> <\body> >|<\doc-subtitle> Lecture7: > on the torus\ II. A priori estimates, UV stability and the limiting FBSDE |>|> We prove the central estimate of the construction: bounds, uniform in the lattice, on the error of the approximate forecast and on the time the forecast spends at large fields. The two halves of the FBSDE give two inequalities which close only together, with small coefficient /q>>, for every coupling and every finite mass. The same bounds hold for the flow restarted from any field at any scale. So the exact flow is an explicit flow with a control of finite relative entropy. This writes the limit measure as an explicit density over an explicit SDE, for the whole lattice sequence, and it gives the FBSDE on the continuum torus, the field equation at every resolution, uniqueness in law for the scalar form of the FBSDE, a renormalised variational formula with no infinite constant, and quartic exponential tails. The results presented in Lectures\U7 were obtained recently, with the use of AI, in collaboration with Sarah-Jean Meyer. > <\table-of-contents|toc> |.>>>>|> The energies> |.>>>>|> Moments at fixed cutoff |.>>>>|> > The field equation |.>>>>|> > |-0.3em|>|0em||0em|>>5 |.>>>>|> > |.>>>>|> > |.>>>>|> > Thresholds and energies |.>>>>|> > |.>>>>|> > Forward coercivity> |.>>>>|> The scalar model |.>>>>|> > |.>>>>|> > The lattice version |.>>>>|> > A pointwise bound |.>>>>|> > |.>>>>|> > Backward feedback and closure> |.>>>>|> The backward representation |.>>>>|> > Three contributions |.>>>>|> > Closing the loop |.>>>>|> > Consequences |.>>>>|> > |.>>>>|> > |.>>>>|> The boundary at infinite resolution> |.>>>>|> The boundary lemma |.>>>>|> > UV stability |.>>>>|> > Restarted laws |.>>>>|> > |.>>>>|> > Tightness and tails> |.>>>>|> One Hilbert space |.>>>>|> > Spatial suprema by a maximum principle |.>>>>|> > The terminal field |.>>>>|> > Exponential moments |.>>>>|> > |.>>>>|> > Identifying the limit> |.>>>>|> The deterministic coefficients |.>>>>|> > The exact flow as a controlled explicit flow |.>>>>|> > The limit law |.>>>>|> > The filtering model and the FBSDE |.>>>>|> > The renormalised field equation |.>>>>|> > Uniqueness in law and the variational formula |.>>>>|> > What is not proved, and where to go next |.>>>>|> > |.>>>>|> > |.>>>>|> > |.>>>>|> 6.> We recall the symbols used throughout; the definitions are in Sections\U, 11 and12 of Lecture. <\itemize> > is the periodic lattice of mesh > in the unit torus, =\>>, f=N*f>, |u,v|\>=u*v>, and |\|\<\|\|\>>> is the > norm for >. For a kernel, |C|\<\|\|\>>>=sup|\|>\z> is the row norm, which bounds |C|\<\|\|\>>p>> for every when is symmetric. =\> is the nearest-neighbour Laplacian. \U. The resolution ,\|)>> is the squared mass of the regulator, and \0> is the squared mass of the Gaussian reference. =q-\> is the precision and =K> the covariance of the scales not yet resolved at resolution ; |C|\<\|\|\>>p>=q>. In particular >> is the covariance of the Gaussian reference and >=q-\> the free field operator. The resolved covariance is =C-C>, with Q=C>, and the bubble =|)>\q> is the variance of the shell at resolution , by . . > is the quartic coupling, |)>=|12>\>, so that U=\*y> and the gradient of the interaction is |3>*y>. =\*q> is the dimensionless coupling and =|)>> the field size at which the cube balances >. >> is the auxiliary scale above which > is small. (Section). > is the Polchinski process, with >=0> and >\\>; is the Brownian motion on > driving it, the Brownian motion of the bridge , and > the conditional expectation given the past of >. The forecast is =\\>>, and > is the conditional covariance of >>. The filtration is =\:s\q|)>> (the same letter as the exact force, as in Lecture; the meaning is always clear from the context). The forecast is a function of the field, =m|)>>, and so is the approximate forecast, =y|)>>, where > solves y+|\>=Kx>. \U, . > is the exact force in the forecast coordinate, \=K\+\|)>>, and |\>=|3>*y+M*y+|~>> the explicit approximate force : the cube, a linear operator =\I+\>, made of the exact running scalar mass and the centred kernel =\>> produced by the iteration, and the clipped corrections |~>>. =+D|\>|)>> is the approximate response, =-H\> the drift produced by the force defect >, and =-H|)>>. In the coordinate equation, =C**y+|~>|)>> collects the linear and the clipped parts of the explicit force. At the end of Lecture we had, at each lattice cutoff , the exact system (Theorem) <\equation> Y>||)>+H|)>*Z|]>\q+H|)>\W,>>|Z>||)>+H|)>*Z|]>\q+\\W,q\q,>>>>> where is the approximate forecast built from the explicit force |\>>, -Y> is its error, and =Y+Z> is the exact forecast, a martingale. The inputs from Lecture, all for q>>, are: <\enumerate-Alpha> (Lemma): > is comparable to +\*y|)>> and <\equation*> |H|\<\|\|\>>p>\|)>/q, =C*+g|)>>. (Proposition): >+b>> with |b>|\<\|\|\>>>\C*\*q> and <\equation*> |b>|\<\|\|\>>\C*\*q|t|\<\|\|\>>, where =y*\r>>. : <\equation*> K*+\|)>|)>+|3>*Y=K*\, with a small Lipschitz correction, |D*\|\<\|\|\>>p>\\>. , used from Section on: the global linear bound |b|\<\|\|\>>\C*\*q|y|\<\|\|\>>> of Proposition; the bound \C*a> of ; below >>, the frozen force \U; and, for the density, the defect bound together with the bound of the Supplement on the large-field defect > before multiplication by . The goal is to show that is small and that rarely visits large fields, >. This is UV stability. Neither equation can be estimated alone: the forward equation contains , and the source of the backward equation is controlled by the large fields of . \; For an FBSDE with Lipschitz coefficients one estimates |Z|\<\|\|\>>> by Itô's formula and Gronwall, and closes either on a short time interval or under a monotonicity condition. Here the time interval is infinite, the coefficients are not uniformly Lipschitz in the relevant norms, and the coupling is not small in a Lipschitz sense. What we have instead is structure: the backward equation is linear in , the forward equation is a memoryless cubic elliptic equation at each resolution, and its noise is sub-Gaussian whatever the interaction (Proposition of Lecture, as far as we know a new observation of these notes). \; The system starts at > with >=Z>=0>. The energies below are integrated from >> (Section), and no value of at >> is used: this is what the memoryless field equation makes possible. Below >> everything follows from the martingale property of the forecast (Section). Sections\U prove these estimates at a fixed lattice and a finite upper scale. Sections\U remove the cutoffs and identify the limit. \; Each estimate has one job. <\itemize> (Lemma): the noise > has free-field tails whatever the interaction. This is where the posterior structure enters. (Proposition): pathwise, at each scale, the cube bounds the large-field part of by the noise tails and . It controls where the explicit force is inaccurate. (Lemma, Proposition): is the transported defect, as the equation is linear in it. Its energy is bounded by the bounded-field source, a boundary term, and /q>> times the large-field action. This factor is the only smallness that closes the loop. (Lemma): qualitative, at fixed . It removes the boundary term, and it is why the upper scale goes first. (Theorem): the result of these four, |Z|\<\|\|\>>\ C*q> uniformly in . It says that the control turning the explicit flow into the exact one has finite entropy. (Proposition): the same from any field at any scale, and below >> by the martingale property of the forecast. (Lemmas, ): the defect is integrable along both flows, and the explicit flow is a strong continuum SDE, continuous in the cutoff. (Section): tightness in >, >. (Proposition, Theorems\U): quartic tails by a linear source, a restart at ,h*C>f|)>>, and Gaussian tails by a quadratic tilt. These are end results, and nothing else depends on them. (Proposition, Lemmas, ): after closure, large fields occur only at finitely many dyadic scales. Neither the loop nor the identification of the limit uses them. So the formula for > (Theorem) needs the first five at the start of the flow, and if \*q>> the deduction below >> of Proposition there, all with first moments of the energies; plus the defect, the coordinate, the tail, the consistency of the coefficients, and moments of the explicit flow, which are Itô's formula. The FBSDE then follows by Girsanov's theorem (Theorem), with no further estimate on the flow, only the convergence of and and lower semicontinuity. The restart at every > gives the kernel >: the formula (), the statement that the forecast is a continuous function of the field, its convergence at each scale, and Section; the sources use it at ,h*C>f|)>>. Higher moments of the energies serve only for the stretched exponential bound, the higher moments in Corollary and in Lemma, and () with 1>. \; Detailed proofs of what these lectures cite or condense are in the .\ \; The results presented in Lectures\U7 are joint work with Sarah-Jean Meyer, obtained recently with the use of AI. M.Gubinelli is partly supported by UK Research and Innovation via the grant \PStochFields\Q EP/Z534328/1 and by the Simons Collaboration on Probabilistic Paths to QFT. S.-J.Meyer is supported by the grant \PStochFields\Q EP/Z534328/1. All estimates below are first proved on a finite interval >,T|]>> and at fixed . There, everything is finite. <\lemma> At fixed and , the maxima over >,T|]>> of ,Y,\,Z> have all moments, and so does >>||\<\|\|\>>>+|\|\<\|\|\>>>|)>\q>. Consequently the energies >, > and > defined below have all moments at fixed . <\proof> > has all maximal moments, since > has all moments. Global coercivity, +D|\>\c*q> and |\>=0>, gives |Y|\<\|\|\>>\C|K\|\<\|\|\>>/q>. Doob's inequality for the martingale =\*\>> gives its maximal moments, and -Y>. For the integrands, |Cov|\<\|\|\>>>\Cov\\|\>|\<\|\|\>>>, and |H|)>|\<\|\|\>>>\C>/q>. The energies involve, besides and , the field > of Section and |)>>; finite-dimensional norm equivalence and |\|\<\|\|\>>\\|y|\<\|\|\>>> give their moments. These constants depend on and . They are used only to justify the absorptions below, never in the final bounds. 