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  <screens|<\hidden>
    <cqft-title-slide|Observables and field equations|Joint work with
    Sarah-Jean Meyer, with the use of AI|SISSA lectures, Lecture<nbsp>5 \V
    notes: Lecture<nbsp>5, Ÿ9, Appendices A\UB>
  </hidden>|<\hidden>
    <mytitle|Where we are>

    \;

    <with|font-series|bold|Lecture<nbsp>4.> The interacting field at
    resolution <math|t> solves the FBSDE

    <\equation*>
      \<mathd\>\<Phi\><rsub|t>=<wide|Q|\<dot\>><rsub|t>*F<rsub|t>\<mathd\>t+<wide|Q|\<dot\>><rsub|t><rsup|1/2>\<mathd\>W<rsub|t>,<space|2em>F<rsub|t>=-\<mathD\>V<rsub|t><around|(|\<Phi\><rsub|t>|)>=\<bbb-E\><rsub|t><around|[|-\<mathD\>V<rsub|T><around|(|\<Phi\><rsub|T>|)>|]>,
    </equation*>

    <math|\<Phi\><rsub|t>\<sim\>\<nu\><rsub|t>\<propto\>\<mathe\><rsup|-V<rsub|t>>\<mu\><rsup|Q<rsub|t>>>.
    Filtering: <math|\<Phi\><rsub|t>=J<rsub|t>\<Phi\><rsub|T>+\<zeta\><rsub|t>>,
    <math|J<rsub|t>=K*Q<rsub|t>>, <math|K=Q<rsub|T><rsup|-1>>; forecast
    <math|\<psi\><rsub|t>=\<bbb-E\><rsub|t>\<Phi\><rsub|T>=\<Phi\><rsub|t>+S<rsub|t>F<rsub|t>>;
    regulator <math|\<cal-R\><rsub|t>=S<rsub|t><rsup|-1>-K>,
    <math|S<rsub|t>=Q<rsub|T>-Q<rsub|t>>; effective average action
    <math|\<Gamma\><rsub|t>>; <math|\<mu\><rsup|\<phi\>><rsub|t>> the law of
    <math|\<Phi\><rsub|T>> given <math|\<Phi\><rsub|t>=\<phi\>>, and
    <math|<math-up|Cov><rsub|t>> its covariance.

    <with|font-series|bold|Today.>

    <\enumerate>
      <item>From Minkowski to Euclidean: the field equation and
      <strong|Nelson's Euclidean noise>.

      <item>The <strong|field equation at every resolution>; DS equations,
      also pathwise.

      <item>The noise is <strong|sub-Gaussian>, whatever the interaction.

      <item>Infinite volume; appendices: algebra of the flow, counterterms,
      localisation.
    </enumerate>
  </hidden>|<\hidden>
    <vspace*|2.5cm>

    \;

    <cqft-part-title|Observables and field equations>

    <vspace|0.3cm>

    <with|color|dark grey|Lecture<nbsp>5>

    \;
  </hidden>|<\hidden>
    <mytitle|Observables of the interacting theory>

    \;

    For <math|G> a functional of the terminal field,
    <math|\<bbb-E\><rsup|\<bbb-P\>><rsub|t><around|[|G<around|(|\<Phi\><rsub|T>|)>|]>=G<rsub|t><around|(|\<Phi\><rsub|t>|)>>,
    <math|G<rsub|t><around|(|\<phi\>|)>=\<mu\><rsub|t><rsup|\<phi\>><around|(|G|)>>,
    and

    <\equation*>
      \<partial\><rsub|t>*G<rsub|t>+<tfrac|1|2><math-up|Tr><around*|(|<wide|Q|\<dot\>><rsub|t>*\<mathD\><rsup|2>*G<rsub|t>|)>-<around|\<langle\>|<wide|Q|\<dot\>><rsub|t>*\<mathD\>V<rsub|t>,\<mathD\>G<rsub|t>|\<rangle\>>=0,<space|2em>G<rsub|T>=G:
    </equation*>

    the backward Kolmogorov equation, a Doob transform of the Gaussian heat
    equation. Any solution gives an observable, closed at finite <math|T>,
    not necessarily as <math|T\<to\>\<infty\>>. The force <math|F<rsub|t>> is the
    observable of <math|G=-\<mathD\>V<rsub|T>>.

