t_n);\mathrm{\Gamma })=\underset{\mathrm{\Gamma }}{\int \cdots \int }\prod _{m=1}^{n}p(t_m-t_{m-1},y_{m-1},y_m)\mu (d{y}_0)d{y}_1\cdots d{y}_n\hfill \\ \hfill \text{ for }n\ge 0,0=t_0<t_1<\cdots <t_n,\text{ and }\mathrm{\Gamma }\in {\mathcal{B}}_{({\mathbb{R}}^N{)}^{n+1}}\end{array}$$ is con­sist­ent, in the sense that if $\{t_1^{\prime },\dots ,t_{n^{\prime }}^{\prime }\}\subset \{t_1,\dots ,t_n\}$ then $P_{\mu }((t_1^{\prime },\dots ,t_{n^{\prime }}^{\prime });\cdot )$ is the mar­gin­al dis­tri­bu­tion of $P_{\mu }((t_1,\dots ,t_n);\cdot )$ on the co­ordin­ates cor­res­pond­ing to $\{t_1^{\prime },\dots ,t_{n^{\prime }}^{\prime }\}$. Hav­ing es­tab­lished this con­sist­ency, Kolmogorov ap­plied his Con­sist­ency The­or­em to pro­duce a meas­ure ${\tilde{\mathbb{P}}}_{\mu }$ on $({\mathbb{R}}^N{)}^{[0,\mathrm{\infty })}$ whose fi­nite-di­men­sion­al mar­gin­als are giv­en by $${\tilde{\mathbb{P}}}_{\mu }(\{\omega :(\omega (t_0),\dots ,\omega (t_n))\in \mathrm{\Gamma }\})=P_{\mu }((t_1,\dots ,t_n);\mathrm{\Gamma }).$$ Fi­nally, us­ing his con­tinu­ity cri­terion, he showed that $C([0,\mathrm{\infty });{\mathbb{R}}^N)$ has out­er meas­ure 1 un­der ${\tilde{\mathbb{P}}}_{\mu }$ and there­fore that the re­stric­tion of ${\tilde{\mathbb{P}}}_{\mu }$ to $C([0,\mathrm{\infty });{\mathbb{R}}^N)$ de­term­ines a unique Borel prob­ab­il­ity meas­ure ${\mathbb{P}}_{\mu }$ there.

The trail which Kolmogorov blazed was widened and smoothed by many people, in­clud­ing W. Feller [e2], J. Doob [e3], G. Hunt [e4], [e5], R. Blu­menth­al and R. Getoor [e10], E. B. Dynkin [e6], and a host of oth­ers. What emerged from this line of re­search, es­pe­cially after the work of Hunt, was an iso­morph­ism between Markov pro­cesses and po­ten­tial the­ory. However, none of these really ad­dressed the ques­tion that was both­er­ing Varadhan and me. Namely, from our point of view, the route mapped out by Kolmogorov was too cir­cuit­ous: one star­ted with $L$, con­struc­ted from it either a trans­ition prob­ab­il­ity func­tion or re­solvent op­er­at­or, and only then char­ac­ter­ized the res­ult­ing Markov pro­cesses in terms of these an­cil­lary ob­jects. What we were look­ing for was a char­ac­ter­iz­a­tion of ${\mathbb{P}}_{\mu }$ dir­ectly in terms of $L$, one from which it would be clear that, if (for each $n\ge 1$) ${\mathbb{P}}_n$ is re­lated to $L_n$ with ini­tial dis­tri­bu­tion ${\mu }_n$, and if ${\mu }_n\to \mu$ weakly and $L_n\phi \to L\phi$ for smooth $\phi$ with com­pact sup­port, then ${\mathbb{P}}_n$ tends weakly to ${\mathbb{P}}_{\mu }$. Ob­vi­ously, the best way to achieve this goal is to phrase the char­ac­ter­iz­a­tion in terms of ${\mathbb{P}}_{\mu }$-in­teg­rals in­volving $L\phi$.

