> <\body> 4 \V notes: Lecture4, ŸŸ7\U8> |<\hidden> \; Stochastic quantisation of the GFF; the field at resolution is a martingale of smooth fields; are martingales in the filtration of finite-resolution measurements. Interactions, still indexed by the >. <\enumerate> Generate the measure : renormalisation, optimal control, filtering. They meet in one system for a and a : the . Potentials: the value function generates them; Legendre; . (Lecture5): observables, and the with its sub-Gaussian noise. Statements marked are exact in finite dimensions, with a terminal resolution \>; the limits \>, \> are the real problems. >> |<\hidden> \; Fix > and \\>\>>. Let =Q-Q>, the covariance still unresolved at resolution . <\itemize> Split the covariance into shells and integrate them out. At every resolution the field is again a Gibbs measure, driven by an effective force. Steer the Gaussian field by a drift, at the least cost. The effective potential is the value function, and the Polchinski equation is the dynamic programming equation. The field at resolution is a noisy observation of the terminal field. The conditional law is the posterior, the force is what the observation reveals, and backward in resolution its dynamics is linear and Gaussian, started from >. \; They produce the same process. |<\hidden> \; 4> \; |<\hidden> \; Finite volume >, terminal resolution \> (UV cutoff), interaction > bounded below, e.g. <\equation*> V|)>=>|12>*H*;c|)>+\*H*;c|)>|]>,\\|)>=Z\|)>>\>\|)>. <\definition*> <\equation*> \|)>>\\*+\|)>>\>\|)>,t\. The interaction felt at resolution once the scales > are integrated out; =-log Z> is the free energy. <\equation*> \|)>>=\>*|)>>|]> |<\hidden> \; <\proposition*> \/\\=Z\|)>>>. Then >|)>=\\\>\>>, and >\|\|)>=\>>, the tilt of >*-X|)>> by >>. <\itemize> At every resolution the field is again a : reference >, potential >. The measure-valued martingale \>> goes from > () to >> (): a of > (Eldan), adapted to the covariance profile. <\equation*> \>\|)>=|)>>|Z>\>*\-X|)> \; <\small> Under >, =X+-X|)>> with independent increment \>>, so <\equation*> \>|)>|]>=Z*\|)>\|)>>|]>=Z*\|)>\|)>>|]>. |<\hidden> \; <\theorem*> Under >, the >, \X>, solves <\equation*> \\=>*F\t+>\W>,F=-\V|)>=\>*V|)>|]>. The is a >-martingale: an observable field of the interacting theory. <\small> Density process =Z\|)>>>. It is a martingale, so by Itô it has no drift, M=-M|\V|)>,>\W|\>>. Girsanov: >=W+>*\V|)>\s>. Finally V|)>=\>*V|)>>. <\equation*> \V|)>=->V*|)>|)>\*|)>>|]>|\*|)>>>=\>V*|)>|]> |<\hidden> -transform> \; Under >, is the , with generator =>*\|)>>. The function <\equation*> h|)>\Z\|)>>=\\|)>>X=\|]>,+L|)>*h=0, is space-time harmonic: this is the . Since \/\\\|>=h|)>>, <\equation*> L*G=h*+L|)>*G|)>-\*G=L*G+|>*\log h,\G|\>,F=\log h|)>. Simplest case: a Brownian motion conditioned on its endpoint, => Gaussian density of the remaining increment, and the drift of the bridge. \; <\myframed> The Polchinski process is the -transform> of the GFF martingale by >>: the free field conditioned to end with the law >. |<\hidden> \; Read as an equation for an unknown pair ,F|)>>: <\equation*> \=>*F\s+>\W,F=\*V|)>|]>,t\. <\itemize> The field