> <\body> 3 \V notes: Lecture3, ŸŸ4\U6> |<\hidden> \; A Euclidean field is a measure on |)>>; the GFF > is its Gaussian part; local operators are functions of >. Wick powers >: are random distributions iff \d>, and otherwise. The GFF as an equilibrium; then index everything by a |)>>. <\enumerate> : the GFF as the stationary marginal of a Langevin dynamics, and as the one-point marginal of an elliptic equation. The GFF as the terminal value of a martingale of smooth fields; martingales. Observables of the GFF: Wick powers in every dimension; . |<\hidden> \; 3> \; |<\hidden> \; Finite dimensions: X=-\*S|)>\t+\B> has generator =\-\*S\\> and <\equation*> \>\=\\\+\\*S|)>=0: the Gibbs measure \x/Z> is invariant. Apply this to the Euclidean action, with a fictitious : <\equation*> \*\=--\|)>*\-V|)>+*\*\\\, > space-time white noise: **\|]>=|u,v|\>*\\|)>>>. (stationary solutions) For : the stochastic heat equation (infinite-dimensional OU process). History: Parisi\UWu 1981 (gauge theories without gauge fixing); physics review Damgaard\UHüffel 1987\U88; first mathematical construction for |)>> on >: Jona-Lasinio\UMitter 1985. Review: Gubinelli, 2025. |<\hidden> > invariance> \; Itô's formula: for |)>,\,\|)>|)>> the generator is <\equation*> \*F|)>=|f,f|\>*\*g|)>-\*|)>*\*g|)>. <\proposition*> >*\*F|]>=-\>|D*G,D*F|\>|)>>|]>> for \*C>>; in particular >**F|]>=0>. Apply DS to \*g> with > and sum over : <\equation*> \*\*g*\*|)>|]>=|f,f|\>*\**k*\*g+k*\*g|]>. |<\hidden> \; <\theorem*> For \>, \|)>> let =*\|)>>, \ =t>*A>f|)>>. <\enumerate-alpha> **\|]>=|f,\*A>C*g|\>>: stationary, and |)>=\> for every . > is a mild and weak solution: for t>, =\-\*\r+*\*>\h|)>>. From any initial datum \\|)>> independent of >: \\>\\*|f,C*f|\>>>. Hence > is the . |K,K|\>=>s>|f,\*A>g|\>\r=|f,\*A>*g|\>>; an >-valued version exists by Itô\UNawata/Mitoma regularisation. |<\hidden> \; >.> The modes |)>> are independent stationary OU processes X=-\*X\t+\B> with variance >. Mild form from the semigroup identity =K*A>h>+>\|)>*A>h>; weak form by stochastic Fubini (Exercise4.3). By independence, <\equation*> \\\>=\\f|)>>|]>\*|f,|)>*C*f|\>>. For 0>, f\0> in |)>>, so f|)>\0> a.s. If \\> the product is exactly *|f,C*f|\>>> (invariance); an invariant > must have the same characteristic functional. |<\hidden> \; <\itemize> As a field on \\>, the stationary solution has covariance +A|)>*+A|)>=2*+A|)>>; its equal-time slice is >+A|)>*\|2*\>=A>. The stochastic time has nothing to do with the Euclidean time >. Interacting case: a . <\itemize> >: Da Prato\UDebussche 2003 (also Jona-Lasinio\UMitter, Albeverio\URöckner). >: Hairer 2014; Gubinelli\UImkeller\UPerkowski 2015; Catellier\UChouk 2018. Global bounds: Mourrat\UWeber 2017; Gubinelli\UHofmanová 2019; construction of Euclidean > via the dynamics: Gubinelli\UHofmanová 2021. |<\hidden> \; 3> \; |<\hidden> \; The elliptic SPDE in dimensions <\equation*> -\\\>|)>*\+V|)>=\,\*\\\, is related to the -dimensional Gibbs measure: formally <\equation*> |)>|)>\exp *|\,A*\|\>+V|)>|]>|)>*\*\. <\itemize> Mechanism: a hidden (Parisi\USourlas; Klein\ULandau\UPerez 1984). Rigorous nonlinear results: (Albeverio\UDe Vecchi\UGubinelli 2020); polynomial on > and exponential on > (ADG 