    <with|font-series|bold|Tangent observables.>
    <math|V<rsub|T><rsup|\<theta\>>> depending on a coupling,
    <math|U<rsub|t>=\<partial\><rsub|\<theta\>>*V<rsub|t><rsup|\<theta\>>>:

    <\equation*>
      U<rsub|t><around|(|\<Phi\><rsub|t>|)>=\<bbb-E\><rsup|\<theta\>><rsub|t>*<around*|[|\<partial\><rsub|\<theta\>>*V<rsub|T><rsup|\<theta\>><around|(|\<Phi\><rsub|T>|)>|]>,<space|2em>\<partial\><rsub|\<theta\>>*\<bbb-E\><rsup|\<nu\><rsup|\<theta\>><rsub|T>><around|[|G|]>=-<math-up|Cov><rsup|\<nu\><rsub|T><rsup|\<theta\>>><around*|(|G,\<partial\><rsub|\<theta\>>*V<rsup|\<theta\>><rsub|T>|)>.
    </equation*>

    E.g. the running <math|:\<varphi\><rsup|4>>: for
    <math|\<theta\>=\<lambda\>>: observables <strong|conjugate to the
    coordinates of theory space>.
  </hidden>|<\hidden>
    <mytitle|Field equations: Minkowski and Euclidean>

    <\with|font-size|0.9>
      Minkowski (Lecture<nbsp>1): <math|<around|(|\<Box\>+m<rsup|2>|)>*\<varphi\>+V<rprime|'><around|(|\<varphi\>|)>=0>
      as an <strong|operator> equation, with the CCR. On time-ordered
      functions <math|\<partial\><rsub|t><rsup|2>> also hits the step
      functions:

      <\equation*>
        <around|(|\<Box\><rsub|x>+m<rsup|2>|)>*\<tau\><around|(|x,x<rsub|1>,\<ldots\>|)>=-\<tau\><around|(|V<rprime|'><around|(|\<varphi\>|)><around|(|x|)>,x<rsub|1>,\<ldots\>|)>-\<mathi\>*<big|sum><rsub|j>\<delta\><around|(|x-x<rsub|j>|)>*\<tau\><around|(|\<ldots\>,<wide|x|^><rsub|j>,\<ldots\>|)>.
      </equation*>

      Contact terms: the canonical commutator, a derivative,
      <math|\<mathi\><around|[|\<partial\><rsub|t>*\<varphi\><around|(|<math-bf|x>|)>,\<cdummy\>|]>=\<delta\>/\<delta\>\<varphi\><around|(|<math-bf|x>|)>>.
      Rotate, <math|t=-\<mathi\>*\<tau\>>: <math|\<Box\>+m<rsup|2>\<to\>A>,
      <math|-\<mathi\>*\<delta\><around|(|t|)>\<to\>\<delta\><around|(|\<tau\>|)>>,
      and we get the DS equation of Lecture<nbsp>2:

      <\equation*>
        \<bbb-E\><rsup|\<nu\>>*<around|[|G<space|0.17em><around|\<langle\>|h,\<omega\>|\<rangle\>>|]>=\<bbb-E\><rsup|\<nu\>>*<around|[|\<mathD\><rsub|h>*G|]>,<space|2em>\<omega\>\<assign\>A*\<varphi\>+V<rprime|'><around|(|\<varphi\>|)>,
      </equation*>

      nonzero: the Euclidean field does not solve <math|A*\<varphi\>+V<rprime|'><around|(|\<varphi\>|)>=0>.

      <\myframed>
        <strong|The noise is the transpose of the derivative:>
        <math|\<mathD\><rsub|h><rsup|\<top\>>=-\<mathD\><rsub|h>+<around|\<langle\>|h,\<omega\>|\<rangle\>>>,
        so <math|<around|\<langle\>|h,\<omega\>|\<rangle\>>=\<mathD\><rsub|h><rsup|\<top\>>1>,
        minus the score of <math|\<nu\>>. Not an independent input: a
        functional of the field, fixed by its law. Free field:
        <math|\<omega\>=A<rsup|1/2>*\<xi\>>, one derivative of white noise.
      </myframed>
    </with>
  </hidden>|<\hidden>
    <mytitle|Characterising the noise: two attempts>