is driven by a force determined from the terminal field. No Polchinski equation needed; =|)>> from the solution. : solve with renormalised >, estimates uniform in and volume, take limits. > is an observable (a semimartingale); observables are built from the noise and from . |<\hidden> <\with|font-size|0.9> \; |fbsde_flow.png>||0.6pag||> Vertical: =-\V|)>>. Dashed: backward the process is (Gaussian blurring). Force to first order, /3|)>*H*;c|)>>. |<\hidden> \; <\theorem*> <\equation*> \*V=|\V,>*\V|\>->*\*V|)>,V>. =\>=\*\>>u>, =|)>>, solves the backward heat equation *u=->*\*u|)>>. Insert *u=V\\V-\*V|)>*u>. |||||||| : generates new, nonlocal interactions (). |<\cell> : contracts pairs of fields with the shell covariance >\t>: . >>>> A backward Hamilton\UJacobi equation; \> is a Cole\UHopf transform. |<\hidden> \\documentclass[tikz]{standalone} \\usepackage[T1]{fontenc} \\usepackage[sc]{mathpazo} \\usepackage{amsmath,amssymb} \\usetikzlibrary{calc,decorations.markings} \\tikzset{ \ blob/.style={circle,draw,thick,fill=#1,minimum size=7mm,inner sep=0pt}, \ blob/.default={black!12}, \ vtx/.style={circle,fill=black,inner sep=1.5pt}, \ prop/.style={thick}, \ full/.style={double,double distance=1.3pt,thick}, \ ins/.style={circle,fill=black,inner sep=1.5pt}, \ cross/.style={circle,draw,thick,fill=white,inner sep=0pt,minimum size=2.4mm,path picture={\\draw[thick] (path picture bounding box.north west)--(path picture bounding box.south east) (path picture bounding box.north east)--(path picture bounding box.south west);}}, } % external legs: \\legs{node}{angles} \\newcommand\\legs[2]{\\foreach \\a in {#2}{\\draw[thick] (#1.\\a)--++(\\a:2.6mm);}} \\newcommand\\rootleg[2]{\\draw[line width=1.4pt,-latex] (#1.#2)--++(#2:3.6mm);} \\begin{document} \\begin{tikzpicture}[font=\\small,baseline] % row 1 \\node at (0,0) {$\\partial_t$}; \\node[blob] (a) at (.65,0) {$V$}; \\legs{a}{60,120,240,300} \\node at (1.65,0) {$=\\ \\tfrac12$}; \\node[blob] (b) at (2.6,0) {$V$}; \\legs{b}{110,180,250} \\node[blob] (c) at (4.3,0) {$V$}; \\legs{c}{70,0,290} \\draw[prop] (b)--node[ins,pos=.5]{} node[above=1pt,pos=.5]{$\\dot Q_t$} (c); \\node at (5.45,0) {$-\\ \\tfrac12$}; \\node[blob] (d) at (6.5,-.15) {$V$}; \\legs{d}{200,240,300,340} \\draw[prop] (d.60) .. controls +(60:1.05) and +(120:1.05) .. node[ins,pos=.5]{} node[above=1pt,pos=.5]{$\\dot Q_t$} (d.120); % row 2 \\begin{scope}[yshift=-2.2cm] \\node at (-.45,0) {$\\partial_t$}; \\node[blob] (a) at (.65,0) {$F$}; \\rootleg{a}{180} \\legs{a}{60,300} \\node at (1.65,0) {$=\\ -$}; \\node[blob] (b) at (2.75,0) {$F$}; \\rootleg{b}{180} \\legs{b}{110,250} \\node[blob] (c) at (4.45,0) {$F$}; \\legs{c}{70,290} \\draw[prop] (b)--node[ins,pos=.5]{} node[above=1pt,pos=.5]{$\\dot Q_t$} (c); \\node at (5.5,0) {$-\\ \\tfrac12$}; \\node[blob] (d) at (6.7,-.15) {$F$}; \\rootleg{d}{180} \\legs{d}{240,300} \\draw[prop] (d.60) .. controls +(60:1.05) and +(120:1.05) .. node[ins,pos=.5]{} node[above=1pt,pos=.5]{$\\dot Q_t$} (d.120); \\end{scope} \\end{tikzpicture} \\end{document} \; |ps>|300pt|||>>> > or of the force. Dotted line: >>. Tree term and loop term; iterating gives the connected Feynman graphs.