5> In the profile , where =C> and the regulator is >, the noise observable of Theorem of Lecture is <\equation> \\K\-|)>*\=\>|]>, the second form by the bridge (), <\equation*> K\=|)>\>+B>. Since =K-\|)>=-\|)>> and =K>+q-q>, the field equation for the forecast of Lecture reads <\equation*> K>\+\|)>=\: an elliptic equation for the forecast at resolution , in which the exact force > acts with no smoothing operator, and whose noise is controlled whatever the interaction (Proposition of Lecture, here in one line). <\lemma> For every convex > on >, *\|)>\\*\>|)>>, whatever the interaction. In particular \C*\> satisfies, for all and \>, <\equation> \*\>\exp |2>*|)>*a|)>,|)>*a\C, uniformly in . <\proof> Conditional Jensen in , with =\|C*\,f|\>>>, and . The exact force > is unknown, and so is its coercivity. We replace it by the explicit force of , <\equation*> |\>=K\+|3>y, and the forecast by the approximate forecast, =Y+Z>. The coordinate equation Y+|\>|)>=K\> and give the exact identity <\equation> K>Y+|\>|)>=\+|)>*Z. This is the forecast equation of Lecture with the explicit force in place of the exact one, and it is the equation we estimate: <\itemize> it is an elliptic equation at each resolution, with : the past enters only through the noise >; the coercivity comes from the cube of |\>>, which is explicit; the price of the replacement is the error |)>*Z>, the only trace of the backward equation, at the same resolution; the identity is pathwise: no expectation has been taken. The noise > is rough, with covariance dominated by that of >>, a white noise; pointwise estimates need a noise which is a function. So we use =C\=\B>|]>>, the filtered version of the Gaussian bridge noise > of Section of Lecture, which by has the tails of the free field at resolution . It is not Gaussian, and it depends on the interaction through >, but its bounds do not. Since >\=\-|)>\>, subtracting it from and writing <\equation> V\Y+\|)>-\, (> is not to be confused with the potentials >, > of Lectures\U and of Sections and18.2; the potentials are functions of a field, > here is a random field) we obtain in the form in which it is estimated, <\equation> K>*V+|3>*Y=|)>*-\|)>+Z|)>,Y=V+\-\|)>. The mass of is only >, so at large resolution all the coercivity must come from the cube. The next section shows that it does. Define the signed excess of over three quarters of the threshold, the large-field part of , and the large values of the noise, <\equation> w=|)>*|\|>-r|)>,t=Y*|\|>\r>,\=\*|\|>\r/32>, all pointwise in space, with =|)>>. The three random energies are <\equation> |>|>*>>q|Z|\<\|\|\>>\q>|,>>|>|>>q*|w|\<\|\|\>>+\|w|\<\|\|\>>+\|t|\<\|\|\>>|]>\q>|,>>|>|>>*|\|\<\|\|\>>+\*q|\|\<\|\|\>>|]>\q>|.>>>>> Only sixth powers are integrated over all scales: |\|\<\|\|\>>> can be of order >, so an unthresholded noise contribution to would not be integrable uniformly in . The weights are matched to each other and to the source bounds: <\padded-center> ||>||Z|\<\|\|\>>>>|>>|\\*q|Z|\<\|\|\>>>>>||t|\<\|\|\>>>>|>>||b>|\<\|\|\>>\C*\\\*q|t|\<\|\|\>>>>>||b>|\<\|\|\>>>>|>>| and >s\s=2*q>>>>>> <\exercise> Check the three weights in the table. Show from that the forward estimate, with the weight > on the large-field action, needs the weight > on |Z|\<\|\|\>>>. Show that the backward estimate of Section also holds with weight >> for \\>, keeping the other weights of the table (> on the large-field action, > on |b>|\<\|\|\>>>) as in the proof. Which term breaks at =>, and how does the feedback coefficient depend on >? (Hint: in Proposition the boundary integral is taken up to >.) <\exercise> Show from Lemma that |\|\<\|\|\>>\15*|)>*a|)>\C*q> and |\|\<\|\|\>>\C*q*exp /\|)>>. For =0>, compute the covariance of > and compare it with the bound. On which modes is the bound sharp? <\exercise> Let =>C\W>, so that *X=W+>X\s>, and =Y+\|)>-X>. Using \=W+>\\s>, derive the equation with memory <\equation*> K*V+|3>*Y=>-\|)>+Z|)>*\s. Below the auxiliary scale > comes from the frozen force, and is not small. Here the noise is exactly Gaussian, but the memory has the size of the mass term. Why would testing it with > need the Hardy inequality |qf|\<\|\|\>>\q|)>>\|f|\<\|\|\>>\q|)>>>, and where would the margin come from? What does trade to avoid this? On one site the Laplacian disappears. We also drop the small Lipschitz correction > (restored in Proposition) and treat the noise and the error as given numbers. Then is a scalar equation at each , and all the constants can be checked by hand. <\lemma> Let ,q\0>, 0>, <\math> 0\\q> and \> with <\equation> a*v+|3>*=\*. Let |)>>, *-*r|)>>, =g*\r/32>>. Then <\equation> *w+|16>*w\54*q*\+*\*\+>*q*z. Moreover *\r>> satisfies \*+|\|>|)>>. <\proof> Multiply by >, where 0>; there =+*r> and have the same sign, so \0>. Since \+|\|>>, *\+*r-|\|>> (we give up *r> to leave room, on the lattice, for >). The cube being increasing, and *r=q> give <\equation*> |3>*w*\|12>*w+*w+-*\*\. This is the large-field coercivity: above the threshold the cube provides a sixth power, a quartic term of the size of a mass , and a cubic term which dominates the noise below the threshold. On the right side, *g*w\+q|\|>>, and the first term is absorbed. By Young's inequality, |\|>\*w+54*q*\> and \|48>*w+>*q*z>. This gives . For the last claim, \+*r++|\|>>, so on \r|}>> we have \-*r\+|\|>>. <\lemma> For \0>, |)>\*x-65*\>. For ,r\0>, <\equation> *u**r-\|)>\*u+*r*u+*r*u-*\. <\proof> *x+64*\-3*x*\=*|)>*|)>\0>, so |)>\x-3*x*\-\\*x-65*\>. Apply it with , *r>: *u*|)>\*u*x-*u*\>, and \u+3*b*u+b> with *b=*r\*r> and *b=*r\*r>. Finally *u*\\*u+*\>, and -=>. The excess of the forecast over the threshold is controlled, at each resolution separately, by two things only: how large the noise is, whose tails are known and exponentially small at large , and the error . There is no smallness in the coupling: the estimate is uniform in > once the threshold scales like =|)>>. It is pathwise, and it needs no integration in . <\proposition> For >> large (depending on >), at every cutoff , pathwise and for every q>>, <\equation> q|w|\<\|\|\>>+\|w|\<\|\|\>>+\|t|\<\|\|\>>\C*|\|\<\|\|\>>+\|\|\<\|\|\>>+\*q|Z|\<\|\|\>>|)>. Consequently, for every , <\equation> J\C*+E|)>,|G|\<\|\|\>>|)>>\C*\*p1|)>, with independent of , and . <\render-proof|Proof: the changes from the scalar case> Test with > in |)>> and proceed as in Lemma, with Hölder's inequality in space. Three new terms appear. |w,K>*V|\>\0>. Indeed =\> for the convex function =*-*r|)>>, \0>, and for any nondecreasing >, <\equation*> |\,-\*V|\>=Nx>|)>-\|)>|)>|)>*-V|)>|)>\0, summing over unordered nearest-neighbour pairs. So the elliptic term only helps. Split =\+\> according to \r> or not (writing =L*y> with the averaged derivative). Then |\|\<\|\|\>>>\\*r\r/32> shifts the threshold, and |\|\<\|\|\>>\\|t|\<\|\|\>>>, . Then \+*r+|\|>+|\|>>; on \r|}>> this gives \+|\|>+|\|>>, so for small > <\equation*> |t|\<\|\|\>>\C*|w|\<\|\|\>>+|\|\<\|\|\>>|)>,|\|\<\|\|\>>\C*\*|w|\<\|\|\>>+|\|\<\|\|\>>|)>. This also supplies the part of . These are operator bounds: > is nonlocal, but small in every >. In the cube and on the right side of the error is =|\|>+|\|>>. The bounded parts on the right cost |w|\<\|\|\>>>, which is absorbed by |w|\<\|\|\>>> from the cube. The cube error is at most *|\|\<\|\|\>>+\|w|\<\|\|\>>|)>>, and Young's inequality gives |\|\<\|\|\>>+\|w|\<\|\|\>>|)>>. The parts are absorbed for small >: these are deterministic smallness conditions on >>>, independent of . |w,|)>*Z|\>|\|>\|48>|w|\<\|\|\>>+>*q|Z|\<\|\|\>>>. Multiplying by > and integrating gives \C*+E|)>>. By Minkowski's inequality in , for , ||\|\<\|\|\>>|\<\|\|\>>|)>>\sup||\|>|\<\|\|\>>|)>>>, and by Hölder and , with threshold /32=*|)>>, this is at most <\equation*> |\|>|)>>*\|\|>\r/32|)>>\C*p*q*\/|)>>, and integrating in with /|)>> gives |G|\<\|\|\>>\C*\*p>. No information on the interacting field has been used, apart from the equation itself and the posterior structure of the noise. The tails of > are those of the free field whatever the interaction: this replaces the subtraction of a Gaussian field. The maximum principle gives more than the integrated estimate: a bound in >> driven by the cube alone. <\proposition> >> coercivity>For >> large, at every cutoff, pathwise and for every q>>, <\equation> |Y|\<\|\|\>>>\C*>|)>|\|\<\|\|\>>>+|\|\<\|\|\>>>+q+q|Z|\<\|\|\>>|]>. In particular there is 0> such that =0>, i.e. |Y|\<\|\|\>>>\r>, whenever |\|\<\|\|\>>>+q|Z|\<\|\|\>>\c*r>: large fields occur only when the noise or the error is large. <\proof> Let =q-q>, -\|)>>, and remove the error by =\*C>*Z>. Then |\|\<\|\|\>>>\\*a|)>|Z|\<\|\|\>>\C*q|Z|\<\|\|\>>>, since the kernel of >> is in > uniformly in by . By , > and +d> satisfy <\equation*> K>*W+|3>*=\*d,Y=W+D. At a positive maximum of , >*W\0>, so |d|\<\|\|\>>>/\|)>> there; similarly at a negative minimum. Hence |Y|\<\|\|\>>>\2|D|\<\|\|\>>>+|d|\<\|\|\>>>/\|)>>. With |\|)>|\<\|\|\>>>\\|Y|\<\|\|\>>>>, the term |Y|\<\|\|\>>>> is absorbed, and by Young's inequality |Y|\<\|\|\>>>/\|)>\|Y|\<\|\|\>>>+C*/\|)>>, where /\\C> since =\*q>. This proves . For the last claim use =r>: the terms of are at most +c+g|)>*r>, and \g>>> is small. <\lemma> Uniformly in and in the interaction, <\equation*> |\|\<\|\|\>>|)>>\C,\\supq>>q*|)>|\|\<\|\|\>>>. <\proof> At fixed the norm |C*f|\<\|\|\>>>> is convex, so by Lemma |\|\<\|\|\>>>> is