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Celebratio Mathematica

S. R. Srinivasa Varadhan

Probability and statistics

Diffusion theory

by Daniel W. Stroock

Giv­en a second-or­der (pos­sibly de­gen­er­ate) el­lipt­ic dif­fer­en­tial op­er­at­or \begin{equation}\tag{1} \label{1} L=\frac12\sum_{i,j=1}^Na_{ij}(x)\,\partial {x_i}\partial _{x_j}+\sum_{i=1}^N b_i(x)\,\partial _{x_i} \end{equation} act­ing on \( C^2(\Bbb R;{\Bbb C}) \) and a Borel prob­ab­il­ity meas­ure \( \mu \) on \( \mathbb{R}^N \), what does it mean for a Borel prob­ab­il­ity meas­ure \( \mathbb{P}_\mu \) on \( C\bigl([0,\infty );\mathbb{R}^N\bigr) \) 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{equation}\tag{2} \label{2} \partial _tp(t,x,\,\cdot\,)=L^*p(t,x,\,\cdot\,)\quad\text{with } p(0,x,\,\cdot\,) =\delta _x, \end{equation} where \( L^* \) is the form­al ad­joint of \( L \) giv­en by \[ L^*\psi (y)=\frac12\sum_{i,j=1}^N\partial _{y_i}\partial _{y_j}\bigl( a_{ij}(y)\,\psi (y)\bigr)-\sum_{i=1}^N\partial _{y_i}\bigl(b_i(y)\,\psi (y)\bigr). \] He then showed that, un­der suit­able reg­u­lar­ity and non-de­gen­er­acy con­di­tions on the coef­fi­cients \( a \) and \( b \), there is one and only solu­tion to \eqref{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 \( (t,x,y)\in (0,\infty )\times \mathbb{R}^N\times \mathbb{R}^N \) and that it sat­is­fies the Chap­man–Kolmogorov equa­tion \begin{equation}\tag{3} \label{3} p(s+t,x,y)=\int p(t,\xi ,y)\,p(s,x,\xi )\,d\xi . \end{equation} Know­ing \eqref{3}, he could check that, for any \( \mu \in\mathbf{M}(\mathbb{R}^N) \), the fam­ily of prob­ab­il­ity meas­ures giv­en by \begin{multline*} P_\mu \bigl((t_1,\dots,t_n);\Gamma \bigr) =\idotsint\limits_\Gamma \prod_{m=1}^np\bigl(t_m-t_{m-1},y_{m-1},y_m\bigr)\,\mu (dy_0)\,dy_1\cdots dy_n \\ \text{ for }n\ge0,\; 0=t_0 < t_1 < \cdots < t_n,\text{ and }\Gamma \in\mathcal{B}_{(\mathbb{R}^N)^{n+1}} \end{multline*} is con­sist­ent, in the sense that if \( \{t^{\prime}_1,\dots,t^{\prime}_{n^{\prime}}\}\subset \{t_1,\dots,t_n\} \) then \( P_\mu(( t^{\prime}_1,\dots,t^{\prime}_{n^{\prime}});\,\cdot\,) \) is the mar­gin­al dis­tri­bu­tion of \( P_\mu ((t_1,\dots,t_n);\,\cdot\,) \) on the co­ordin­ates cor­res­pond­ing to \( \{t^{\prime}_1,\dots,t^{\prime}_{n^{\prime}}\} \). Hav­ing es­tab­lished this con­sist­ency, Kolmogorov ap­plied his Con­sist­ency The­or­em to pro­duce a meas­ure \( \widetilde{\mathbb{P}}_\mu \) on \( (\mathbb{R}^N)^{[0,\infty )} \) whose fi­nite-di­men­sion­al mar­gin­als are giv­en by \[ \widetilde{\mathbb{P}}_\mu\bigl( \{\omega :\,(\omega (t_0),\dots,\omega (t_n))\in\Gamma \}\bigr)=P_\mu \bigl((t_1,\dots,t_n);\Gamma \bigr). \] Fi­nally, us­ing his con­tinu­ity cri­terion, he showed that \( C\bigl([0,\infty );\mathbb{R}^N\bigr) \) has out­er meas­ure 1 un­der \( \tilde{\mathbb{P}}_\mu \) and there­fore that the re­stric­tion of \( \tilde{ \mathbb{P}}_\mu \) to \( C\bigl([0,\infty );\mathbb{R}^N\bigr) \) 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 \( n\ge1 \)) \( \mathbb{P}_n \) is re­lated to \( L_n \) with ini­tial dis­tri­bu­tion \( \mu _n \), and if \( \mu _n\to \mu \) weakly and \( L_n\varphi \to L\varphi \) for smooth \( \varphi \) 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 \( L\varphi \).