>> |<\hidden> \; >: adapted drifts with *|u|\<\|\|\>>\\>; =X+>>*u\s>. <\theorem*> HJB> <\equation*> -log \|)>>|]>=V=inf\> \*|)>+*|u|\<\|\|\>>\s|]>. Optimal feedback >=->*\V>>|)>>, >>|)>=\>. > is the ; Polchinski is the . (verification). Itô Polchinski: <\equation*> \*|)>+|u|\<\|\|\>>\t|]>=V+\*|2>|u+>*\V|)>||2>>\t. |<\hidden> Perturb u+\*v>: <\equation*> u=->*\*V|)>|]>=>*F: \\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} \\definecolor{navy}{HTML}{1B2A41} \\definecolor{accent}{HTML}{B5402E} \\definecolor{SNavy}{HTML}{1B2A49} \\definecolor{STeal}{HTML}{1F7A72} \\definecolor{SGold}{HTML}{B8862B} \\begin{document} \\begin{tikzpicture}[box/.style={draw=navy,fill=navy!6,rounded corners,inner sep=6pt,align=center,font={}}, \ \ lab/.style={above,font={},align=center}] \\node[box] (pde) {\\textbf{PDE}\\\\Polchinski}; \\node[box,right=2.4cm of pde] (sde) {\\textbf{dynamics}\\\\FBSDE}; \\node[box,right=2.4cm of sde] (var) {\\textbf{variational}\\\\Boué--Dupuis}; \\draw[\<-\>,thick,navy] (pde) -- node[lab]{decoupling\\\\field} (sde); \\draw[\<-\>,thick,navy] (sde) -- node[lab]{Euler--\\\\Lagrange} (var); \\draw[\<-\>,thick,navy] (pde.south) to[bend right=15] node[below,font={}]{HJB / value function} (var.south); \\end{tikzpicture} \\end{document} |ps>|0.814par|||>>> *\*|u|\<\|\|\>>\|)>\||)>>: a pathwise lift of the Gibbs principle <\equation*> -log Z=inf>|V\\+\|\>|)>|>. Useful because adapted drifts can be , as functionals of . |<\hidden> \; Generating functional: <\equation*> \=-log \\\=inf \*|)>|)>+|u|\<\|\|\>>\s|]>-V. Bounds uniform in > tightness, moments, identification of limits. >.> The leading drift \-|3>*>:X>: has , <\equation*> \*||3>*>:X:|\<\|\|\>>\s=|12>*L*Q\z\T. Change variables |3>*>:X:+r>, then paracontrolled changes of variables: the singular terms become explicit polynomials in Wick powers of , , and the remainder is coercive. Uniform control in Besov/paracontrolled norms construction of >. > |<\hidden> Minimise <\equation*> J=g|)>+\,u|)>*\s subject to >=b>, =x>. <\equation*> \=|p,b|\>-\. <\theorem*> Along an optimal pair there is a with <\equation*> >=\*\,>=-\*\,p=-D*g|)>,u\arg max \|)>. <\small> <\itemize> State forward, costate backward: a . The costate is the Lagrange multiplier of the constraint >=b>; an optimal control is a critical point of the Hamiltonian. Next: why this holds, and what is. > |<\hidden> <\with|font-size|0.85> Perturb u+\*v>; the state varies by =\>*x>, <\equation*> |\>=\*b*\+\*b*v,\=0. The costate equation >=-\*\> is exactly what makes <\equation*> |\s>|p,\|\>=|\*\,\|\>+|p,\*b*v|\>. Integrating, with =-D*g|)>>: <\equation*> |\\>J*v|)>=0>=-|\*\,p,u|)>,v|\>*\s. Dynamic programming over t|]>>: <\equation*> V=min*\t+Vt,x+b*\t|)>|]>\\*V=\>V|)>,\>=max \. Hamilton's equations for >> are its characteristics, and along them =-\V|)>>: the costate is the force the future exerts on the present. |<\hidden> <\with|font-size|0.9> State field, >*u>, =|u|\<\|\|\>>>, >. > is maximised at >*p>: <\equation*> \>=|p,>*p|\>: a free particle with mass matrix >>, and no dependence on the state . <\itemize> >=0>, >=>*p>: the :\*p,p|)>>, with \Q-Q> the covariance still unresolved. All interaction is in