For­mu­lat­ing such a char­ac­ter­iz­a­tion was re­l­at­ively easy. Namely, no mat­ter how one goes about its con­struc­tion, ${\mathbb{P}}_{\mu }$ should have this prop­erty:1 $$\begin{array}{c}\text{(4)}& {\mathbb{E}}^{{\mathbb{P}}_{\mu }}[\phi (\omega (t_2))|{\mathcal{B}}_{t_1}]-\phi (\omega (t_1))={\mathbb{E}}^{{\mathbb{P}}_{\mu }}[{\int }_{t_1}^{t_2}L\phi (\omega (\tau ))d\tau |{\mathcal{B}}_{t_1}]\text{for all }\phi \in C_{\mathrm{c}}^{\mathrm{\infty }}({\mathbb{R}}^N;\mathbb{C}),\end{array}$$ where ${\mathcal{B}}_t$ is the $\sigma$-al­gebra gen­er­ated by $\{\omega (\tau ):\tau \in [0,t]\}$. More suc­cinctly, $$\begin{array}{c}\text{(5)}& (\phi (\omega (t))-{\int }_0^{t}L\phi (\omega (\tau ))d\tau ,{\mathcal{B}}_t,{\mathbb{P}}_{\mu })\text{is a martingale for all }\phi \in C_{\mathrm{c}}^{\mathrm{\infty }}({\mathbb{R}}^N;\mathbb{C}).\end{array}$$ Thus, we said that ${\mathbb{P}}_{\mu }$ solves the mar­tin­gale prob­lem for $L$ with ini­tial dis­tri­bu­tion $\mu$ if $\mu$ is the ${\mathbb{P}}_{\mu }$-dis­tri­bu­tion of $\omega ⇝\omega (0)$ and $\text{(5)}$ holds.

Hav­ing for­mu­lated the mar­tin­gale prob­lem, our first task was to prove that, in great gen­er­al­ity, solu­tions ex­ist. Of course, see­ing as all roads lead to solu­tions to the mar­tin­gale prob­lem (in situ­ations to which they ap­plied), we could have bor­rowed res­ults from earli­er work (for ex­ample, work of either the Kolmogorov or Itô schools). However, not only would that have made no con­tri­bu­tion to known ex­ist­ence res­ults, it would have been very in­ef­fi­cient. At least when $a$ and $b$ are bounded and con­tinu­ous, ex­ist­ence of solu­tions to the mar­tin­gale prob­lem is an easy ap­plic­a­tion of com­pact­ness, com­pletely ana­log­ous to the proof of ex­ist­ence of solu­tions to or­din­ary dif­fer­en­tial equa­tions with bounded con­tinu­ous coef­fi­cients. Our second task was to show that, un­der ap­pro­pri­ate con­di­tions, solu­tions are unique, and we knew that unique­ness posed a chal­lenge of an en­tirely dif­fer­ent or­der from the one posed by ex­ist­ence. In­deed, if all roads lead to solu­tions to the mar­tin­gale prob­lem, then unique­ness would prove that all roads must have the same des­tin­a­tion.

Our ap­proach to unique­ness de­pended on the as­sump­tions be­ing made about the coef­fi­cients $a$ and $b$. One ap­proach turned on the ob­ser­va­tion that any solu­tion to the mar­tin­gale prob­lem is the dis­tri­bu­tion of a solu­tion to an Itô stochast­ic in­teg­ral equa­tion. When the coef­fi­cients are reas­on­ably smooth, this ob­ser­va­tion al­lowed us to prove unique­ness for the mar­tin­gale prob­lem as a con­sequence of unique­ness for the as­so­ci­ated stochast­ic in­teg­ral equa­tion. However, when the coef­fi­cients are not smooth, Itô’s the­ory can­not cope. Spe­cific­ally, be­cause his the­ory ig­nores el­lipt­i­city prop­er­ties, it is dif­fi­cult to see how one would go about bol­ster­ing it with power­ful res­ults from the vast and beau­ti­ful PDE lit­er­at­ure.

The key to over­com­ing the prob­lem just raised is to real­ize that unique­ness for the mar­tin­gale prob­lem is a purely dis­tri­bu­tion­al ques­tion, where­as the unique­ness found in Itô’s the­ory is a path-by-path state­ment about a cer­tain trans­form­a­tion (the “Itô map”) of Browni­an paths. Thus, to take ad­vant­age of the PDE ex­ist­ence the­ory, we aban­doned Itô and ad­op­ted an easy du­al­ity ar­gu­ment which shows that unique­ness for the mar­tin­gale prob­lem re­quires only a suf­fi­ciently good ex­ist­ence state­ment about the ini­tial-value prob­lem for the back­ward equa­tion ${\partial }_{t}u=Lu$. Namely, if $u$ is a clas­sic­al solu­tion2 to the back­ward equa­tion with ini­tial data $f$, then one can show that, for any solu­tion $\mathbb{P}$ to the mar­tin­gale prob­lem for $L$, $$(u((t_2-t{)}^{+},\omega (t)),{\mathcal{B}}_t,\mathbb{P})$$ is a mar­tin­gale and there­fore that, for all $0\le t_1<t_2$, $${\mathbb{E}}^{\mathbb{P}}[f(\omega (t_2))|{\mathcal{B}}_{t_1}]=u(t_2-t_1,\omega (t_1)).$$ Hence, if such clas­sic­al solu­tions ex­ist for enough of the $f$, then the con­di­tion­al dis­tri­bu­tion of $\omega ⇝\omega (t_2)$ un­der $\mathbb{P}$ giv­en ${\mathcal{B}}_{t_1}$ is uniquely de­term­ined for all $0\le t_1<t_2$. From this, to­geth­er with the con­di­tion that the $\mathbb{P}$-dis­tri­bu­tion of $\omega ⇝\omega (0)$ is $\mu$, it fol­lows that there is only one solu­tion to the mar­tin­gale prob­lem. This ar­gu­ment works without a hitch when, for ex­ample, $a$ and $b$ are bounded and uni­formly Hölder con­tinu­ous and $a$ is uni­formly el­lipt­ic (that is, $a\ge \epsilon I$ for some $\epsilon >0$), since in that case PDE the­or­ists had long ago provided the ne­ces­sary ex­ist­ence state­ment about the back­ward equa­tion.