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{gather} \tag{4} \label{4} \mathbb{E}^{\mathbb{P}_\mu } \bigl[\varphi \bigl(\omega (t_2)\bigr)\,\big|\,\mathcal{B}_{t_1}\bigr] -\varphi \bigl(\omega (t_1)\bigr)= \mathbb{E}^{\mathbb{P}_\mu }\Bigl[\int_{t_1}^{t_2}L\varphi \bigl(\omega (\tau )\bigr)\,d\tau \,\Big|\,\mathcal{B}_{t_1}\Bigr] \\ \quad \text{for all }\varphi \in C_{\mathrm{c}}^\infty (\mathbb{R}^N;{\Bbb C}), \nonumber \end{gather} where \( \mathcal{B}_t \) is the \( \sigma \)-al­gebra gen­er­ated by \( \{\omega (\tau ):\,\tau \in[0,t]\} \). More suc­cinctly, \begin{gather} \tag{5} \label{5} \Bigl(\varphi \bigl(\omega (t)\bigr)- \int_{0}^{t}L\varphi \bigl(\omega (\tau )\bigr)\,d\tau , \mathcal{B}_t, \mathbb{P}_\mu \Bigr) \\ \text{is a martingale for all }\varphi \in C_{\mathrm{c}}^\infty (\mathbb{R}^N;{\Bbb C}). \nonumber \end{gather} 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 \( \omega \rightsquigarrow \omega (0) \) and \eqref{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 \( \partial _tu=Lu \). 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 \( L \), \[ \Bigl(u\bigl((t_2-t)^+,\omega (t)\bigr), \mathcal{B}_t, \mathbb{P}\Bigr) \] is a mar­tin­gale and there­fore that, for all \( 0\le t_1 < t_2 \), \[ \mathbb{E}^\mathbb{P}\bigl[f\bigl(\omega (t_2)\bigr)\,\big|\,\mathcal{B}_{t_1}\bigr]=u\bigl(t_2-t_1,\omega (t_1)\bigr). \] Hence, if such clas­sic­al solu­tions ex­ist for enough of the \( f \), then the con­di­tion­al dis­tri­bu­tion of \( \omega \rightsquigarrow\omega (t_2) \) 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 \( \omega \rightsquigarrow\omega (0) \) 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, \( a\ge \varepsilon I \) for some \( \varepsilon > 0 \)), 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 \( L^p(\mathbb{R}^N;{\Bbb C}) \) for \( p\in(1,\infty ) \). 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 \( p\in(1,\infty ) \) 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 \eqref{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 \( x\rightsquigarrow \mathbb{P}_x \) is weakly con­tinu­ous and \( \{\mathbb{P}_x:\,x\in\mathbb{R}^N\} \) is a strong Markov fam­ily.

Concluding remarks

There are sev­er­al ad­di­tion­al com­ments which may be help­ful.