    <\with|font-size|0.9>
      <strong|Nelson (1975): condition in real space.> On the lattice <math|\<varepsilon\>\<bbb-Z\><rsup|d>>,
      <math|\<omega\>\<assign\>A*\<varphi\>+V<rprime|'><around|(|\<varphi\>|)>>.
      By the Markov property <math|\<varepsilon\><rsup|d>*\<omega\><around|(|x|)>>
      is minus the score of the law of <math|\<varphi\><around|(|x|)>> given
      its neighbours:

      <\equation*>
        \<bbb-E\><around|[|\<omega\><around|(|x|)>\|\<varphi\><around|(|y|)>,y\<neq\>x|]>=0,<space|2em>\<bbb-E\><around|[|\<omega\><around|(|x|)>*\<omega\><around|(|y|)>|]>=<around|(|-\<Delta\>+m<rsup|2>+\<bbb-E\>*V<rprime|''><around|(|\<varphi\>|)>|)>*\<delta\><around|(|x-y|)>,
      </equation*>

      not Gaussian. A random variable iff
      <math|\<bbb-E\>*V<rprime|''><around|(|\<varphi\>|)>\<less\>\<infty\>>
      in the limit (heuristically): unlikely in <math|d=3> (cf.
      Albeverio\ULiang\UZegarlinski 2006). <math|\<omega\>>: <strong|Nelson's Euclidean noise> (we keep his letter); as far as we know, not studied since
      as a noise.

      <strong|Albeverio\UYoshida (2002): prescribe a Gaussian noise.> Solve
      <math|A*\<psi\>+V<rprime|'><around|(|\<psi\>|)>=A<rsup|1/2>*\<xi\>>:
      <math|\<psi\>+A<rsup|-1>*V<rprime|'><around|(|\<psi\>|)>> is a free
      field, and formally

      <\equation*>
        <math-up|Law><around|(|\<psi\>|)>\<propto\><around|\||det<around|(|1+A<rsup|-1>*V<rprime|''>|)>|\|>*\<mathe\><rsup|-<around|\<langle\>|\<psi\>,V<rprime|'>|\<rangle\>>-<frac|1|2><around|\<langle\>|V<rprime|'>,A<rsup|-1>*V<rprime|'>|\<rangle\>>>*\<mu\><rsup|A<rsup|-1>>\<neq\>\<mathe\><rsup|-V>*\<mu\><rsup|A<rsup|-1>>:
      </equation*>

      nonlocal, not Markov, <strong|no reflection positivity> (only on
      functionals of <math|\<psi\>+A<rsup|-1>*V<rprime|'><around|(|\<psi\>|)>>, a free field).

      <\myframed>
        <strong|Here: condition on the scales.> The noise at resolution
        <math|t> is the conditional expectation of <math|\<omega\>> given the
        coarse field: not Gaussian, but a genuine random variable at each
        resolution, with Gaussian tails.
      </myframed>
    </with>
  </hidden>|<\hidden>
    <mytitle|Dyson\USchwinger equations at every resolution>

    \;

    All operators are functions of <math|A> (so they commute);
    <math|K=Q<rsub|T><rsup|-1>>, <math|J<rsub|t>=K*Q<rsub|t>> (heat profile:
    <math|J<rsub|t>=\<mathe\><rsup|-A*<around|(|t<rsup|-2>-T<rsup|-2>|)>>\<to\>\<mathe\><rsup|-A/t<rsup|2>>>
    as <math|T\<to\>\<infty\>>).

    The noise observable is the <strong|filtered bridge noise>: with
    <math|\<Phi\><rsub|t>=J<rsub|t>\<Phi\><rsub|T>+\<zeta\><rsub|t>>,
    <math|\<omega\><rsub|t>=K\<Phi\><rsub|t>-J<rsub|t>F<rsub|t>=S<rsub|t><rsup|-1>\<bbb-E\><rsub|t>\<zeta\><rsub|t>>.