dominated by the supremum of the Gaussian field *B>>. Its variance is at most >, and for \1> its increments have variance at most *|)>>> uniformly in , so Dudley's bound gives the mean q>> and Gaussian concentration gives fluctuations at the scale >. For the supremum over >, write =C*K*C>, where |K*C|\<\|\|\>>\\>\2>. Since \=|)>|)>\W> by and , > is a martingale, |C*\|\<\|\|\>>>> is a submartingale, and Doob's inequality gives, for 2>, |sup>|\|\<\|\|\>>>|\<\|\|\>>>\C*Q*>+|)>>. Markov's inequality with \*log > bounds the probability that the normalised supremum over > exceeds > by >>, for \\>. This is summable over dyadic , and |\|\<\|\|\>>\C> follows. <\exercise> Replace |3>*> by in . Show that *a*>, so that the estimate fails as \> at fixed mass . Which term of replaces the missing mass, and how does the threshold |)>> enter? <\exercise> Prove that |\,-\*u|\>\0> for every convex \C> and H>, and that equality holds iff |)>=\|)>|)>> on every edge x> with \u|)>>. <\exercise> Solve numerically for random ,q,a,\,g,z>, and check . Take Gaussian with variance >, , =1>, and compare > with /\|)>> and with . Is the exponent of |)>\q> attained on one site? What should improve it on the lattice? The backward equation in is linear in . Its propagator is the adapted operator solution of <\equation> \*\=\*H|)>,\=I,s\q. By (A), |\|\<\|\|\>>2>\exp |u>\u\L*>, with =exp >>>\*u|u>|)>\exp >>+g>>|)>|)>>. <\lemma> For >\q\T>, at fixed , <\equation> Z=\**Z+\*b|)>\s|]>. <\proof> By the product rule and, *Z|)>=\*H*Z\s-\*+H*Z|)>\s+\*\\W=-\*b\s+\*\\W>. The stochastic integral is a true martingale by Lemma and the deterministic bound on >. Integrate from to and take >. So is the expected source accumulated on the unresolved scales, transported back to , plus the boundary error. This is the conditional-expectation form of a backward equation, as in =\|]>> for the force (Theorem of Lecture); here the source is the defect. Split >+Z>+Z>> according to the terminal term and the two parts of the source. We estimate each in the scalar manner: norms of the propagator times norms of the source. By (B), <\equation*> |Z>|\<\|\|\>>\>L**C*\*s\s=2*L*C*\*q,>*>>>q|Z>|\<\|\|\>>\q\C*\*q>. This is where the two correction steps of Lecture are used: with one step the source would be |)>> and the first integral would diverge logarithmically. |Z>|\<\|\|\>>\L*\*|T*Z|\<\|\|\>>>, so, with \|T*Z|\<\|\|\>>*;L|)>>>, <\equation*> >*>>q|Z>|\<\|\|\>>\q|\|>|)>>\|\>*>>>q\q*\=|\*q>>*\. At fixed , \0> as \> (Lemma). Let >\>>s|b>|)>|\<\|\|\>>\s\C*\*s|t|\<\|\|\>>\C*\*J> by (B). By Cauchy\USchwarz in , <\equation*> |Z>|\<\|\|\>>\|q>*\*s|b>|\<\|\|\>>\s\|q>>s\s|)>*\>|)>|]>, so that, by Jensen's inequality, |Z>|\<\|\|\>>\2*L*q*\*A>>. Integrating, and using that > is a contraction on |)>>, <\equation*> >*>>q|Z>|\<\|\|\>>\q|\|>>\|\>*>>>q\q|A>|\<\|\|\>>>\|q>>|J|\<\|\|\>>>. <\proposition> At every cutoff, <\equation> |E|\<\|\|\>>>\C*\*q>+C*\*q>*\+C*\*q>|J|\<\|\|\>>>, with independent of , and . <\proof> |Z|\<\|\|\>>\3*|Z>|\<\|\|\>>+|Z>|\<\|\|\>>+|Z>|\<\|\|\>>|)>> and the three estimates above. <\key-result> <\theorem> There is >>, depending on >, such that for every , and 1>, <\equation> |E|\<\|\|\>>>+|J|\<\|\|\>>>\C>*+\|)>. <\proof> Let |E|\<\|\|\>>> and |J|\<\|\|\>>>. By Propositions and, <\equation*> y\A**p+x|)>,x\B+B*\+\*y,\=|q>>, with ,B,B> and the constant in > independent of , and bounded for all >> beyond the thresholds already required. Choose >> such that also *\\>. Then A**p+B+B*\|)>+*y>. Since \> by Lemma, it can be absorbed, and follows. The structure of the argument is the point: <\padded-center> <\script-output|tikz|default> \\documentclass[tikz]{standalone} \\usepackage[T1]{fontenc} \\usepackage[sc]{mathpazo} \\usepackage{amsmath,amssymb,mathtools,bm} \\usepackage[mathscr]{eucal} \\usetikzlibrary{decorations.pathreplacing,arrows.meta,positioning} \\usepackage[dvipsnames]{xcolor} \; \\begin{document} \\begin{tikzpicture}[ box/.style={draw=MidnightBlue,rounded corners=2pt,align=center,inner sep=4pt,font=\\small}, arr/.style={-{Stealth[length=2.2mm]},thick,draw=MidnightBlue!80}] \\node[box] (J) at (0,0) {large-field action $J_T$\\\\(forward, pathwise)}; \\node[box] (E) at (7,0) {backward energy $E_T$\\\\(backward, in $L^p$)}; \\draw[arr] ([yshift=0.2cm]E.west) -- node[above,font=\\small]{$J\\le C(G+E)$} ([yshift=0.2cm]J.east); \\draw[arr] ([yshift=-0.2cm]J.east) -- node[below,font=\\small]{$E\\le B+\\frac{C\\lambda^2}{q_\\ast}J$} ([yshift=-0.2cm]E.west); \\end{tikzpicture} \\end{document} |ps>||||>> The smallness is in the , /q>>, not in the coupling of either equation, and it is independent of the moment order, so a single >> closes the loop for every . The growth > is then inherited from the tails of the noise, >, and gives stretched exponential moments *exp >+J>|)>|}>\\> once the boundary term is removed. Note that /q>=g>>>. The boundary term > is removed in Section, at each fixed , by letting \>. Assuming this for the moment, we obtain <\equation> sup|E>|\<\|\|\>>>+sup|J>|\<\|\|\>>>\C>*|)>,sup \|Z|\<\|\|\>>\C>*q. The second bound follows from the representation formula on |)>>: the three contributions are, up to the factor , |)>>, and, by the computation above, *q*\*A>>. So the approximate forecast differs from the exact one by |)>> in ;L|)>> (root mean square), uniformly in the lattice. Since |H|\<\|\|\>>\C/q>, the coupling term *Z> in the forward equation is |)>> in ;L|)>>, integrable in : the forward equation is, up to an integrable drift, the explicit SDE Y=b*\q+H*\W>. The same computation holds along the path. At > the boundary part vanishes (Lemma) and, with >> now integrated up to > (it has all moments, since >\C*\*J>>), |Z|\<\|\|\>>\3**q>+2*L*\*A>|)>>. Since *A>> is a martingale, Doob's inequality gives, for 2>, <\equation> |supq>>q|Z|\<\|\|\>>|\<\|\|\>>|)>>\C>*|)>. <\exercise> On one site, with =*y|)>>, show that <\equation*> \=*exp *Y|u**Y|)>>\u|)>\. In which sense do large fields make the backward equation more contractive? Redo the three estimates of Section on one site. <\exercise> Give an example of nonnegative numbers (possibly infinite) with 1+x>, 1+*y> and >. Where is Lemma used in the proof of Theorem? <\exercise> From |E>+J>|\<\|\|\>>\C*|)>> for all 1>, deduce *exp >+J>|)>|}>\\> for small . Sections\U proved () at every lattice cutoff and every upper scale . Three things remain. <\enumerate-roman> Show that the boundary term vanishes: \0> as \> at fixed (Section). Turn the uniform energy bounds into tightness of the measures (Section). Identify the limit: the deterministic coefficients; then, from the restarted flow, the limit as an explicit density over an explicit flow, along the whole sequence; then the equations, read off by Girsanov's theorem (Section). At a fixed lattice, the terminal resolution is >, and the field at resolution , >, converges to the terminal field >>. The approximate force |\>> was built to converge to the bare force: \\>, \0>, and the nonlinear corrections were integrated from >, so that they vanish there. This is the boundary condition of the backward equation. <\lemma> At fixed , for every \>, as \>, <\equation> |Y-\>|\<\|\|\>>;H|)>>+|q*Z|\<\|\|\>>;H|)>>\C*q,\\C*T. The constants depend on ; this is not a uniform estimate. <\proof> All norms on > are equivalent, and we do not track the dependence on . .> By , =|)>*C*\>+C*B>>, and |)>*C-I=-C*K>=O|)>>, |C*B>|\<\|\|\>>>=O|)>>. So |\-\>|\<\|\|\>>>\C*q>. Global coercivity gives |Y|\<\|\|\>>\C|K\|\<\|\|\>>/q>, bounded in every > uniformly in . The coordinate equation =Y+C*|\>|)>> and the bounds |\|>\C>, |\|\<\|\|\>>>=O|)>> and ||~>|\<\|\|\>>>\C>|)>|y|\<\|\|\>>>\C*\*q|y|\<\|\|\>>>> (Taylor from the four-leg vertex, since the low jet vanishes) show that |\>|)>> is bounded in >, so |Y-\|\<\|\|\>>>=O|)>>. Moreover -\|\|>=O|)>>, and therefore <\equation> ||\>|)>-D*V>|)>|\<\|\|\>>>\C*q. Thus Subtract the approximate coordinate equation \=K*Y+|\>|)>> from the exact one ( and Proposition(b)), \=K*\+\*>|)>|]>>: <\equation> q*Z=q*C*|\>|)>-\*D*V>|)>|]>. Let >|)>> and =D*V|)>>, which is >-measurable. Then |\*G-G|\<\|\|\>>>\|\*|)>|\<\|\|\>>>+|G-G|\<\|\|\>>>\2*|G-G|\<\|\|\>>>=O|)>>, by the cubic Lipschitz bound for > and the first step. With and |q*C|\<\|\|\>>\1> this proves the claim. The proof uses no estimate uniform in , and no concentration of the posterior: only that the Polchinski process >, which is >-measurable, converges to the hidden field. In Lecture language, the forecast and the field at resolution have the same limit at infinite resolution, and the force =K*-\|)>> converges to the bare force >|)>> (Exercise; recall =\*>|)>|]>> by , so this is the conditional-expectation step of the proof). Now let \> at fixed in: the paths and the law > do not depend on , > and > increase to >> and >>, and \0> by Lemma. Only then take the supremum over . <\key-result> <\theorem> For >> large enough, depending on >, <\equation> sup|E>|\<\|\|\>>>+|J>|\<\|\|\>>>|)>\C>*|)>,sup \|Z|\<\|\|\>>\C>*q, and <\equation*> sup \*exp >+J>|)>|}>\\ for small 0>. <\proof> Monotone convergence in, then the representation formula on |)>>, whose terminal term vanishes by Lemma. The exponential moment follows from the polynomial growth by Markov's inequality with R>. The coordinate is Lipschitz for the uniform norm, by the maximum principle. <\lemma> Uniformly in , |y-y|)>|\<\|\|\>>>\3|x-x|\<\|\|\>>>> for q>> and with the constant for *q>>. So the drift in () is Lipschitz on >> with constant . <\proof> With =y-y>, >, +\-\|)>>, gives +|3>-|)>|)>=0>. At a positive maximum of , \ s\0>, so \|)>> there, i.e. \0> and |\-\|)>|\<\|\|\>>>+|h|\<\|\|\>>>>; similarly at a negative minimum. Since > is >-Lipschitz on >>, |\|\<\|\|\>>>\2|h|\<\|\|\>>>+2\|\|\<\|\|\>>>>. For *q>> use and |C>>K|\<\|\|\>>\\>\2>. Finally |C|\<\|\|\>>\\>=1/s>. The estimates above use the start of the flow only through the noise. So they hold, with the changes below, for the flow restarted from any field at any scale, and these estimates are the input of Section. Let >=\\>C\ W|)>>>, and define the exact force =-D*V>, and let > be the exact flow conditioned on =x>, \\=-CD*V|)>\ s+C\ W>. Fix , q> and H>, |x|\<\|\|\>>>>. By the Markov property of the flow, > is the law of =Cx+|^>+B|)>>, q>, where > is a Brownian motion independent of |^>>, and |^>> has the law <\equation> \>\ z|)>\exp|z,Kx|\>-|z,Kz|\>-V|)>\ z. Note that ,0|]>>=\>, so ,0>> is the flow itself. The posterior given > is exp|z,K\|\>-|z,Kz|\>-V|)>\ z>, the same function of > as for >. So =m|)>>, and >, =\-Y> and the system () are unchanged, with the same coefficients and with =y>, |Y|\<\|\|\>>>\6R> by Lemma. Let =K\-\=Kx+\B>, =C\>. Then <\equation> \=d+\, d=CKx,\=C\B. Here K=I-C> and C> has a positive kernel of total mass at most , so |d|\<\|\|\>>>\2R>. By conditional Jensen, > satisfies Lemma, with a\ C>. With the field =Y+\|)>-\> of () (not the potential >, always written with its argument), the computation of () with > replaced by gives the field equation <\equation> KV+|3>Y=-\|)>+Z|)>. It has the form of (), with mass \0> and t-q\ t>, and it holds for q>> also when *q>>, as it uses only =K+t-q> and the coordinate equation at . <\proposition> Let >=>|)>>. With energies on >,\|)>> and as in (), for 1>, with independent of , <\equation> |E>|\<\|\|\>>|)>>+|J>|\<\|\|\>>|)>>\ C+pR|)>, <\equation> \|Z|\<\|\|\>>\ C|)>s,\|Y|\<\|\|\>>\ C|)>+s|)>, s\ q>. Below the auxiliary scale, if *q>>, for 1>, <\equation> sup s\ q>>\|\|\<\|\|\>>+|Y|\<\|\|\>>>+|Z|\<\|\|\>>|)>\ C. <\proof> Proposition applies pathwise to (): its proof uses only a nonnegative mass, a right-side coefficient >, the positivity of |w,KV|\>> and the bounds on >. So \ C+E|)>> with > built on =\1|\|>\*r/32>>. Let =\ R>, so that r/64> for q>. For such , |\|>\*r/32> forces |\|>\*r/64> and |\|>\2|\|>>, and the proof of () applies to > with this threshold. For t\*q> use |\|\<\|\|\>>\ C+|\|\<\|\|\>>|)>> and ||\|\<\|\|\>>|\<\|\|\>>>\ C*pt>. Then the part of on >,q|)>> has > norm at most <\equation*> C>>>+pt|)>+\ t+pt|)>|)>\ t\ C*p|)>, the constant depending on >. So |G|\<\|\|\>>\ C+pR|)>>. Lemma holds as >> has all moments. Lemma and Proposition hold on >,T|]>>, as they use only the propagator (), the deterministic bounds (B) on >,b>> and the conditional expectations; the feedback coefficient is /q>\ C\/q>>. No initial value of at >> is used. So Theorem holds with > computed under >. the proof of Lemma applies at fixed , with \|^>> and the coordinate equation \=K\+\D*V|^>|)>>, so \0>. This proves (). The bound on follows from the representation () as in (). The bound on follows from Itô's formula for |Y|\<\|\|\>>> in the system (), with |Y|\<\|\|\>>\3R> if q>> and the moments of >>> below if *q>>, the linear bound on (Proposition), |H|\<\|\|\>>\ C/s> and H\ C*s> (). The coupling term needs care: the naive bound |Y|\<\|\|\>>|Z|\<\|\|\>>C/s> would give the Gronwall factor C*s\s>, which is not bounded. Split instead |Y,HZ|\>|\|>\s|Y|\<\|\|\>>+C*s|Z|\<\|\|\>>>: the first part gives the bounded factor s\s>, and by the bound on the second contributes at most |)>s\s\\>. Let *q>>. By the representation () at >> and conditional Jensen, |Z>>|\<\|\|\>>> has all moments. By () at the scale >>, pathwise, |t>>|\<\|\|\>>\ C>|\|\<\|\|\>>+\|\|\<\|\|\>>+\q>|Z>>|\<\|\|\>>|)>>, and |Y>>|\<\|\|\>>\ r>>+|t>>|\<\|\|\>>>. So >>=Y>>+Z>>> has all moments in >. As > is a martingale, |\|\<\|\|\>>\\|\>>|\<\|\|\>>> for q>>, by Jensen. Now =y>>|)>> with =C>>K\=C>>Kx+C>>\+C>>\B>. By Lemma, |Y|\<\|\|\>>>\3|x|\<\|\|\>>>>, as >>=0>, and >>:L\ L>> is bounded uniformly in , while |C>>\B|\<\|\|\>>>> is dominated in every convex moment by the supremum of a Gaussian field at the fixed scale >>, by conditional Jensen as in Lemma. This bounds |Y|\<\|\|\>>>>. Then =\-Y>. This proves (). <\exercise> Show that, at fixed , =K*-\|)>\-D*V>|)>> in every >, and that =K*Z-|\>|)>>. Interpret the last identity: the force observable of Lecture is the explicit force at the approximate forecast, corrected by *Z>. <\exercise> Using the bound on the boundary term in the proof of Proposition, explain why \0> alone would not make it vanish, and why the rate |T*Z|\<\|\|\>>=o> of Lemma is what is needed. To compare lattices, partition > into cubic cells > of side > centred at \>, and let <\equation*> *u|)>=u*\|)>,*f|)>=N*>f,=\*j. :H\H=L|)>> is an isometry, =\>>, and \I> strongly. Fields are embedded by >, operators by \j>. The embedded innovation has covariance >; in the limit it becomes a cylindrical Brownian motion on , obtained as the limit of the lattice innovations. The energy bounds control > and > norms of large fields. Spatial suprema at finite resolution come from the >> coercivity of Proposition. <\lemma> Let =supq>>q*|)>|Y|\<\|\|\>>>> and =supq>>q|Z|\<\|\|\>>>. For >> large, with > the supremum of the noise of Lemma, <\equation> \\C>*+\|)>,|\|\<\|\|\>>>\C,|\|\<\|\|\>>>\C>*|)>, uniformly in . <\proof> By the definitions, |\|\<\|\|\>>>\q*|)>*\> and |Z|\<\|\|\>>\q*\>. Inserting these in , the first term gives *q*|)>*\>, and the others are of lower order. The bound on > is , whose boundary term vanishes by Lemma. Any exponent below > keeps below the threshold > at large . The exponent comes from the cube alone. The memory form of Exercise, with a weight in absorbing the memory, gives every exponent above >, at the price of that absorption. At the terminal resolution there is no such bound: >> is a distribution. The >> bound cannot replace Proposition in the loop: it is linear in |Z|\<\|\|\>>>, so its sixth power is not controlled by the energy , while the test with > gives a bound quadratic in . Once the loop has closed it does give more. Almost surely |Y|\<\|\|\>>>\ r> past a random scale, so large fields occur only at finitely many dyadic scales, and |)>> is then the bounded-field defect alone. Suprema over finite intervals of scales q>> have all moments uniformly in , which is the boundedness part of the hypothesis of Section along the exact flow, though Section does not need it, as it passes through the explicit flow. The same maximum principle, in the form of Lemma, gives the estimate below >> in Proposition. The approximate forecast converges at infinite resolution in negative Sobolev spaces, uniformly in (to the same limit as the forecast >). The martingale part of has bracket H>, and the positive inequality \*C> gives <\equation> >||)>*H|\<\|\|\>>>\q\C*>*|)>|q+c>\q\C*Q|)>>,\s\, where > bounds the lattice symbol from below (aliases of the cell embedding only increase the frequency). The drifts are integrable at infinity in : the bounded-field drift is |)>>, the large-field drift is controlled by q|b>|\<\|\|\>>\C*\*J>> and Cauchy\USchwarz, and |H*Z|\<\|\|\>>\q\C*q|Z|\<\|\|\>>\C*\*E>>. More precisely, the squared drift tails beyond are |)>> (bounded-field), |)>*\*J>> (large-field) and |)>*\*E>> (the term), all at most |)>>> for >. By Doob's inequality, for \s\>, <\equation> sup \*supQ>|Y-Y|\<\|\|\>>>\C*Q|)>>. Letting \> at fixed (Lemma) this holds with > replaced by >>. As in the proof of Proposition, |Y>>|\<\|\|\>>> has all moments, uniformly in . So \|\\>|\<\|\|\>>>\\> for \s'\>, and these spaces embed compactly in > for s'>: <\corollary> For every > the embedded laws \\> are tight in |)>>, and \*|\*\>|\<\|\|\>>>\\> for every . For the moments: the drift integrals have all moments by, and the > martingale has a deterministic bracket bound by at >>, the field at the lower scale has all moments by the paragraph above, so the Burkholder\UDavis\UGundy inequality gives all moments. The threshold > is that of the free field in : the interacting field is no rougher than the Gaussian one. At every finite resolution its part is smoother (>). Theorem gives only stretched exponential moments of the energies. Hairer and Steele showed that the measure has quartic exponential tails, as its formal density suggests, by tilting it and using a priori bounds for the tilted dynamics. Here is a proof in the language of this lecture, through linear sources: the filter is available for every prior, and the noise bound does not depend on the prior. Let H>, \>, and let |^>\\|\,f|\>>\>, which has all moments at fixed . Let > be the Polchinski process with terminal law |^>>, i.e. the bridge () built on |^>\|^>>, \=|)>|^>+B>>, and |^>=\|^>>. <\equation> KY+|\>|)>=K\+h*f, Z=|^>-Y. <\proposition> For q>>: the pair > solves () with the same ; \0> in every > at fixed ; the field equation () holds with the extra term on the right side. If \ q>> and \70\|f|\<\|\|\>>>>, then the conclusion of Theorem holds on ,\|)>>, with constants independent of >. <\proof> This is the flow restarted at ,x|)>> with >f>: then ,x|]>>=|^>> and \+h*f=K>x+|)>|^>+B>>. So the system and the boundary are as in Section; the extra term in the field equation is |)>d+K>d=h*f> with =h*Cf>. Since |d|\<\|\|\>>>\|f|\<\|\|\>>>/q\ r/64> for q>, as 70>, the proof of Proposition applies with no range below the threshold. The backward estimate and the closure only improve with >. There is also an exact identity with no cube and no derivative. As =K\-|)>|^>> and \=|)>|^>+B>> (here > is the unshifted noise, =\>|]>>), for every prior and *q>, <\equation> |^>=Y+Z+-B>|q-q>. <\theorem> Let |f|\<\|\|\>>>=|f|\<\|\|\>>+|f|\<\|\|\>>+|f|\<\|\|\>>>>. There are ,C> independent of and such that <\equation> \|\>,f|\>|\|>\ C|f|\<\|\|\>>>,\exp|\>,f|\>/|f|\<\|\|\>>>|)>\ C. Consequently, \exp>|\\>|\<\|\|\>>>>|)>\\> for \>. The same bounds hold for the limit, by Theorem. <\proof> Let |f|\<\|\|\>>>=1> and =max>,2q+1,|)>|)>>. Put ||^>,f|\>>. By (), \ r+|\|>> and \ q/2>, <\equation*> \ r|f|\<\|\|\>>+|t|\<\|\|\>>|f|\<\|\|\>>+|Z|\<\|\|\>>|f|\<\|\|\>>+|\-B>,f|\>|\|>. Average over ,2q|]>>. By Proposition, |^>J>\ C> on ,\|)>>; since |)>>>|t|\<\|\|\>>\ q\ J>>, Hölder's inequality gives the average of |^>|t|\<\|\|\>>> at most /\|)>>. Also |^>|Z|\<\|\|\>>\ C*q>, and the last term has expectation at most > by Lemma. So |^>\ C*r>+C\ C|)>>. With =log\\|\>,f|\>>> we have =|^>X> at fixed , so \ C+|)>>, and Chernoff's inequality gives |\>,f|\>|\|>\*R|)>\2\>> for R>. For the norm, take a real Fourier basis > and =|)>>>, with w\\>. As |\\,f|\>=|\,jf|\>> and |jf|\<\|\|\>>>\ C>, the bound applies to =|\\>,f|\>>. Cauchy\USchwarz gives wz|)>\ W wz>, and Jensen's inequality concludes. The scale \> is where the cube at its threshold, >, dominates the source; it plays the role of the time in coming down from infinity. On one site \/3|)>h>, so the powers are sharp there. The moments with \> are at a regularity far from the free threshold. At the threshold we have the Gaussian order, by a quadratic tilt. <\theorem> For > there is \0> with \exp|\\>|\<\|\|\>>>|)>\\>. <\proof> Let \*s\>, |2>|\\|\<\|\|\>>>>, |^>\\\>, with no shift. As in the proof of Lemma, with -W> in place of >, () gives \ D*W|^>|)>> at fixed , with |D*W|\<\|\|\>>\\|\\|\<\|\|\>>>>. Let |^>|\|^>|\<\|\|\>>>\\>. Then () with gives |^>>+J>|)>\ Cx|)>>, and the argument for the field at the lower scale gives |^>|Y>>|\<\|\|\>>\ Cx|)>>. Integrating the forward equation on >,\|)>>, with () for the martingale and Cauchy\USchwarz in for the drifts, Cx|)>>. So 2C> for small >. Thus |^>\> for the ball of radius , =4C>, in >. Since |^>\\\ K/2>>, this bounds =\\\2\ K/2>>. <\exercise> On one site, with \exp|12>x-|2>x|)>>, show that exp|12>x|)>> is finite if and only if \\>. Compute > asymptotically and compare with Theorem. <\exercise> Show directly on one site that |^>-y|)>\ h> if > solves |3> y=q*\>, and that \0> with (). Why does a quartic tilt have no such shift, and which power of the field does its boundary term produce? <\exercise> Deduce Carleman's condition for |\,f|\>> from Theorem. Why does the theorem exclude a nondegenerate Gaussian law for |\,f|\>>? <\exercise> On >,Q|]>>, test >V=|)>-\|)>+Z|)>-|3>Y> with >, and show from the energy bounds that \|V|\<\|\|\>>>\ q\\>. In which sense is the field at finite resolution smoother than the free field? The coefficients >, ,N>>, >, > are constructed at each lattice. Their uniform bounds (Proposition) do not by themselves identify a limit: a weakly convergent vertex cannot in general be contracted against a varying covariance. The three tools are: <\itemize> Derivative kernels with bounded rooted norms are signed measures with bounded mass; they converge weakly along subsequences, and the centred kernel > converges in a weak topology controlled by its first moment. The limiting scalar mass solves the same scalar fixed-point problem, whose solution is unique in its ball, and the centred kernels are explicit outputs of the iteration; so the whole sequence converges. If \G> in Hilbert\USchmidt norm, and > are symmetric, uniformly bounded and converge weakly to >, with |G|\<\|\|\>>\\\1> and |G|\<\|\|\>>\\>, then |)>*G\|)>*G> in Hilbert\USchmidt norm. The free resolvent is the compact factor that turns weak convergence of the derivative of the force into strong convergence of the response. The rows of the inverses are Hölder (any exponent >); with the Hilbert\USchmidt convergence given by the sandwich, the squared kernels |)>> converge uniformly to a continuous kernel. Weakly convergent vertices of bounded variation can therefore be contracted against them, and the loop steps of the recursion pass to the limit. With these, if \ Y> uniformly on > for almost every I>, with |Y|\<\|\|\>>>> bounded, in probability, then |)>\ H|)>> and |)>\ b|)>> in |)>> and >, in probability. By the resolvent identity =D|\>|)>C>, for bounded , > is small in Hilbert\USchmidt norm on high modes, uniformly in ; so the Fourier projection and > give the same limit, and \H|)>\ W\H\ W> in , in probability, when > is the projection of , and the processes are adapted to a filtration for which is a Brownian motion. For *q>> this holds with the frozen coefficients, as > in is the loop part of the defect, and is continuous across >> although jumps. Compactness would give limit points, and nothing in the FBSDE selects among them. A change of measure does better, following the idea of Gubinelli and Meyer for sine\UGordon: the exact flow is an explicit flow with control , and that control has finite relative entropy. Let > solve y+|\>=Kx>, let >=K> be the explicit force in the field variable, and recall the exact force =-D*V> of Section. Differentiating the coordinate equation in and gives the Burgers equation with a source, <\equation> \>+>D>+D>>|]>=-KH\\ y. Up to here we have needed only forces. To compare the laws of the two flows we need, for the first time, a scalar. Since |\>> is symmetric, |\>> is a gradient, and so are >> and the defect > (as >D|\>=-D H>). Let <\equation> e=|\*y|)>,y|\>\ \,>=-|>*x|)>,x|\>\ \, so that =\>, >=->>, =>=0>. Then (), together with K>, is the gradient of <\equation> \>+D>|)>-|CD>|\<\|\|\>>=e\ y+c, c=-H|)>, where the constant is read off at , as >=K-KHK>. Here >> is the value function of the explicit flow, in the sense of Section of Lecture, and is its running cost. This is the only place where a potential enters, and it is unavoidable: Girsanov alone gives a density in terms of the unknown , and only the identity =D*e> makes it explicit. So >> solves the Polchinski equation up to its defect. Since D>=C|)>>, the > from at scale is <\equation> \ x=-C|)>\ s+C\ W, x=x. Its coordinate |)>> solves \ Y=b\ s+H\ W>, >: indeed \x|)>=Y\ s+\ W>, and one applies Itô's formula to Y+|\>=Kx>. It contains no . All this holds for all q>; below >> the defect is . <\lemma> Let > be the pointwise projection on ,r|]>>. For all and q>>, <\equation> |\|>\ C\s|\y|\<\|\|\>>+C\s|\y|\<\|\|\>>+C\s|t|\<\|\|\>>. <\proof> On the ball =\>, so y|)>|\|>\ C\s|\y|\<\|\|\>>+C\s|\y|\<\|\|\>>> by the primitive formula and : the quartic term comes from the cubic part of the defect, the quadratic one from its centred linear part, as |y,\y|\>|\|>\|\|\<\|\|\>>>|y|\<\|\|\>>>. With >=\y+\y|)>>, -ey|)>=|\>|)>,y-\y|\>\\>, and y> is supported on the set > where \*r>, with norm at most |t|\<\|\|\>>>. Write =D\*\>z|)>+\>. Then >|\|>\ >|)>|)>/6+|\|\<\|\|\>>>2r\ C\s> and |\|>\ r|t|\<\|\|\>>>, so the first term contributes at most s|t|\<\|\|\>>>. By the term-by-term bounds behind Proposition (the four groups of Section of Lecture, proved in Section of the Supplement), which bound > before multiplication by (the bound stated for >=-H\> would not suffice, as > is unbounded), |\|\<\|\|\>>\ C*g|t|\<\|\|\>>+r|z\|\<\|\|\>>|)>>; since >|\|>\> and >> has size above > only on >, the second term contributes at most |t|\<\|\|\>>+r|t|\<\|\|\>>|t|\<\|\|\>>|)>>. On the support of > we have |\|>\ r>, so |t|\<\|\|\>>\ r|t|\<\|\|\>>> and |t|\<\|\|\>>\ r|t|\<\|\|\>>>. All these are at most r|t|\<\|\|\>>=C\s|t|\<\|\|\>>>. With =V->> (> is not the Laplacian in this section), =>c\ s> and =V->-\>, subtracting () from the Polchinski equation gives <\equation> \\+D\|)>-|CD>,D\|\>-|CD\|\<\|\|\>>+e\ y+c=0. This is a Hamilton\UJacobi\UBellman equation with the control , added as a drift u> to the explicit flow, \ Y=+Hu|)>\ s+H\ W>, and running cost |u|\<\|\|\>>>, with optimal feedback >=-CD\>. Moreover D\=C->|)>=Z>, since -x|)>> and >=-D>=-K>. So the exact flow is the explicit flow with control . <\theorem> At fixed , > and > are equivalent on the paths on |)>>, and <\equation> \\|\\>=\>exp>e|)>\ s|)>,\>=\>exp>e|)>\ s|)>. Further, <\equation> \|\|)>=>\|Z|\<\|\|\>>\ s, <\equation> A=\|\|)>+\>>e|)>\ s, and so <\equation> \>>e|)>\ s\ A\\>>e|)>\ s. Uniformly in , for |x|\<\|\|\>>>\ R> and q>>, <\equation> \>>|)>|\|>\ s+\>>|)>|\|>\ s\ CT. The integrals over >|]>>, if *q>>, are at most >. In particular |\|>\ C>. <\proof> >. Both are absolutely continuous with respect to the free flow from , \\=C\ W>. As > is harmonic the first density is |)>+V>>. By () and Itô's formula the second is >|)>+>+|)>|)>+c|)>\ s|)>>; it is a true martingale since the drift is Lipschitz at fixed . So <\equation*> M:=\\|\\>|\|>>=exp-\|)>-|)>+c|)>\ s|)>. >. Under > the explicit flow has the extra drift Z>, so =|Z,\ W|\>+|Z|\<\|\|\>>\ s>, and =\|Z|\<\|\|\>>>, as the stochastic integral is a true martingale at fixed . By Proposition and |Z|\<\|\|\>>\ Cs>, (), \ C>. So > is uniformly integrable under >, and \\> on |)>> with density >=M>, and () holds. >. Both laws have, at fixed , \ x>> almost surely. Under >, \|^>>, from =Cx+|^>+B|)>>. Under >, \ Y=b\ s+H\ W> with |b|\<\|\|\>>\ C\ s|Y|\<\|\|\>>> and |H|\<\|\|\>>\ C/s>, so |Y|\<\|\|\>>> is finite and > converges almost surely, the drift being integrable and the martingale bounded in >; and +|3>Cy=O|y|\<\|\|\>>>|)>|)>\0>. Then |)>\0>: at fixed the coefficients converge with rates and |x|\<\|\|\>>>+|x|\<\|\|\>>>|)>|)>>, so >\ V>, and \ V> by Gaussian integration, uniformly on bounded sets. By the bounds below >|)>|\|>\ s\\> almost surely under both laws, and is integrable at fixed . So >=exp-\->e|)>=\>\ e>>, which is positive; hence the equivalence and (), the second formula by >M>=1>. Take >> of >> in (): this is (), the integrability coming from (). The left inequality in () is \0>, the right one Jensen's inequality in (). So the exact and explicit potentials are compared using only expectations of the defect and the entropy of the control. For *q>> use , so |\|>\ C|y|\<\|\|\>>> by (): under > by (); under > by Itô's formula for |Y|\<\|\|\>>>, the