2021); elliptic > in 4 and 5 dimensions (Gubinelli\UHofmanová 2019). For it is elementary, and it explains the >. |<\hidden> \; <\theorem*> Let =m-\\\>=-\>+A> and > white noise on \\>. <\enumerate-alpha> \=>*\> has a unique tempered solution =>*\u|)>>. =\f|)>\L>, continuous in ; =>*\|)>> is the section of > at . **\|]>=|f,C*g|\>>: every section has law >, and |)>|)>\>> is stationary and isotropic. > maps > onto >, so u|)>=0> u> forces . |<\hidden> > comes from> \; Fourier transform in only: |^>=\q\z>+A|)>*f>. By Plancherel and the spectral theorem, <\equation*> |K,K|\>=|f,>q||)>>*+A|)>|]>*g|\>=>*|f,A*g|\>, since <\equation*> >q||)>>*+a|)>>=>*>\r|+a|)>>=*a>. Multiply by >. Stationarity and isotropy: the covariance *\q\|)>>|f,+A|)>*g|\>q||)>>> depends only on |\|>>. |<\hidden> \; Field on \\> with covariance >+A|)>>; its slice \\> has covariance <\equation*> c*A,c=*|)>||)>*\>. >>||>|>>|*A>>||)>>>>|>>||)>*A>>|>|, A\A>>|+A|)>\A>>|>|>>|A>>|>>>> The GFF arises iff >. For =\>, every codimension-two slice of the elliptic field is a GFF. |<\hidden> \; <\itemize> Wilson\UKadanoff: physics is organised . The measure-on-> picture hides this, and forces singular objects from the start. Observe the field at resolution : a function >. Information at resolution : =\:s\t|)>>. A physical quantity is not a random variable but a , i.e. a martingale. <\myframed> Observables are martingales in the filtration of finite-resolution measurements. They need not have a terminal value. |<\hidden> \; 3> \; |<\hidden> \; <\definition*> A is a family |)>0>> of symmetric, translation invariant operators with =0>, >\\*Q\0>, \C=-\|)>=A>, and smooth kernels for \>. <\equation*> Q=C\>,>=>\>,>*=\/t>||)>>\*/4>\. >> lives on distances 1/t>; > keeps the modes \t>: is a . Others: sharp Fourier cutoff \t>*+m|)>>; decompositions, >*=0> for \1/t>. Notation: =Q*>, =Q-Q>. For 2>, \\> (>*log t> in , t> in 3>). |<\hidden> > \; cylindrical Brownian motion on |)>>, <\equation*> X\>\W. <\proposition*> > is a smooth Gaussian field, |)>=\>>; |)>> is a continuous martingale with |X,X|\>=>*\t>; \X>> in >>, -1>, with >|)>=\>; -X\\>, with law >>. The GFF is the . (torus). =>\B|)>*e>: independent Gaussian martingales, variance *\*q|)>>; then as in Lecture2. |<\hidden> : a picture> \; |multiscale_gff.png>|0.95par|||>Heat profile on >, : and the lattice cutoff, same white noise. The martingale reveals finer and finer details; the large scales are fixed at low resolution. |<\hidden> \; =\\\|\|]>>. <\definition*> An is an |)>>-martingale |)>0>>. Its expectation is |O|\>=\*O>. An observable field is a family |)>\>>. <\itemize> L>|)>> gives a observable =\*Y>. The point: observables need be closed. >, 2>, diverges, yet is a perfectly good family of estimates. <\lemma*> (a) |O-O|\<\|\|\>>>\0\O=O>.(b) adapted, |N-O|\<\|\|\>>>\0\O=lim \*N>. > is an > contraction. |<\hidden> <\with|font-size|0.9> Two integrable adapted processes are if <\equation*> N\N\|N-N|\<\|\|\>>>\0\|)>. <\itemize> A class contains : if O> are martingales, then |O-O|\<\|\|\>>>\|O-O|\<\|\|\>>>\0>. When it exists, it is recovered by looking back: =lim\> \*N> in >, for any in the class. An a class containing a martingale; its members are , the martingale is the canonical representative. For L>|)>>, <\equation*> Y\*Y|)>0> is injective (*Y\Y> in >), with image the uniformly integrable martingales: the closed observables. <\myframed> > random variables.