    <\theorem*>
      <with|font-shape|right|(formal)> The <strong|noise observable>
      <math|\<omega\><rsub|t>=K*\<Phi\><rsub|t>-J<rsub|t>*F<rsub|t>> is a
      centred martingale, <math|\<mathd\>\<omega\><rsub|t>=K*<wide|Q|\<dot\>><rsub|t><rsup|1/2>\<mathd\>W<rsup|\<bbb-P\>><rsub|t>-J<rsub|t><space|0.17em>\<mathd\>F<rsub|t>>,
      and

      <\equation*>
        \<bbb-E\><rsup|\<bbb-P\>>*<around*|[|\<mathD\><rsub|J<rsub|t>*h>*G<around|(|\<Phi\><rsub|t>|)>|]>=\<bbb-E\><rsup|\<bbb-P\>><around*|[|G<around|(|\<Phi\><rsub|t>|)><space|0.17em><around|\<langle\>|h,K*\<Phi\><rsub|t>-J<rsub|t>*F<rsub|t>|\<rangle\>>|]>.
      </equation*>

      At <math|t=T>: the DS equation <math|\<bbb-E\><rsup|\<nu\><rsub|T>>*<around|[|\<mathD\><rsub|h>*G|]>=\<bbb-E\><rsup|\<nu\><rsub|T>><around|[|G<around|\<langle\>|h,K*\<varphi\>+\<mathD\>V<rsub|T>|\<rangle\>>|]>>.
    </theorem*>

    <with|font-series|bold|Proof.> <math|<wide|J|\<dot\>><rsub|t>=K*<wide|Q|\<dot\>><rsub|t>>:
    drifts cancel. IBP for <math|\<mu\><rsup|Q<rsub|t>>> in direction
    <math|J<rsub|t>*h>, using <math|\<Phi\><rsub|t>\<sim\>\<nu\><rsub|t>>.

    At <math|t=T>: <math|\<omega\><rsub|T>=K*\<Phi\><rsub|T>+\<mathD\>V<rsub|T><around|(|\<Phi\><rsub|T>|)>>,
    the Euclidean noise, and <math|\<omega\><rsub|t>=\<bbb-E\><rsub|t><around|[|\<omega\><rsub|T>|]>>:
    <strong|what an observer at resolution <math|t> knows of it>.
  </hidden>|<\shown>
    <mytitle|Dyson\USchwinger, pathwise>

    \;

    <math|G> a terminal functional, <math|G<rsub|t><around|(|\<Phi\><rsub|t>|)>=\<bbb-E\><rsub|t><around|[|G<around|(|\<Phi\><rsub|T>|)>|]>>
    its observable. Then

    <\equation*>
      N<rsub|t>\<assign\>N<rsub|t><around*|(|\<Phi\><rsub|t>|)>=<around*|(|\<mathD\><rsub|J<rsub|t>*h>*G<rsub|t>|)><around|(|\<Phi\><rsub|t>|)>-G<rsub|t><around|(|\<Phi\><rsub|t>|)><space|0.17em><around|\<langle\>|h,\<omega\><rsub|t>|\<rangle\>><space|1em><text|is
      a martingale, >N<rsub|0>=0.
    </equation*>

    <\itemize>
      <item><strong|Itô>: the drift of <math|\<mathD\><rsub|J<rsub|t>h>G<rsub|t>>
      is <math|<around|\<langle\>|\<mathD\>G<rsub|t>,<wide|Q|\<dot\>><rsub|t><around|(|K-\<mathD\>F<rsub|t>J<rsub|t>|)>h|\<rangle\>>\<mathd\>t>,
      exactly the covariation of the two martingales
      <math|G<rsub|t><around|(|\<Phi\><rsub|t>|)>> and
      <math|<around|\<langle\>|h,\<omega\><rsub|t>|\<rangle\>>>
      (<math|\<mathD\>F<rsub|t>> symmetric). <math|N<rsub|0>=0> since
      <math|J<rsub|0>=0>, <math|\<omega\><rsub|0>=0>.

      <item><math|N> is the <strong|observable of the DS defect>
      <math|\<mathD\><rsub|h>G-G<around|\<langle\>|h,\<omega\><rsub|T>|\<rangle\>>>;
      <math|\<bbb-E\>*N<rsub|t>=0> is DS at resolution <math|t>, and at
      <math|T> the DS equation of <math|\<nu\><rsub|T>>.