When uni­form Hölder con­tinu­ity is re­placed by simple uni­form con­tinu­ity, the ex­ist­ence the­ory for the back­ward equa­tion is less sat­is­fact­ory. To be pre­cise, al­though solu­tions con­tin­ue to ex­ist, they are no longer clas­sic­al. In­stead, their second de­riv­at­ives ex­ist only in $L^p({\mathbb{R}}^N;\mathbb{C})$ for $p\in (1,\mathrm{\infty })$. As a con­sequence, the pre­ced­ing du­al­ity ar­gu­ment breaks down un­less one can show a pri­ori that every solu­tion to the mar­tin­gale prob­lem is suf­fi­ciently reg­u­lar to tol­er­ate lim­it pro­ced­ures in­volving $L^p({\mathbb{R}}^N)$-con­ver­gence of the in­teg­rands. Without ques­tion, the proof of such an a pri­ori reg­u­lar­ity res­ult was the single most in­tric­ate part of our pro­gram. In broad out­line, our proof went as fol­lows: First, we de­veloped ma­chinery which al­lowed us to re­duce everything to the case when $b=0$ and $a$ is an ar­bit­rar­ily small per­turb­a­tion of the iden­tity. Second, we again used the con­nec­tion with the Itô stochast­ic in­teg­ral equa­tions to show that any solu­tion $\mathbb{P}$ can be ap­prox­im­ated by ${\mathbb{P}}_n$’s which, con­di­tioned on ${\mathcal{B}}_{m/n}$, are Gaus­si­an dur­ing the time in­ter­val $[m/n,(m+1)/n]$, and there­fore have the re­quired reg­u­lar­ity prop­er­ties. Fi­nally, as an ap­plic­a­tion of the Cal­der­on–Zyg­mund the­ory of sin­gu­lar in­teg­ral op­er­at­ors, for each $p\in (1,\mathrm{\infty })$ we were able to show that, when the per­turb­a­tion is suf­fi­ciently small, the $L^p({\mathbb{R}}^N)$-reg­u­lar­ity prop­er­ties of the ap­prox­im­at­ing ${\mathbb{P}}_n$’s can be con­trolled in­de­pend­ent of $n$ and are there­fore en­joyed by $\mathbb{P}$ it­self.

The steps out­lined above led us to a proof of the fol­low­ing the­or­em, which should be con­sidered the center­piece of [3], [2].

The­or­em. Let $L$ be giv­en by $\text{(1)}$, where $a$ and $b$ are bounded, $a$ is con­tinu­ous, $b$ is Borel meas­ur­able, and $a(x)$ is sym­met­ric and strictly pos­it­ive def­in­ite for each $x\in {\mathbb{R}}^N$. Then, for each Borel prob­ab­il­ity meas­ure $\mu$ on ${\mathbb{R}}^N$, there is pre­cisely one solu­tion to the mar­tin­gale for $L$ with ini­tial dis­tri­bu­tion $\mu$. Moreover, if ${\delta }_x$ is the unit mass at $x$ and ${\mathbb{P}}_x={\mathbb{P}}_{{\delta }_x}$, then $x⇝{\mathbb{P}}_x$ is weakly con­tinu­ous and $\{{\mathbb{P}}_x:x\in {\mathbb{R}}^N\}$ is a strong Markov fam­ily.

There are sev­er­al ad­di­tion­al com­ments which may be help­ful.