V*p|)>>: . HJB Polchinski without the trace. (Bismut 1973, Peng 1990): the costate solves a BSDE <\equation*> \p=-\*\\s+\\W, with *\W> the price of adaptedness. Here *\=0>: the costate is a , =-\V|)>=F>. The Itô term turns HJB into : quadratic part trees, trace term loops. <\myframed> The FBSDE is the stochastic Hamiltonian system: field state, force costate. |<\hidden> \; \; Here >=->>, =0>. Optimal control =>*p>, with costate =F>: <\equation*> \F=\\W,\=\F|)>*>. <\itemize> |)>> is the , =F|)>>; Burgers is its quasilinear PDE, and > the martingale representation integrand of the costate. In Burgers: transport term F*>*F|]>> > drift of the state; Itô term >*\*F|)>> > noise. The shear > is the flow of the transport part: its characteristics. |<\hidden> All covariance operators are functions of ; >, =K*Q>. Since \/\\> depends on > only, given > the interacting field is the : <\equation*> \=J\+\,\\=>S-\|)>\t+>\B, with >, independent of >. >, with prior >; |)>t>>, pulled towards the hidden field. <\proposition*> <\itemize> Posterior: >y|)>\\|S\,y|\>-|y,\y|\>>\y|)>>, =S-Q> the precision gained: the . =\-\|)>|]>>: >F> is the posterior mean of the pull, =\+SF>, and is the . The Polchinski SDE is the observation equation in its own filtration. Filter equation: \|)>|]>=||)>,S>\W|\>>. |<\hidden> Reverse the resolution, |^>=\>: <\equation*> \|^>=->Q|^>\s+>\,|^>=\\\. <\with|font-size|0.9> <\itemize> \/\\> depends only on the >: the transition kernels are those of the Gaussian reference, only the initial law changes. A linear process with Gaussian transitions, not a Gaussian process. : blur the interacting field with Gaussian noise, =J\+\>; this does not know the interaction. : Polchinski undoes the blurring, all the interaction is in the drift. Time reversal (0>, finite cutoff): forward drift =>F>, backward drift =->Qx>, and +\=>\log p>: reversal of the density at that scale (Nelson's osmotic velocity). So is the score of > relative to >>, > is the denoiser (Tweedie), and the noise is =-J\log p|)>>. \; <\myframed> The Polchinski process is a whose data distribution is the interacting measure. |<\hidden> \; In the stochastic localisation literature the observation is written as a linear measurement: with =S\>, <\equation*> j=\\+N,N=S\,\\ N|]>=\ t>, signal-to-noise ratio >. The posterior is an exponential family in , with log-partition function =\|j,y|\>-|y,\y|\>>\ y|)>>: <\itemize> =\\|)>>, =\\|)>>, j=|\>\\ t+|\>\W>; \+|\>\\|)>+|\\,|\>\\|\>=0>: Polchinski in the variable . Eldan's basic scheme is the profile with =t>, a Brownian motion, run to >: the profile of Lectures6\U7. Alaoui\UMontanari 2022; Bauerschmidt\UBodineau\UDagallier 2024>> |<\hidden> \; <\padded-center> ||||||| \; |<\cell> |<\cell> |<\cell> >| > |<\cell> field at resolution |<\cell> state |<\cell> observation; posterior mean, Gaussian prior >| > |<\cell> effective force |<\cell> costate |<\cell> posterior mean of the terminal force >| |<\cell> fresh fluctuation of the shell |<\cell> noise of the controlled field |<\cell> innovation >| > |<\cell> unresolved covariance |<\cell> free motion from to |<\cell> posterior covariance, Gaussian prior >| =\+SF> |<\cell> mean