      <item><math|N<rsub|t>> uses only resolution-<math|t> objects:
      <math|G<rsub|t>>, its derivative along <math|J<rsub|t>h>,
      <math|\<omega\><rsub|t>>. No terminal value needed.
    </itemize>

    <\myframed>
      For <math|\<Phi\><rsup|4><rsub|3>>, <math|\<omega\><rsub|T>> is not
      expected to have a limit, but DS survives as an <strong|identity between observables>:
      <math|N> is a martingale null at <math|0>, for every <math|G>,
      <math|h>.
    </myframed>
  </shown>|<\hidden>
    <mytitle|The field equation at resolution <math|t>>

    \;

    As <math|T\<to\>\<infty\>>, <math|K\<to\>A>, and
    <math|\<omega\><rsub|t>\<assign\>K*\<Phi\><rsub|t>-J<rsub|t>*F<rsub|t>>
    becomes

    <\equation*>
      A*\<Phi\><rsub|t>=A*Q<rsub|t>*<space|0.17em>F<rsub|t>+\<omega\><rsub|t>*<space|2em><around|(|A*Q<rsub|t>=\<mathe\><rsup|-A/t<rsup|2>><text|
      for the heat profile>|)>:
    </equation*>

    the Euclidean field equation <math|A*\<varphi\>=-V<rprime|'><around|(|\<varphi\>|)>+>noise,
    <strong|seen at resolution <math|t>>. No products of distributions: the
    nonlinearity enters only through the force <strong|observable>, smeared
    at scale <math|1/t>. For <math|\<Phi\><rsup|4><rsub|3>>, with
    <math|F<rsub|t>> in the role of the renormalised cube,
    <math|t\<to\>\<infty\>> corresponds to the renormalised DS equation
    (Gubinelli\UHofmanová 2021); for the limit see Lecture<nbsp>7.

    <\myframed>
      <strong|Compare: stochastic quantisation (Lecture<nbsp>3)>

      There the field equation is imposed in an <strong|auxiliary time>; here
      it holds at <strong|each resolution>, and the auxiliary parameter is
      the resolution itself.
    </myframed>

    Conditioning an SPDE solution on a scale filtration gives identities of
    the same shape but with a different noise: the martingale
    <math|J<rsub|t><space|0.17em>\<mathd\>F<rsub|t>> in
    <math|\<mathd\>\<omega\><rsub|t>> is essential.
  </hidden>|<\hidden>
    <mytitle|The field equation for the forecast>

    <\with|font-size|0.9>
      Substitute <math|\<Phi\><rsub|t>=\<psi\><rsub|t>-S<rsub|t>F<rsub|t>>;
      since <math|K*S<rsub|t>=1-J<rsub|t>>, the smoothing disappears:
      <math|K*\<psi\><rsub|t>=F<rsub|t>+\<omega\><rsub|t>>. As
      <math|T\<to\>\<infty\>>, with <math|F<rsub|t>=-\<mathD\>\<Gamma\><rsub|t><around|(|\<psi\><rsub|t>|)>=-\<mathD\>V<rsub|t><around*|(|\<Phi\><rsub|t>|)>>:

      <\equation*>
        <wide*|A*\<Phi\><rsub|t>+\<mathe\><rsup|-A/t<rsup|2>>\<mathD\>V<rsub|t><around*|(|\<Phi\><rsub|t>|)>=\<omega\><rsub|t>|\<wide-underbrace\>><rsub|<text|Polchinski
        process>><space|3em><wide*|A*\<psi\><rsub|t>+\<mathD\>\<Gamma\><rsub|t><around|(|\<psi\><rsub|t>|)>=\<omega\><rsub|t>|\<wide-underbrace\>><rsub|<text|forecast>>
      </equation*>

      <\itemize>
        <item>For the forecast the force acts on the same variable, with no
        smoothing: the Euler\ULagrange equation of
        <math|<frac|1|2><around|\<langle\>|\<psi\>,A*\<psi\>|\<rangle\>>+\<Gamma\><rsub|t><around|(|\<psi\>|)>>,
        forced by <math|\<omega\><rsub|t>>.