1. Al­though, with 20/20 hind­sight, the char­ac­ter­iz­a­tion giv­en by \eqref{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{multline*}\tag{6} \label{6} \Bigl(\exp\Bigl[\Bigl(\xi ,\omega (t)-\int_0^tb\bigl(\omega (\tau )\bigr)\,d\tau \Bigr)_\mathbb{R}^N-\tfrac12\int_0^t\bigl(\xi ,a(\tau )\xi \bigr)_\mathbb{R}^N\,d\tau\Bigr], \mathcal{B}_t, \mathbb{P}_\mu \Bigr)\\ \text{is a martingale for all }\xi \in{\Bbb C}^N. \end{multline*} The equi­val­ence of \eqref{4} and \eqref{6} is a quite easy ap­plic­a­tion of ele­ment­ary Four­i­er ana­lys­is and the ob­ser­va­tion that, aside from in­teg­rabil­ity is­sues, \[ \Bigl(M(t)\,V(t)-\int_0^tM(\tau )\,dV(\tau ), \mathcal{B}_t, \mathbb{P}\Bigr) \] is a mar­tin­gale when \( \bigl(M(t),\,\mathcal{B}_t,\,\mathbb{P}\bigr) \) is a con­tinu­ous mar­tin­gale and \( V(t) \) is a con­tinu­ous, \( \{\mathcal{B}_t:\,t\ge0\} \)-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.

Works

[1]S. R. S. Varadhan: “On the be­ha­vi­or of the fun­da­ment­al solu­tion of the heat equa­tion with vari­able coef­fi­cients,” Comm. Pure Ap­pl. Math. 20 : 2 (May 1967), pp. 431–​455. MR 0208191 Zbl 0155.​16503 article

[2]D. W. Stroock and S. R. S. Varadhan: “Dif­fu­sion pro­cesses with con­tinu­ous coef­fi­cients. II,” Comm. Pure Ap­pl. Math. 22 (July 1969), pp. 479–​530. MR 0254923 Zbl 0167.​43904 article

[3]D. W. Stroock and S. R. S. Varadhan: “Dif­fu­sion pro­cesses with con­tinu­ous coef­fi­cients. I,” Comm. Pure Ap­pl. Math. 22 (May 1969), pp. 345–​400. MR 0253426 Zbl 0167.​43903 article

[4]D. W. Stroock and S. R. S. Varadhan: “Dif­fu­sion pro­cesses with bound­ary con­di­tions,” Comm. Pure Ap­pl. Math. 24 (March 1971), pp. 147–​225. MR 0277037 Zbl 0227.​76131 article

[5]D. W. Stroock and S. R. S. Varadhan: “Dif­fu­sion pro­cesses and mar­tin­gales. II,” pp. 67–​75 in Mar­tin­gales (Ober­wolfach, May 17–30, 1970). Edi­ted by H. Dinges. Lec­ture Notes in Math­em­at­ics 190. Spring­er (Ber­lin), 1971. MR 0359025 incollection

[6]D. W. Stroock and S. R. S. Varadhan: “Dif­fu­sion pro­cesses and mar­tin­gales. I,” pp. 60–​66 in Mar­tin­gales (Ober­wolfach, May 17–30, 1970). Edi­ted by H. Dinges. Lec­ture Notes in Math­em­at­ics 190. Spring­er (Ber­lin), 1971. MR 0359024 incollection

[7]D. W. Stroock and S. R. S. Varadhan: “Dif­fu­sion pro­cesses,” pp. 361–​368 in Pro­ceed­ings of the sixth Berke­ley sym­posi­um on math­em­at­ic­al stat­ist­ics and prob­ab­il­ity (Berke­ley, CA, June 21–Ju­ly 18, 1970), vol. III: Prob­ab­il­ity the­ory. Edi­ted by L. M. Le Cam, J. Ney­man, and E. L. Scott. Univ. Cali­for­nia Press (Berke­ley, CA), 1972. MR 0397899 Zbl 0255.​60055 inproceedings