> Removing uniform integrability enlarges the space; an almost sure limit alone need not determine the observable. |<\hidden> For a standard Brownian motion and the Hermite polynomials > of variance , the martingales <\equation*> B,B-t,B-3*t*B,\,H;t|)> are observables without a terminal value, and not even bounded in >: <\equation*> \-t|\|>=t*\-1|\|>\\,Z\N. They sum up to the exponential martingale (\\, \\0>): <\equation*> Z=\*B-\*t/2>=0>|n!>*H;t|)>,Z\0,\*Z=1, since *B-\*t/2\-\*t/2\-\>. The mass escapes to rarer and rarer paths. <\myframed> A point value > of the GFF is a Brownian motion run with the clock >: these are the prototypes of the Wick powers ;c|)>>. |<\hidden> <\proposition*> Let =N+\*\s+M>, with \L>, a martingale and >|\|>*\s\\>. The only observable with |N-O|\<\|\|\>>>\0> is <\equation*> O=N+\>\*\s|]> =N+M+\>\*\s> is a martingale, and |O-N|\<\|\|\>>>\\>|\|>*\s\0>; uniqueness is asymptotic uniqueness. =B+\*sin B>. Itô: =-*\*sin B>, and *sin B=\/2>*sin B>, so <\equation*> \>\*\s=-*sin B*>\/2>*\s=-\*sin B,O=B. <\myframed> This is the germ of the forward\Ubackward equations: in Lecture4 the forecast =\*\> is the field plus the expected drift still to come. |<\hidden> Fix 0>, \>, the heat kernel =*s|)>*\/2*s>>, and <\equation*> \\p-y|)>,0\t\T. <\itemize> *\=\> for t>, by Chapman\UKolmogorov \p=p>. >:> g*\*\y=\*g|)>> for bounded , by the Markov property. at fixed , \0> almost surely as T>, since \y> a.s.; but *\=p\0>. \> is a probability density for every T>, so |)>> is a measure-valued observable, although each > tends to zero. <\myframed> -y|)>> is not a random variable, but it is an observable:> the family of its conditional densities. Its derivatives *\> are observables too. T>; reparametrise by > to get ,\|)>>. The same works for any Markov process with a transition density.>> |<\hidden> \; An observable is determined by its . <\itemize> A of : functionals >> of smooth fields with |O->|)>|\<\|\|\>>>\0>. A germ determines . > is if it has germs <\equation*> >|)>=|\>,\*X,\,\*X|)>. Germs are not unique: this freedom the freedom of choosing counterterms. :=H*;c|)>> is a local observable in (more later), although >: does not exist as a random distribution when \d>. |<\hidden> \; 3> \; |<\hidden> \; Recall *H*=-*\*H>, *H=n*H>. <\proposition*> For every , , : :\H*;c|)>> is a continuous martingale, :X:=n:X:\X>, and <\equation*> \*::X:|]>=\*n!Q*|t\\>\*n!G**y|)>. Itô: the drift is *\+\|)>*H*>=0>. <\itemize> Lecture2: >: exists iff \d>. As , Wick powers exist in every , unsmeared. For \d>, f\0>, :> is unbounded in > and (Paley\UZygmund) not tight: ; the observable is the whole family. The singularity has moved to . |<\hidden> \; <\definition*> \*O> is the observable with | \*O|)>-O O|\<\|\|\>>>\0>; unique, and \*O|)>=lim \ O|]>>. <\proposition*> If ,O> are continuous, square integrable on bounded intervals, and *>|O,O|\>|\|>\\>, then <\equation*> \*O|)>=O O+\>\|O,O|\>|]>. O-|O,O|\>> and *|O,O|\>>> are martingales. |<\hidden> <\with|font-size|0.9> For z>, T>, >=> and =*s|)>>, the