of the conditional law |<\cell> terminal state, free prediction |<\cell> posterior mean >| > |<\cell> covariance of the conditional law |<\cell> -S\VS> |<\cell> posterior covariance >| > |<\cell> effective potential |<\cell> value function |<\cell> log-likelihood ratio of the observation, up to > >>>>> -\=SF>: , in units of the free propagator >, and > replaces > as the propagator. |<\hidden> \; At finite the force is a observable, =\|]>>, and for > it is : <\equation*> F=-|3>*H*;c|)>-2*\*H*;c|)>. \; <\myframed> \>> should converge . In it is closed, >=-|3>:\>>:. In a closed limit =\*F>> would let the DS identity pass to \>, and conditioning on >|)>> would give an integrable logarithmic derivative of > as a function of the field: not expected (Albeverio\ULiang\UZegarlinski 2006: the regularised ones do not converge in >). Locality of the limiting force is the crux of the construction. |<\hidden> \; |polchinski_0d.pdf>|1par|0.55pag||> =1-\>, , =*x-*x>. Left: =-V>, almost linear at low resolution, double well as T>. Centre: FBSDE paths split between the wells (stochastic localisation). Right: histogram of > vs. the density of >. |<\hidden> \; 4> \; |<\hidden> \; Differentiating Polchinski, =-\V> solves the backward viscous <\equation*> \*F=-\F*>*F|]>->*\*F|)>,F=-\V. <\itemize> Cole\UHopf log h> linearises it (\> solves a heat equation). Uses only the force: , while > is extensive (> volume). Matters in infinite volume. Flows for different covariances commute when F> is symmetric, i.e. a gradient. Non-gradient forces (general Wilson\UItô diffusions): curvature. |<\hidden> \; <\proposition*> The conditional mean of the terminal field is <\equation*> \\\|]>=\+S*F,,F|)>=\,F|)>, a martingale with <\equation*> \\=>\W+S\F=+S*\F|)>|)>*>\W. <\small> martingale \*>*F\s=S*F>. In the differential, >*F\t> cancels with >*F\t>. <\itemize> =\-S*F>: a martingale (the ) minus the drift still expected on the unresolved scales. Noise of >: fresh shell fluctuation . The forward\Ubackward coupling moves from the drift to the volatility. > is the w.r.t. the classical dynamics. |<\hidden> Invert \=x+S*F>: +S*\|)>>, |)>\-F>. Response and regulator: <\equation*> H|)>=+\\|)>|)>,\=S-Q,|\>=S*>*S\0. <\theorem*> <\equation*> \\=H|)>*S*>\W,\|)>=\*V|)>|]>, =\>. ; covariance =H*|\>*H>>. The decoupling field solves the equation <\equation*> \*\+*\*\|)>=0,\=\V. +S*\F=H*S>; |)>> is a martingale and > has no drift: Itô. |<\hidden> \; Compare with Burgers: <\equation*> *F=-\F*>*F|]>->*\*F|)>|\>>\*\=-*\*\|)>|\>> <\itemize> The transport term is : dropping the Itô term, =\V> does not move. All live in the coordinate change +S*\|)>>. come only from the quadratic variation > of the forecast; the nonlinearity has moved into its field dependence. No gradient structure, no Gaussian convolution used: valid for any FBSDE with a Markov martingale force, whenever the coordinate change is invertible. |<\hidden> <\with|font-size|0.9> u+u*\u=0>, |)>=g>. Along the >, is constant: in the Lagrangian label nothing moves. With G>, the <\equation*> G=min+|2*s>|}>,u=|s>, the minimiser being the label of ; characteristics cross (shocks) once > vanishes