bounds and |Y|\<\|\|\>>\6R>. So these contribute at most >. For q>> use Lemma, |\|>\ C\s|\y|\<\|\|\>>+C\s|\y|\<\|\|\>>+C\s|t|\<\|\|\>>>. Under >: the third term integrates on |)>> to at most TJ>>. For the first two, with |\Y|\<\|\|\>>\ r|Y|\<\|\|\>>>, both are at most s|Y|\<\|\|\>>>, and () gives a bound T>. Under >: with =Y-Y>, by the linear bound |b|\<\|\|\>>\ C\ s|Y|\<\|\|\>>> of Proposition, valid for every , H\ C*s> and Itô's formula, |\|\<\|\|\>>\ C+>>R|)>|)>> for , with depending on >> if *q>>, by on >|]>>; in that case read > below as >>>, only > being claimed. The first two terms give TR+C\TR+C\TRT|)>>. For the third, when q/C>, |t|\<\|\|\>>\ C*r|\|\<\|\|\>>> as before, giving T> and terms in R> of higher decay. When *q/C>, |t|\<\|\|\>>\ C+|\|\<\|\|\>>|)>>, whose integral with the weight s> on /C|]>> is at most R+R|)>T+log/T|)>|)>>. Since /T|)>\ CT>, this is (). Couple the lattices by common Fourier Gaussians, > being the projection of a cylindrical on a fixed probability space. The consistency of the fixed-scale coefficients of Section gives the following, along ,N>\ b>>. <\lemma> If \ H>, is continuous, and the interpolations x\ x> uniformly, then for each , y|)>\ y> uniformly, and |)>|)>\ e|)>>. <\proof> For : boundedness by the maximum principle, equicontinuity by Lemma and translation covariance, uniqueness of the coordinate equation among continuous by testing with the difference, as H>. For : =0>, so =|\,y|\>\ t>. By the consistency of the coefficients (Theorem of the Supplement), |)>> converges weakly in > at each for smooth bounded >, and uniformly in , |\-\|\<\|\|\>>\ C|z|\<\|\|\>>>|)>|y-z|\<\|\|\>>+|y-z|\<\|\|\>>|)>>. A three-term approximation extends the weak convergence to uniformly convergent fields, and dominated convergence in gives the claim. Moreover |\|>\ C*g|y|\<\|\|\>>> by the global linear bound on > behind Proposition (Lemma of the Supplement), which dominates on >. For *q>> the same applies to > in , and |y|\<\|\|\>>> is continuous; for > use and the convergence of the kernel of >>> in >. By Lemma, +v> with =x+C\ W> and solving an ODE on |)>>, uniquely, for each path \ C;L>|)>>. So the explicit flow > is a continuum strong solution, adapted to , for continuous . By Lemma and the continuity of the ODE, if x\ x> uniformly, then along the coupling T>|\x-x|\<\|\|\>>>\0> almost surely, since sup T>|\\-\|\<\|\|\>>>\ CN p>> by Kolmogorov interpolation between the covariance difference and the Hölder bounds, and Borel\UCantelli. Testing, |Y,h|\>> converges as \> in > for L>, uniformly in , since |Hh|\<\|\|\>>\ C*s|h|\<\|\|\>>> and |b|\<\|\|\>>\ C\ s|Y|\<\|\|\>>>. So |Y>,h|\>> is defined; by () and the Itô bound on |Y|\<\|\|\>>>, > converges also in >>, \>, in |)>>, uniformly in . <\theorem> Let x\ x> uniformly, continuous. The densities () converge in > along the coupling, |)>\ A>, and for bounded continuous cylinder functions of the terminal field <\equation> >>O|^>|)>\\O:=>|>\>>|)>exp>e|)>\ s|)>|]>,>>|>=>|>exp>e|)>\ s|)>.>>>>> The same holds for bounded functionals of the path |)> T>> and finitely many |\,h|\>> jointly, continuous for the uniform norm, and for |)>=|\,h|\>>, L>. So and O> are jointly continuous in \,\|)>\ C|)>>, also for |\,h|\>>. We write |m,h|\>=\|\,h|\>>, the ; |m|\<\|\|\>>\ C> by Proposition. <\proof> Let =exp>e|)>\ s|)>>, =\>a>. On > the integrals converge almost surely by Lemma, since the paths are bounded; the tail on |)>> is small in > uniformly in by (). So \ a=exp>e|)>> in probability. By (), () and (), *flog f\ C>, so |)>> is uniformly integrable by de la Vallée Poussin's criterion, and |\|>\ C> by Theorem. Along any subsequence with \ A>, \\a> in >, so =\*a>. Thus the limit is unique and \ A>. As |Y>,h|\>\|Y>,h|\>> in probability, > convergence of > gives () for bounded continuous and, the paths converging uniformly along the coupling, for the joint functionals. For |\,h|\>> use also the uniform second moment of ||^>,h|\>> under >>, from the system () and (). Continuity: the limit of > along every uniformly convergent sequence exists, so the limit function is continuous in . If ,x|)>\>, the flows driven by the same converge uniformly on compact scales by Lemma, q> q>|)>|\|>\ s\0> by the bounds on the defect, and the tails and the entropy are uniform; the same argument gives joint continuity. <\theorem> For bounded continuous cylinder , the whole sequence converges, and with > the explicit flow \ Y=b\ s+H\ W> on ,\|)>>, >=0>, <\equation> O\\=>>|)>\>>e|)>\ s>|]>|\>>>e|)>\ s>|]>>. With =0> and > the Gaussian measure of covariance >>, the renormalised free energy converges: <\equation> -log\>\\-\|)>\ A,0|)>=-log\>\>>e|)>\ s>. <\proof> This is Theorem at =,0|)>>, as ,0>> is the flow for >, and >>=0>, >=-log\>\\>. With the tightness of Corollary this identifies the limit along the whole sequence. The divergent constant has not disappeared: it is |)>> in (), and it does not enter (). Exercise computes it in the Gaussian case. We now read off the limit FBSDE from the density. Let =\,0>>, =f\> with ,0|)>>\>>e>>, on the paths of . Let =Y>>, =x>, tested with smooth functions, and =\:s\ q|)>>. <\key-result> <\theorem> >>Let ,q\0>, \>, with the counterterm of Theorem (or >> large and ,N>\ b>>, ,N>|\|>\ B\ q>>). Under >: <\enumerate-alpha> \\>, and =|)>C\+CB>> with a cylindrical Brownian motion independent of >. The conditional law of > given > is ,\|)>>. So for smooth , <\equation> |\,h|\>:=\|\,h|\>\|\|]>=|m|)>,h|\>, which is a function of >, and > is the conditional mean of the terminal field. =Kx->\\ s> is a cylindrical Brownian motion for |)>>. With =y|)>>, -Y>, and for q>, <\equation> =>|>|)>+H|)>Z|)>\ s+>H|)>\ W>>|=>|>|)>+H|)>Z|)>\ s+>\\ W,>>>>> where =-H>, =\ W> with no orthogonal part, and > is the conditional covariance, <\equation> |h,k|\>=\|\,h|\>|\,k|\>\|\|]>-|\,h|\>|\,k|\>. Moreover >=Z>=0>, and |Z|\<\|\|\>>\ C*q> for large . \\> in > in mean square, for every >. Along the whole sequence, in law and jointly with >>: > converges uniformly on compact sets of scales and space, > uniformly in space at each , |\,h|\>>, |Z,h|\>> at each , the innovations > in >|)>>, \>, to . The energies >,J>> satisfy the bounds of Theorem in the limit. <\proof> By Theorem at ,0|)>>, the pair ,\>|)>> under the exact law converges in law to |)>> under >. At each , \-|)>\>> is a Brownian motion independent of >>; test with h>, as Kjh> converges in every Sobolev norm. So |B,h|\>> for smooth are jointly Gaussian, independent of >, with covariance t|)>|h,k|\>>; by density is a cylindrical Brownian motion, with paths in >>, \>. Moreover \\,B:r\ q-q|)>>, as =|)>C\+CB>>. Under > the flow is Markov, and by Theorem at |)>>, <\equation> D:=\>|]>=exp,0|)>-A|)>->e|)>\ s|)>, and >|)>\|\|]>=\>|)>\|\|]>/D=\O|)>>. The right side of () is continuous in , so () holds for all simultaneously, and is continuous and positive. By Lemma the flow is a strong solution, and and >> determine each other; so > is a Brownian filtration under >. As 0> has finite entropy, by lower semicontinuity from Theorem, martingale representation gives \ D=D|Z,\ W>|\>> with >|Z|\<\|\|\>>=2\|\|)>>, and >- Z> is a >-Brownian motion for > by Girsanov's theorem. Under >, the lattice equation \ Y=b\ s+H\ W>> passes to the limit by Section, along the convergence of the coupling at each scale. So under >, \ Y=+HZ|)>\ s+H\ W>, and x|)>=+Z|)>\ s+\ W>. By (a), the increments of after are independent of > and of >, so |Kx-Kx,h|\>\|\|]>=|\,h|\>> for smooth . As > is a martingale, |Kx,h|\>->|\,h|\>\ u> is an >-martingale. So is |W,h|\>=|Kx,h|\>->|Y+Z,h|\>\ u>, where > is locally in \ u\\;L|)>>, by (), () and lower semicontinuity. Their difference is a continuous martingale of finite variation, null at >, so it vanishes. With a countable dense set of , =Y+Z> for almost every , almost surely, where =m|)>> is defined at every scale. So -Y=m-y> is defined at every scale, =Z> almost everywhere, and is the innovation. Moreover |\|\<\|\|\>>\\|\|\<\|\|\>>\\>, by Fatou's lemma along (e) and the same bounds, so > is square integrable. Let |\,h|\>> and >|]>>. Under >, >|]>> is a Brownian integral, \ N=|n,\ W>|\>>, and , so Itô's formula gives \ M=|n/D-M*Z,\ W>-Z\ s|\>=|n/D-M*Z,\ W|\>>. This defines h>. Here L|)>> by Theorem and Fatou's lemma, so is a true >-martingale and |h|\<\|\|\>>\ s=S>. Summing over an orthonormal basis, >||\<\|\|\>>>\ s=\|\|\<\|\|\>>>. For bounded and >-measurable and smooth, -W> tested with is |\,k|\>+|B>-B>,k|\>-|\,k|\>\ u>, and the term in is independent of >. So |W-W,k|\>|]>=\|\,k|\>-|\,k|\>|)>|]>\ u>. By the representation and Itô's isometry the left side is \|h,k|\>|]>\ u>. On the right, condition on \\>. So the integral over ,\|]>> of the difference of the two sides of () is a continuous martingale of finite variation, hence null. This is () for almost every , almost surely, and > is symmetric and nonnegative, Hilbert\USchmidt almost everywhere. The equation for is the difference of those for > and . At >, >=0>: so >=0> since |\>> is odd, and >=\>=\>\=0>. The bound on is weak lower semicontinuity from Theorem, by (e). () passes to the limit by lower semicontinuity. For the path and > this is Theorem; for =m|)>> and > use continuous convergence in the same theorem and the uniform bound of Proposition. Then =Kx->\\s> converges tested, and > is tight in >|)>> as a Brownian motion. For the energies, =\-|)>C\> and =C|)>\-|3>Y|]>> converge uniformly, since the rows of the kernel of > converge uniformly in >, and along the coupling =m|)>> is bounded in > and converges weakly. The integrands of and are lower semicontinuous in these variables, and Fatou's lemma applies. <\remark> Let be a real Brownian motion and =|)>\ B>. Then ,B|)>> converges in law to ,B|)>>, with > an