product is exact: <\equation*> \*\=\*\/4*>*p/2>->|)>. Complete the square: -y|)>+-z|)>=2*->|)>+|2>>, and use *|)>=\**|)>>. The product is 0>, so its expectation is its > norm (\N>): <\equation*> |\*\|\<\|\|\>>>=\*\/4*>*p/2>>|)>\0T|)>. So *\\0>: the zero observable is the martingale in its class, hence \\=0> for z>. At the exponential suppression disappears, *|)>=\*p/2>\\>, and no martingale is in the class: its constant expectation would have to be this divergent limit. <\myframed> The product depends dramatically on whether the singular supports meet: separated supports multiply to zero, coincident ones call for renormalisation. |<\hidden> Fine resolution >, coarse resolution u>, >=*\|)>>. Since >=\>*p/2>>, Chapman\UKolmogorov gives the exact identity <\equation*> \>|)>|]>=\>*p/2>-y|)>. Taylor expand in the time variable, with >*p>=*\*p>>: <\equation*> \>|)>|]>=\>*\-*\>|4>*\*\+*\>|32>*\*\+O,s>|)>. <\itemize> Singular coefficients (>\\>; *\>,\*\>\0>) times observables ,\*\,\>: an . The order of limits matters: fixed coarse resolution , then \0>. Without conditioning, |>|)>-\>*\>|\<\|\|\>>>\\>. |<\hidden> \; |,|\>=Q*> is deterministic. Heat profile, 2>: the condition holds iff y>, and then <\equation*> *\|)>=X*X+G*-Q*,|*\|\>=G*. The correct two-point function, >>>. For : no product, since \\>. At every fixed resolution, as x>, <\equation*> *\|)>-G*\X-c=:X:, i.e. *\=G**+::+o>: the Wick square is the regular part of the product. |<\hidden> \; For y>, <\equation*> ::*\::=n\k>j!G*::. <\itemize> Wick's theorem at the level of observables. The > carry all singularities; the observables on the right have regular limits as x>. |<\hidden> \; <\itemize> ,O> with germs supported in >, >, ,U|)>\0>: the covariation |D*O,>*D*O|\>\t> sees >> only between the supports. Finite range decomposition: it vanishes for large , so the product exists and is asymptotically the pointwise product. Heat profile: exponentially small. Observables on multiply freely; products at the need renormalisation. <\myframed> This is the structure of a of observables (precise axioms not established here). >> |<\hidden> <\with|font-size|0.9> \; |locality.png>|1par|0.66pag||> Left: kernels of >> at and ; their overlap integrates to >*=\|,|\>/\s>. Right: negligible for 1/>; total >. As x> the mass escapes to \>. |<\hidden> \; <\itemize> : the GFF is the stationary marginal of the parabolic Langevin dynamics and the one-point marginal of the elliptic SPDE. The GFF is the terminal value of a martingale > of smooth fields; , possibly without terminal value. Wick observables exist in every dimension; products and the OPE live at . <\myframed> Lecture4: interactions, the Polchinski equation and its FBSDE, the Wetterich equation. |<\hidden> \; |||||||| (numbering of the notes) <\itemize> 4.1 Infinitesimal invariance characterises >. 4.4 Slices of Gaussian fields. 5.1 The field at resolution ; 5.2 divergence of >. 5.3 Brownian observables; 5.4 Brownian deltas. 6.2 Products of Wick observables; 6.3 exponentials. 6.4 The Brownian OPE; 6.5 point splitting. |<\cell> <\itemize> Da Prato\UZabczyk, . Albeverio\UDe Vecchi\UGubinelli, , AoP 2020. Bailleul\UChevyrev\UGubinelli, , in prep. >>>> > <\initial> <\collection> >>