somewhere. Viscous case: Cole\UHopf. Dictionary: <\padded-center> |||>|>>||+S\V|)>>>>|>|>||>|>|>>|=>>|>|G>>|=\V>>|>|>>>>>>> With noise the label is the of under the Cole\UHopf measure, a martingale; the Moreau formula is Lax\UOleinik with \\>, and at finite cutoff no shocks form. |<\hidden> <\proposition*> If =-\V>, then <\equation*> H|)>=\>\|\|)>=S-S*\*V|)>*S. So \0>, +S*\F> is always invertible, and \S> for convex > (Brascamp\ULieb). <\small> The forecast coordinates are the first two conditional moments of the stochastic localisation: ,|)>=\|\|]>,\|\|)>|)>>. The second moment <\equation*> \*\\|]>=\\\+ is a martingale, hence <\equation*> \|\,\|\>=|\>\t,=-|\>\t+\|)>. Gaussian reference: *|\>*S=>>. |<\hidden> Conditional law >\\>>, *\=S+\*V>. Then <\equation*> +\*V|]>|)>\Cov\\+\*V|)>|]>. <\small> <\itemize> Cramér\URao, no convexity needed: \\\|]>=1> and Cauchy\USchwarz. Brascamp\ULieb (1976), for *\\0>: \\|\f,*\|)>*\f|\>>. Equal for quadratic >. Via =S-S*\*V*S>: two-sided bounds on *V>, i.e. on the Wetterich propagator >. > convex > \S>, *V\0>: convexity propagates along Polchinski. *V\\\Cov\+\|)>>. \; <\myframed> >: Wick ordering gives > the curvature *c+2*\\-\> at =0> (already for >). The upper bound is lost, the conditional law may spread over several wells: the (clipping and global coercivity, Lecture6). |<\hidden> <\with|font-size|0.9> \; |localisation.pdf>|1par|0.66pag||> Two-component field, > a tilted double well (contours: >). >>, =\+S*F>>, >> and > ellipse of >>; open circle: >. The conditional law localises on one well. |<\hidden> \; Phase space of pairs (field, force) \\\\>, with the symplectic form <\equation*> \=\x\\p. <\itemize> (dimension ) is if =0>. A graph > is Lagrangian > F> symmetric > is a gradient. Symplectic maps send Lagrangian submanifolds to Lagrangian ones. The shear =*p,p|)>> is symplectic since > is symmetric: >*\=\+S*\p\\p=\>. \; <\myframed> So > maps the graph of =-\V> to a Lagrangian submanifold; if it is a graph over the forecast >, it is the graph of a gradient. |<\hidden> \; The cheapest way to reach > at from at , without interaction, costs the <\equation*> \|)>=|\-x,S-x|)>|\>, p=S-x|)> <\itemize> \=-p>, >\=p>: so > is the of the shear >. \=|\\,>\\|\>>: a complete integral of the noiseless Polchinski equation. Jacobi: in the new coordinates the Hamiltonian vanishes, and >, are constants of motion. \; <\myframed> ,F|)>> are the Jacobi coordinates of the free Hamiltonian of the field. With noise, the constants of motion become . |<\hidden> =-\V>. The map x+S*F> is > times the gradient of <\equation*> |x,S*x|\>-V,SS\0, so it is injective: the image of the graph of > is a graph over >, that of > with <\equation*> \=\\,H=+\*\|)>,\=V. Generating function: the <\equation*> V=inf>|)>+|x-\,S*|)>|\>|}>, optimality: +S*\\|)>>, V=\\|)>>. No convexity of > or > is used: only +\*\=Cov\0>. Without noise, caustics and shocks. |<\hidden> \; |shear.pdf>|1par|0.66pag||> Left: > maps the graph of =-V> to the graph of > over the forecast. Right: >, > and > (, ). |<\hidden> \; <\theorem*> <\equation*> \*\=|\>*+\*\|)>-S|)>|]>,\=V. Equivalently, for the >=\+*|\,Q*\|\>>, <\equation*> \*\>=|\>**\>+\|)>|]>+c, from