independent Brownian motion: the integrands converge weakly to , but their squares converge weakly to >. A limiting FBSDE obtained by compactness could in this way have an extra martingale, orthogonal to the driving noise. A compactness proof must exclude this by strong compactness of >, for which the compactness route of Section of the Supplement uses the Riccati equation of the conditional covariance (Exercise). Here the density does it: the filtration is Brownian under the reference. What remains open is whether > is generated by (Section). The terminal law satisfies an integration by parts formula in which the renormalised cube appears only tested in expectation. <\theorem> Let ,\,h> be smooth and |)>=g|\,h|\>,\,|\,h|\>|)>> with C>|)>>, and let *G|)>=|\a>*G*+a*h|)>|\|>>. Along the sequence of Theorem (recall -\|)>> with =q>), <\equation> lim \**\>|)>|\*\>+|3>>|)>,j*h|\>>|]>=\>**G|)>-G|)>|\,-\|)>*h|\>|]>. <\proof> At fixed , Gaussian integration by parts in the density -*|\,K>*\|\>>> gives, with >> and =j*h>, <\equation*> \**x|)>|\*x+|3>*x,h|\>|]>=\*>*\|)>|]>-\**x|)>|x,K>*h|\>|]>. Cell averaging commutes with lattice translations, so >*j*h=j*-\>|)>*h> with >> the finite-difference Laplacian acting on smooth functions, which converges to *h> in every Sobolev norm. The right side converges by Corollary (weak convergence in > and uniform moments), and the derivative term is a finite sum with converging coefficients |j*h,j*h|\>\|h,h|\>>. This is the terminal case of the Dyson\USchwinger equation of Theorem, of Lecture, for the limiting theory. It does not say that *\+|3>*\> converges as a random distribution: only its expectation against cylinder functions is identified. This is consistent with the expectation of Lecture (Remark) that the limiting force is an observable without a terminal representative; it is not proved here that no such representative exists. At every resolution, on the other hand, the force is explicit: =K*Z-|\>|)>> (Exercises and). <\theorem> For the limit of Theorem, Theorem and Corollary of Lecture hold at every resolution, with =K-\|)>>, =K\-|)>\> and =|)>C\ I>. That is, for all smooth and cylinder , almost surely as distributions, and with no renormalisation constant, <\equation> K>\=JF+\, <\equation> \*Dh>G|)>=\|)>|\,h|\>|]>=\|)>|\,K>h|\>-|F,Jh|\>|)>|]>, and |F,h|\>,|\,h|\>> are centred martingales, with <\equation*> |F,h|\>=>|K-I|)>h,\ W|\> and \|\,h|\>>\\|)>|h|\<\|\|\>>/2>>. Moreover: <\enumerate-alpha> For large , with the explicit force <\equation*> ||\>|)>,h|\>:=|K\,h|\>-|Y,Kh|\>, <\equation> ||F,h|\>+||\>|)>,h|\>|\<\|\|\>>|)>>\ C|h|\<\|\|\>>>q. So |F,h|\>> is the unique martingale asymptotically equivalent to the germ <\equation*> -||\>|)>,h|\>, in the sense of Lemma of Lecture. As \>, <\equation> \>\|)>||\>|)>,Jh|\>|]>=\>G|)>-G|)>|\,-\|)>h|\>|]>, the right side of (). For C>|)>>, |\>,h|\>,\,|\>,h|\>|)>>, \ q>, <\equation> \\g|J>h,h|\>=\|\,K>h|\>+||\>|)>,Jh|\>|)>|]>+O|g|\<\|\|\>>>|h|\<\|\|\>>>q|)>, so at every large resolution the renormalised cube closes the hierarchy, up to an error which vanishes as \>. <\proof> The algebra and the integration by parts are those of Lecture, since by Theorem(a),(b), =\>|]>>, with =J\+CB>> and independent of >. The martingale representation is =\ W>, and the exponential bound is conditional Jensen. For (), =KZ-|\>|)>>, so its left side is at most |Kh|\<\|\|\>>|Z|\<\|\|\>>|)>>, and Theorem(c) applies. For (): by () with Jh=|)>h>, replacing > by |\>|)>> in () costs |)>>. Moreover |\,k|\>=|\,Jk|\>+|\,k|\>> with |\,k|\>\|k|\<\|\|\>>/q>, and \\> tested in mean square, by the martingale property. So \\> tested with smooth functions, and h\ h> in every Sobolev norm. () follows from () with h>, as |Jh|\<\|\|\>>>\|h|\<\|\|\>>>>. This is the renormalised cube: the cube of the forecast with its counterterms, at resolution . By () the two orders of limits, \> at the terminal resolution and \> after , give the same right side. What is not proved is that is closed: () gives no bound on |F,h|\>> uniform in . Along the exact flow the value =A|)>> solves a scalar backward equation, the integrated form of the vector one and the analogue of (4.13) of Gubinelli\UMeyer. Indeed, by () and \ D=D|Z,\ W>|\>>, <\equation> A=>|)>+|Z|\<\|\|\>>|)>\ s+>|Z,\ W|\>. <\definition> Fix q> and a continuous . A is a filtered probability space, of law >, with an >-cylindrical Brownian motion and adapted processes >, with paths in |)>;C|)>|)>>, with =y|)>>, =x>, \ x=-C\ s+C\ s+\ W|)>>, with >|Z|\<\|\|\>>\ s+\>|)>|\|>\ s\\>, satisfying (), > being deterministic, and such that =exp|Z,\ W|\>-|Z|\<\|\|\>>|)>> is a uniformly integrable martingale. The forward equation here is x|)>=+Z|)>\ s+\ W>: the forward equation of (). The limit law is a solution, by the proof of Theorem, at > in place of ,0|)>>: =D> is uniformly integrable under the exact law, closed by \\/\\>, and |)>\0> as \ f\0>. <\theorem> For every solution, the law of on paths is the limit of >>, =A>, and is the adapted functional of the path of of the exact solution. So the terminal law is |)>>. <\proof> Let \\=\>\\>. Under >, =W+ Z> is a Brownian motion, and \ x=-C\ s+C\ W>. By pathwise uniqueness (Lemma), under > has law >. By () at , >=exp>|Z,\ W|\>+>|Z|\<\|\|\>>|)>=exp->e|)>\ s|)>>, a functional of the path of . So the law of under > is >\ e>\>, and normalisation gives =A>. By Theorem this is the limit of >>. For : under >, for the filtration >, > is a Brownian motion and is Markov with respect to >, so the density process on > is =\->e>\|\|]>>, an adapted functional of the path of . The drift under > is determined by the covariation of with >, which is computed from the path of only, since \ W> is a differential of and its drift. So if the backward equation is written for the scalar value , the solution is unique in law. Its is the feedback |)>-y|)>>, and by Theorem it coincides with -Y> of (); differentiability of in is not claimed. <\key-result> <\theorem> For q>, continuous, and bounded measurable on paths, under the reference >, <\equation> -log\>\>e|)>\ s>=inf\\, \|\|)>\\>>>|)>+|u|\<\|\|\>>|)>\ s|]>|]>, where, for > of finite entropy, is the drift of under > added to () as u\ s>, so that solves \ Y=+Hu|)>\ s+H\ W>> and \|\|)>=\>|u|\<\|\|\>>>. For the unique minimiser is , and the value is >. In particular, at =,0|)>>, for a bounded continuous cylinder function of the terminal field, <\equation> -log\\\=\-\,\=inf\>|)>+>>|)>+|u|\<\|\|\>>|)>\ s|]>, Y>=0, the infimum being over the laws of finite entropy with respect to >, and =A,0|)>> is the renormalised free energy (). The minimiser for is . No infinite constant occurs in >, and there is no initial law to optimise over. <\proof> The Gibbs variational principle: for a probability > and a measurable > with >\>\\>, for every > with \|\|)>\\>, >\\ \|\|)>+log\>\>>; where >\1+\>>. This gives that >\> is well-defined in ,\|]>> and >\+\|\|)>\-log\>\>>, with equality exactly at \\\\>\\> if >|\|>\\> there. Apply this with =O+>e>. The integrability >\>\\> is Theorem. At the minimiser for , |\|>\\> passes to the limit from () by Fatou; for bounded the same holds as the tilted law has a bounded density with respect to it. Since > is the strong solution driven by , every > of finite entropy has a drift with =\>|u|\<\|\|\>>> by Föllmer's theorem. At , the minimiser has control , by Theorem ( is the Föllmer drift of >). The equation for holds under > by the proof of Theorem, hence under \\> with \ W=\ W>+u\ s>. () is () at ,0|)>>, with (). <\remark> () with reads, for every control, |)>+|u|\<\|\|\>>|)>\ s\ A\-C>. This is the coercivity +|2>|u|\<\|\|\>>|)>\ s\-C> with constant =1>. It has been obtained from the exact solution without any estimate on the controlled equation. An independent proof with \1>, as suggested by the test with > in the memory form, would give the exponential moments >\ e>>, 1>, which are not needed here. <\remark> In the Barashkov\UGubinelli functional the Lagrangian contains |u|\<\|\|\>>> for a control singular at the cutoff, and renormalisation is a change of variables in the cost, subtracting the Wick cube of the free field (|3>>:X:+r> in Lecture ). Here the reference process carries the renormalisation, the control is the remainder , it is in > uniformly in (of size >), and the only interaction left in the cost is the defect , with > small at large . The coercivity is used pathwise, the remainder in expectation. <\remark> In the smoothed force and the defect are bounded uniformly in the field, there is a spatial cut-off, and the same formula follows from boundedness; () is the analogue of their (4.13). Here the defect is unbounded. What replaces the bound is the entropy of the control: UV stability of the restarted laws makes the densities uniformly integrable, and exponential moments of the defect are not needed. The backward error alone does not give the result: from a given law at scale , Girsanov's theorem reduces the terminal law to that one, which is the unknown. Only the start at >, or the restart from a fixed field, removes it. The limit can also be identified as the invariant measure of the > dynamics, where quartic tails hold also at >> on the literature's lattices; we have not used this. <\itemize> Theorem gives uniqueness in law for the scalar form. A solution of () alone need not have of gradient form. It would suffice that any solution has a Lipschitz decoupling field, which needs a pointwise bound on >, or high-scale contraction with non-degeneracy on compact intervals; only |)>> control of > is available. The same bound is what is missing for the Polchinski process to be adapted to the innovation : the explicit flow is a strong solution, the exact one is not known to be. The Allinger\UMitter theorem