1. Al­though, with 20/20 hind­sight, the char­ac­ter­iz­a­tion giv­en by $\text{(4)}$ is the most ob­vi­ous one, it is not the one which we chose ini­tially. In­stead, be­cause we were in­flu­enced at the time both by Henry McK­ean’s treat­ment [e12] of Itô’s the­ory of stochast­ic in­teg­ral equa­tions and by a pos­sible ana­logy with char­ac­ter­ist­ic func­tions, we chose to char­ac­ter­ize ${\mathbb{P}}_{\mu }$ by say­ing that $$\begin{array}{c}(\exp [(\xi ,\omega (t)-{\int }_0^{t}b(\omega (\tau ))d\tau {)}_{\mathbb{R}}^N-\frac{1}{2}{\int }_0^t(\xi ,a(\tau )\xi {)}_{\mathbb{R}}^{N}d\tau ],{\mathcal{B}}_t,{\mathbb{P}}_{\mu })\hfill \\ \text{(6)}& \hfill \text{is a martingale for all }\xi \in {\mathbb{C}}^N.\end{array}$$ The equi­val­ence of $\text{(4)}$ and $\text{(6)}$ is a quite easy ap­plic­a­tion of ele­ment­ary Four­i­er ana­lys­is and the ob­ser­va­tion that, aside from in­teg­rabil­ity is­sues, $$(M(t)V(t)-{\int }_0^{t}M(\tau )dV(\tau ),{\mathcal{B}}_t,\mathbb{P})$$ is a mar­tin­gale when $(M(t),{\mathcal{B}}_t,\mathbb{P})$ is a con­tinu­ous mar­tin­gale and $V(t)$ is a con­tinu­ous, $\{{\mathcal{B}}_t:t\ge 0\}$-ad­ap­ted pro­cess of loc­ally bounded vari­ation.

2. The pre­ced­ing the­or­em is not the one on which we had ori­gin­ally hoped to settle. When we star­ted, we had hoped for a state­ment in which $a$ needed only be Borel meas­ur­able and loc­ally uni­formly el­lipt­ic. Us­ing spe­cial, heav­ily di­men­sion-de­pend­ent ar­gu­ments, we did prove such a res­ult when $N\in \{1,2\}$, and we some­what na­ively as­sumed that it was only the lim­its on our own tech­nic­al powers which pre­ven­ted us from do­ing so in high­er di­men­sions. However, a few years ago Nikolai Nadir­ashvili [e11] gave a beau­ti­ful ex­ample that showed that the lim­its on our powers were not re­spons­ible for our in­ab­il­ity to go to high­er di­men­sions. Namely, he pro­duced a uni­formly el­lipt­ic, bounded, Borel meas­ur­able $a$ on ${\mathbb{R}}^3$ for which the as­so­ci­ated mar­tin­gale prob­lem ad­mits more than one solu­tion.

3. In ret­ro­spect, there are many as­pects of this work which look rather crude and un­ne­ces­sar­ily cum­ber­some. For in­stance, around the same time, first P. A. Mey­er [e7] and then N. Kunita and S. Wa­tenabe [e9] gave a much bet­ter and clean­er treat­ment of the stochast­ic in­teg­ral cal­cu­lus which we de­veloped. More sig­ni­fic­ant, had we been aware of the spec­tac­u­lar work be­ing done by N. Krylov, in par­tic­u­lar the es­tim­ate in [e8], we could have vastly sim­pli­fied the reg­u­lar­ity proof just de­scribed.

4. In a later work, we ex­ten­ded the mar­tin­gale prob­lem ap­proach to cov­er dif­fu­sions with bound­ary con­di­tions. The res­ult­ing pa­per [4] has, for good reas­ons, been much less in­flu­en­tial: it is nearly un­read­able. On the oth­er hand, it con­tains in­form­a­tion which, so far as I know, is avail­able nowhere else. In par­tic­u­lar, I do not think the the­or­ems there deal­ing with ap­prox­im­a­tions by dis­crete Markov chains have ever been im­proved.

5. Varadhan and my joint work on dif­fu­sion the­ory cul­min­ated in our pa­pers [8] and [9] on de­gen­er­ate dif­fu­sions. The first of these con­tains our ini­tial ver­sion of “the sup­port the­or­em,” which has been giv­en sev­er­al oth­er proofs over the years. Among oth­er things, the second pa­per con­tains an ex­ten­sion of “the sup­port the­or­em” to cov­er situ­ations in which the dif­fu­sion mat­rix $a$ is smooth but does not ad­mit a smooth square root. It seems that most prob­ab­il­ists are not par­tic­u­larly im­pressed by this ex­ten­sion, but we were very proud of it at the time.