the bare action (, =\>) to the 1PI effective action of > (, =0>). |\>*H|)>=-*\*\|)>> by symmetry of *\>, so the force equation reads *\-|\>*H|)>|)>=0>. The constant: envelope theorem Polchinski. |<\hidden> \; \\documentclass[tikz]{standalone} \\usepackage[T1]{fontenc} \\usepackage[sc]{mathpazo} \\usepackage{amsmath,amssymb} \\usetikzlibrary{calc,decorations.markings} \\tikzset{ \ blob/.style={circle,draw,thick,fill=#1,minimum size=7mm,inner sep=0pt}, \ blob/.default={black!12}, \ vtx/.style={circle,fill=black,inner sep=1.5pt}, \ prop/.style={thick}, \ full/.style={double,double distance=1.3pt,thick}, \ ins/.style={circle,fill=black,inner sep=1.5pt}, \ cross/.style={circle,draw,thick,fill=white,inner sep=0pt,minimum size=2.4mm,path picture={\\draw[thick] (path picture bounding box.north west)--(path picture bounding box.south east) (path picture bounding box.north east)--(path picture bounding box.south west);}}, } % external legs: \\legs{node}{angles} \\newcommand\\legs[2]{\\foreach \\a in {#2}{\\draw[thick] (#1.\\a)--++(\\a:2.6mm);}} \\newcommand\\rootleg[2]{\\draw[line width=1.4pt,-latex] (#1.#2)--++(#2:3.6mm);} \\begin{document} \\begin{tikzpicture}[font=\\small,baseline] \\node at (0,0) {$\\partial_t\\,\\Gamma^{\\rm EAA}_t$}; \\node at (1.35,0) {$=\\ \\tfrac12$}; \\draw[full] (2.45,0) circle (.5); \\node[cross] at (2.45,.5) {}; \\node[right] at (3.05,.42) {$\\dot{\\mathcal R}_t$}; \\begin{scope}[yshift=-1.7cm] \\node at (0,0) {$\\partial_t\\,D\\Gamma^{\\rm EAA}_t$}; \\node at (1.35,0) {$=\\ -\\tfrac12$}; \\node[blob] (a) at (2.65,-.3) {\\scriptsize $D^3\\Gamma$}; \\legs{a}{270} \\draw[full] (a.60) .. controls +(60:1.25) and +(120:1.25) .. node[cross,pos=.5]{} (a.120); \\end{scope} \\begin{scope}[yshift=-3.6cm] \\node at (0,0) {$\\partial_t\\,D^2\\Gamma^{\\rm EAA}_t$}; \\node at (1.35,0) {$=\\ -\\tfrac12$}; \\node[blob] (a) at (2.65,-.3) {\\scriptsize $D^4\\Gamma$}; \\legs{a}{230,310} \\draw[full] (a.60) .. controls +(60:1.25) and +(120:1.25) .. node[cross,pos=.5]{} (a.120); \\node at (3.85,0) {$+$}; \\node[blob] (b) at (4.75,0) {\\scriptsize $D^3\\Gamma$}; \\legs{b}{180} \\node[blob] (c) at (6.75,0) {\\scriptsize $D^3\\Gamma$}; \\legs{c}{0} \\draw[full] (b.50) .. controls +(40:.5) and +(140:.5) .. node[cross,pos=.5]{} (c.130); \\draw[full] (b.310) .. controls +(-40:.5) and +(-140:.5) .. (c.230); \\end{scope} \\begin{scope}[yshift=-5.3cm] \\draw[full] (.25,0)--(1.05,0); \\node at (1.65,0) {$=$}; \\draw[prop] (2.1,0)--(2.9,0); \\node[right] at (2.95,0) {$S_t$}; \\node at (3.7,0) {$-$}; \\draw[prop] (4.05,0)--(4.65,0); \\node[blob,fill=black!12] (e) at (5.0,0) {\\scriptsize $D^2\\Gamma$}; \\draw[prop] (5.35,0)--(5.95,0); \\node at (6.35,0) {$+$}; \\node at (6.85,0) {$\\cdots$}; \\end{scope} \\end{tikzpicture} \\end{document} \; |ps>|210pt|||>>> |\>>. Exactly one loop: the Legendre transform has absorbed the tree term.