does not apply: its finite-energy hypothesis is on the signal >, and |\|\<\|\|\>>=\> here. Theorem gives quartic moments in >> for \>, and Theorem Gaussian moments for >. The quartic moment at near > would need spatially resolved sources. Reflection positivity holds at every lattice, for reflections in lattice and mid-lattice planes (nearest-neighbour action), and passes to > for every reflection in a coordinate plane, by approximating the plane with lattice planes (standard; Exercise). Rotation invariance (absent on the torus) and clustering need infinite volume. The force, its Burgers flow, and differences of free energies are intensive (Section of Lecture). The estimates of this lecture must then be made local, with weights in space and constants per unit volume. A new difficulty appears in the backward equation: its integrand > is a nonlocal operator, and the backward error at a point can receive noise from arbitrarily far away. Uniform-in-volume coefficient bounds and local energy moments are themselves open; they could give an infinite-volume Gibbs subsequence. Controlling the \Pnoise from infinity\Q (together with local compactness of the blocks of the derivative of the force) is the main additional step needed to retain the full FBSDE without an orthogonal martingale. The lectures went from the Wightman axioms to the Euclidean formulation (Lecture), to the free field (Lecture), to stochastic quantisation and the scale decomposition (Lecture), and to three views of an interacting field, renormalisation, optimal control and filtering, meeting in the force coordinates (Lecture), and to the field equation at every resolution, whose noise, a conditional expectation of Nelson's Euclidean noise, is sub-Gaussian whatever the interaction (Lecture). In Lectures\U7 the FBSDE in forecast coordinates, together with an explicit solution of the force equation up to an integrable defect, gave a construction of > in which every object has a probabilistic meaning: the forecast is a conditional mean, the response is a conditional covariance, the Brownian motion is an innovation, and UV stability is the statement that an explicit force predicts the conditional mean up to an error that is small at high resolution. In the end the exact flow is the explicit flow with control the error, the measure is an explicit density over the explicit flow, and the renormalised variational problem has that error as its optimal control. <\exercise> Show that |F,j*h|\>=|\,K*j*h|\>-|W,j*h|\>->|\,j*h|\>\s>. Deduce that, along the sequence of Theorem, these converge, with uniformly bounded moments, to |F,h|\>=>|K-I|)>*h,\W|\>>, a martingale for the limiting filtration (note >=0>, since > is even). (Cross-check: =K*Z-|\>|)>>, with Theorem, gives a uniform second-moment bound.) Why does this not give a terminal value >>? <\exercise> Prove the claim of Remark. In the proof of Theorem, which step excludes it? <\exercise> At fixed , show that the conditional covariance > satisfies =-\ q+\ W|]>>, where > is the third conditional cumulant of >> (the conditional law is an exponential family, Proposition(b) of Lecture). For a Gaussian =\>, solve this and check that =C>. For a general >, show that > is a supermartingale, and compare with (). <\exercise> Show that > is reflection positive for reflections in lattice and mid-lattice planes (nearest-neighbour Gaussian times a local potential). Deduce reflection positivity of > for every coordinate plane =a|}>>: take cylinder functions of tests supported in x\a+|}>> (one of the two components of the torus minus the fixed planes =a|}>> and =a+|}>> of >), cell-average them, and use lattice planes \a>, so that for large the averaged tests lie on the correct side. Since lattice and mid-lattice reflections map cells to cells, >*j*f=j*\>*f>, so the reflected tested variables are |\*\,\>*f|\>>, with >*f\\*f> in every >. <\exercise> (a) At fixed , check that is Theorem of Lecture in the profile of Lecture, and compare the proofs of Theorem and of Corollary () of Lecture: score of the law of > against the filtered noise. (b) For =0>, write and check directly. (c) Show that |F,h|\>> has bracket |*K-I|)>*h|\<\|\|\>>\s>, and that does not bound it uniformly in . What would a bound imply? <\exercise> Verify, and show that at > the bound becomes *log >. <\exercise> On one site take =s>, |\>=|3>y+a*y> independent of , with 0> on the range considered, and =|\>>. Show that =|\>''|\>'|)>> and +c=-|\>'|)>|)>>. Simulate \ x=-s|)>\ s+s\ W>, and check () and () numerically. How small is the entropy? <\exercise> In the setting of the previous exercise freeze the force below >>, and check () and () numerically for several >>. Which quantities change with >>, and which do not? <\exercise> For =|2>|\|\<\|\|\>>> and |\>=\ y> for q>>, show that =0> above >> and =-y> below, so |y|\<\|\|\>>>. Compute the density () mode by mode, and show that the constant diverging with is carried by >. <\bib-list|HMa18> N.Barashkov, M.Gubinelli. A variational method for >. 169 (2020), 3339\U3415. M.Gubinelli, M.Hofmanová. A PDE construction of the Euclidean > quantum field theory. 384 (2021), 1\U75. M.Gubinelli, S.-J.Meyer. The FBSDE approach to sine\UGordon up to >. Preprint, arXiv:2401.13648, 2024. M.Hairer, J.Mattingly. The strong Feller property for singular stochastic PDEs. 54 (2018), 1314\U1340. M.Hairer, K.Matetski. Discretisations of rough stochastic PDEs. 46 (2018), 1651\U1709. M.Hairer, P.Schönbauer. The support of singular stochastic partial differential equations. 10 (2022), e1. M.Hairer, R.Steele. The > measure has sub-Gaussian tails. 186 (2022), art. 38. J.-C.Mourrat, H.Weber. The dynamic > model comes down from infinity. 356 (2017), 673\U753. A.Moinat, H.Weber. Space-time localisation for the dynamic > model. 73 (2020), 2519\U2555. J.Ma, J.Yong. . Lecture Notes in Math. 1702, Springer, 1999. <\initial> <\collection> <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> MY99 HSt22 GM24 BG20 GM24 HSt22 HMt18 HMa18 HSc22 MW17 MW20 GH21 <\associate|toc> |math-font-series||Introduction> |.>>>>|> |math-font-series||13The energies> |.>>>>|> |13.1Moments at fixed cutoff |.>>>>|> > |13.2The field equation |.>>>>|> > |The forecast equation of Lecture |->|-0.3em|>|0em||0em|>>5 |.>>>>|> > |The explicit force |.>>>>|> > |The smoothed noise |.>>>>|> > |13.3Thresholds and energies |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||14Forward coercivity> |.>>>>|> |14.1The scalar model |.>>>>|> > |What the scalar lemma says |.>>>>|> > |14.2The lattice version |.>>>>|> > |14.3A pointwise bound |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||15Backward feedback and closure> |.>>>>|> |15.1The backward representation |.>>>>|> > |15.2Three contributions |.>>>>|> > |15.3Closing the loop |.>>>>|> > |15.4Consequences |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||Removing the cutoffs> |.>>>>|> |math-font-series||16The boundary at infinite resolution> |.>>>>|> |16.1The boundary lemma |.>>>>|> > |16.2UV stability |.>>>>|> > |16.3Restarted laws |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||17Tightness and tails> |.>>>>|> |17.1One Hilbert space |.>>>>|> > |17.2Spatial suprema by a maximum principle |.>>>>|> > |17.3The terminal field |.>>>>|> > |17.4Exponential moments |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||18Identifying the limit> |.>>>>|> |18.1The deterministic coefficients |.>>>>|> > |18.2The exact flow as a controlled explicit flow |.>>>>|> > |18.3The limit law |.>>>>|> > |18.4The filtering model and the FBSDE |.>>>>|> > |18.5The renormalised field equation |.>>>>|> > |18.6Uniqueness in law and the variational formula |.>>>>|> > |18.7What is not proved, and where to go next |.>>>>|> > |Summary of the course |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||References> |.>>>>|> <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> MY99 HSt22 GM24 BG20 GM24 HSt22 HMt18 HMa18 HSc22 MW17 MW20 GH21 <\associate|toc> |math-font-series||Introduction> |.>>>>|> |math-font-series||13The energies> |.>>>>|> |13.1Moments at fixed cutoff |.>>>>|> > |13.2The field equation |.>>>>|> > |The forecast equation of Lecture |->|-0.3em|>|0em||0em|>>5 |.>>>>|> > |The explicit force |.>>>>|> > |The smoothed noise |.>>>>|> > |13.3Thresholds and energies |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||14Forward coercivity> |.>>>>|> |14.1The scalar model |.>>>>|> > |What the scalar lemma says |.>>>>|> > |14.2The lattice version |.>>>>|> > |14.3A pointwise bound |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||15Backward feedback and closure> |.>>>>|> |15.1The backward representation |.>>>>|> > |15.2Three contributions |.>>>>|> > |15.3Closing the loop |.>>>>|> > |15.4Consequences |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||Removing the cutoffs> |.>>>>|> |math-font-series||16The boundary at infinite resolution> |.>>>>|> |16.1The boundary lemma |.>>>>|> > |16.2UV stability |.>>>>|> > |16.3Restarted laws |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||17Tightness and tails> |.>>>>|> |17.1One Hilbert space |.>>>>|> > |17.2Spatial suprema by a maximum principle |.>>>>|> > |17.3The terminal field |.>>>>|> > |17.4Exponential moments |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||18Identifying the limit> |.>>>>|> |18.1The deterministic coefficients |.>>>>|> > |18.2The exact flow as a controlled explicit flow |.>>>>|> > |18.3The limit law |.>>>>|> > |18.4The filtering model and the FBSDE |.>>>>|> > |18.5The renormalised field equation |.>>>>|> > |18.6Uniqueness in law and the variational formula |.>>>>|> > |18.7What is not proved, and where to go next |.>>>>|> > |Summary of the course |.>>>>|> > |Exercises |.>>>>|> > |math-font-series||References> |.>>>>|>