[8]D. W. Stroock and S. R. S. Varadhan: “On the sup­port of dif­fu­sion pro­cesses with ap­plic­a­tions to the strong max­im­um prin­ciple,” pp. 333–​359 in Pro­ceed­ings of the sixth Berke­ley sym­posi­um on math­em­at­ic­al stat­ist­ics and prob­ab­il­ity (Berke­ley, CA, June 21–Ju­ly 18, 1970), vol. III: Prob­ab­il­ity the­ory. Edi­ted by L. M. Le Cam, J. Ney­man, and E. L. Scott. Uni­versity of Cali­for­nia Press (Berke­ley, CA), 1972. MR 0400425 Zbl 0255.​60056 inproceedings

[9]D. Stroock and S. R. S. Varadhan: “On de­gen­er­ate el­lipt­ic-para­bol­ic op­er­at­ors of second or­der and their as­so­ci­ated dif­fu­sions,” Comm. Pure Ap­pl. Math. 25 : 6 (November 1972), pp. 651–​713. MR 0387812 Zbl 0344.​35041 article

[10]D. W. Stroock and S. R. S. Varadhan: “Mar­tin­gales. I, II, III,” pp. 113–​161 in Top­ics in prob­ab­il­ity the­ory (Cour­ant In­sti­tute, New York, 1971–1972). Edi­ted by D. W. Stroock and S. R. S. Varadhan. Cour­ant In­sti­tute of Math­em­at­ic­al Sci­ences (New York), 1973. MR 0410912 incollection

[11]D. W. Stroock and S. R. S. Varadhan: “Prob­ab­il­ity the­ory and the strong max­im­um prin­ciple,” pp. 215–​220 in Par­tial dif­fer­en­tial equa­tions (Berke­ley, CA, 9–27 Au­gust, 1971). Edi­ted by D. C. Spen­cer. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 23. Amer­er­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1973. MR 0380109 Zbl 0262.​35028 incollection

[12]G. C. Papan­ic­ol­aou and S. R. S. Varadhan: “A lim­it the­or­em with strong mix­ing in Banach space and two ap­plic­a­tions to stochast­ic dif­fer­en­tial equa­tions,” Comm. Pure Ap­pl. Math. 26 : 4 (July 1973), pp. 497–​524. MR 0383530 Zbl 0253.​60065 article

[13]D. W. Stroock and S. R. S. Varadhan: Mul­ti­di­men­sion­al dif­fu­sion pro­cesses. Grundlehren der Math­em­at­ischen Wis­senschaften 233. Spring­er (Ber­lin), 1979. MR 532498 Zbl 0426.​60069 book

[14]G. C. Papan­ic­ol­aou and S. R. S. Varadhan: “Dif­fu­sion in re­gions with many small holes,” pp. 190–​206 in Stochast­ic dif­fer­en­tial sys­tems (Vil­ni­us, Lithuania, 28 Au­gust–2 Septem­ber, 1978). Edi­ted by B. Gri­geli­onis. Lec­ture Notes in Con­trol and In­form­a­tion Sci­ences 25. Spring­er (Ber­lin), 1980. MR 609184 Zbl 0485.​60076 incollection

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[19]J. Quastel and S. R. S. Varadhan: “Dif­fu­sion semig­roups and dif­fu­sion pro­cesses cor­res­pond­ing to de­gen­er­ate di­ver­gence form op­er­at­ors,” Comm. Pure Ap­pl. Math. 50 : 7 (July 1997), pp. 667–​706. MR 1447057 Zbl 0907.​47040 article

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[21]S. R. S. Varadhan: “Stochast­ic ana­lys­is and ap­plic­a­tions,” Bull. Amer. Math. Soc. (N.S.) 40 : 1 (2003), pp. 89–​97. MR 1943135 Zbl 1012.​60004 article

[22]D. W. Stroock and S. R. S. Varadhan: Mul­ti­di­men­sion­al dif­fu­sion pro­cesses, Re­print edition. Clas­sics in Math­em­at­ics 233. Spring­er (Ber­lin), 2006. MR 2190038 Zbl 1103.​60005 book

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