>> |<\hidden> the backward half of the FBSDE> \; Every object in the Wetterich equation is an observable of the Polchinski process. \; |>|>>|=\|]>>>>|\|)>>>|*V|)>|]>>>>|*\>+\|)>>>|>>>|=S-Q>>|>|*\>>||\>*-S|)>|]>>>>>>>> \; Driven by conditional covariance; , with the full propagator; in the Gaussian case only the constant flows. > |<\hidden> \; At the optimal forecast, =\|)>+\|)>>: the cost attached to the forecast plus the cost of reaching it by free motion. This transports the generating function along >: <\equation*> \ V=|\\,\\|\>+|p,\\|\>-|p,\ x|\>=-|F,\ x|\>. With x|)>=|x,Sx|\>-V> up to a function of , <\equation*> \>|)>=\|)>+|\,S\|\>. <\itemize> > in the field ; > in the coordinate x>; > in the forecast >. Polchinski is the Hamilton\UJacobi equation of >; Wetterich is its Legendre dual. The regulator is the information gained. |<\hidden> \; |||||||| For >, =C*>|)>> and <\equation*> \=>-1>. <\itemize> t> (a mass term) on the modes \t>; exponentially small on \t>. The of the functional RG. Other profiles give other regulators, e.g. the mass regulator =t>. |<\cell> |exp_regulator.pdf>|0.480par|0.62pag||> >>>> |<\hidden> > as a rate function> At : the 1PI effective action of > (Jona-Lasinio 1964), <\equation*> \>|)>=V+sup|j,\|\>-log \|j,\|\>>*\\|}>. <\itemize> : the mean of independent samples is \> with probability \>|)>-V|)>>>: a . Convex even when > is not; minimal at the mean; Hessian there inverse covariance. Formally, the same for the field averaged over a large volume: > per unit volume (the effective potential of QFT, not >) is the rate function of the magnetisation; flat parts may appear (Maxwell). The renormalisation group as a probabilistic object: limit theorems for strongly dependent variables. > |<\hidden> \; Gibbs variational principle for the conditional law: <\equation*> \|)>=inf V\\+\|\>*-\|)>|)>:\\\=\|}>: the free energy of the unresolved shell, . > is the value function; > is its Moreau transform. Drop the Itô term: =V>, and the Moreau transform becomes the <\equation*> V>=inf+|y-x,S*|\>|}>, which solves *V=*|\V,>*\V|\>>: the saddle point of the Gaussian convolution. The minimiser solves the classical field equation *+\V=0> in the shell >; expanding in > gives exactly the tree diagrams. |<\hidden> \; <\itemize> of the interacting measure, renormalisation, control and filtering, produce one forward\Ubackward system. Backward in resolution it is a linear Gaussian blurring of >. In the system has no drift, trees are integrated exactly, and the propagator is the conditional covariance. the value function generates the canonical transformation to these coordinates; the Legendre transform of the filter gives > and the equation. \; <\myframed> Lecture5: observables, the field equation at every resolution and its sub-Gaussian noise. |<\hidden> <\with|font-size|0.9> \; |||||||| (numbering of the notes) <\itemize> 7.1 Quadratic interactions; 7.2 zero dimensions; 7.3 stochastic maximum principle; 7.5 cost of the leading drift. 8.1 Characteristics; 8.4 the constant; 8.6 1PI effective action. 8.7 Trees versus loops; 8.8 regulators; 8.9 Brascamp\ULieb; 8.10 the free action. |<\cell> <\itemize> Polchinski, Nucl. Phys. B 1984; Wetterich, Phys. Lett. B 1993; Jona-Lasinio, Phys. Rep. 2001; Eldan, ICM 2022. Bauerschmidt\UBodineau\UDagallier, , 2024. Barashkov\UGubinelli, >>, Duke 2020. Berges\UTetradis\UWetterich, Phys. Rep. 2002. Ma\UYong, , LNM 1702. >>>> > <\initial> <\collection> >>