        <item><math|\<Phi\><rsup|4>>, one loop (finite <math|T>, first order in <math|\<lambda\>>):
        <math|\<Gamma\><rsub|t>\<approx\>V<rsub|T>+<frac|1|2><math-up|Tr><around|(|S<rsub|t>\<mathD\><rsup|2>V<rsub|T>|)>>
        moves the Wick ordering to <math|c<rsub|t>>:

        <\equation*>
          <around|\<langle\>|\<psi\><rsub|t>,K*\<psi\><rsub|t>|\<rangle\>>+<frac|\<lambda\>|3>*<big|int><around|(|\<psi\><rsub|t><rsup|4>-3*c<rsub|t>*\<psi\><rsub|t><rsup|2>|)>=<around|\<langle\>|\<psi\><rsub|t>,\<omega\><rsub|t>|\<rangle\>>+\<ldots\>,
        </equation*>

        and <math|a<rsup|4>-3*c<rsub|t>*a<rsup|2>\<geqslant\><frac|1|2>*a<rsup|4>-<frac|9|2>*c<rsub|t><rsup|2>>
        (since <math|<around|(|a<rsup|2>-3*c<rsub|t>|)><rsup|2>\<geqslant\>0>):
        the <strong|coercivity of the cube> is visible (heuristically; in
        <math|d=3> the mass counterterm still diverges). For
        <math|\<Phi\><rsub|t>> it is hidden behind
        <math|\<mathe\><rsup|-A/t<rsup|2>>>.

        <item>IBP: <math|\<bbb-E\><around|[|G<around|(|\<psi\><rsub|t>|)><around|\<langle\>|h,K*\<psi\><rsub|t>-F<rsub|t>|\<rangle\>>|]>=\<bbb-E\><around|[|\<mathD\><rsub|<math-up|Cov><rsub|t>\<cal-R\><rsub|t>h>G<around|(|\<psi\><rsub|t>|)>|]>>.
      </itemize>
    </with>
  </hidden>|<\hidden>
    <mytitle|The noise as a posterior mean>

    <\with|font-size|0.85>
      One mode, <math|Q<rsub|t>=t>. Known endpoint <math|y>: a Brownian
      bridge, <math|\<Phi\><rsub|t>=<frac|t|T>*y+\<zeta\><rsub|t>> and
      <math|\<omega\><rsub|t>=\<zeta\><rsub|t>/<around|(|T-t|)>\<sim\>N<around|(|0,\<cal-R\><rsub|t>|)>>.
      Random endpoint <math|\<Phi\><rsub|T>\<sim\>\<nu\><rsub|T>>:
      <math|\<omega\><rsub|t>=\<bbb-E\><around|[|<around|(|\<Phi\><rsub|t>-<frac|t|T>*\<Phi\><rsub|T>|)>/<around|(|T-t|)>\|\<Phi\><rsub|t>|]>>.

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          (5.408,-1.914) -- (5.415,-1.899) -- (5.423,-1.885) --
          (5.430,-1.871) -- (5.437,-1.856) -- (5.443,-1.841) --
          (5.448,-1.827) -- (5.453,-1.812) -- (5.458,-1.798) --
          (5.462,-1.783) -- (5.465,-1.769) -- (5.468,-1.754) --
          (5.470,-1.740) -- (5.471,-1.725) -- (5.472,-1.711) --
          (5.472,-1.696) -- (5.471,-1.682) -- (5.470,-1.667) --
          (5.469,-1.653) -- (5.467,-1.638) -- (5.464,-1.624) --
          (5.461,-1.609) -- (5.457,-1.595) -- (5.454,-1.580) --
          (5.449,-1.566) -- (5.445,-1.551) -- (5.440,-1.537) --
          (5.434,-1.522) -- (5.429,-1.508) -- (5.423,-1.493) --
          (5.417,-1.479) -- (5.411,-1.464) -- (5.405,-1.450) --
          (5.399,-1.435) -- (5.393,-1.421) -- (5.386,-1.406) --
          (5.380,-1.392) -- (5.374,-1.377) -- (5.368,-1.363) --
          (5.362,-1.348) -- (5.355,-1.334) -- (5.349,-1.319) --
          (5.344,-1.305) -- (5.338,-1.290) -- (5.332,-1.276) --
          (5.327,-1.261) -- (5.321,-1.247) -- (5.316,-1.232) --
          (5.311,-1.218) -- (5.306,-1.203) -- (5.301,-1.189) --
          (5.297,-1.174) -- (5.292,-1.160) -- (5.288,-1.145) --
          (5.284,-1.131) -- (5.280,-1.116) -- (5.276,-1.102) --
          (5.273,-1.087) -- (5.269,-1.073) -- (5.266,-1.058) --
          (5.263,-1.044) -- (5.260,-1.029) -- (5.257,-1.015) --
          (5.254,-1.000) -- (5.251,-0.986) -- (5.249,-0.971) --
          (5.247,-0.957) -- (5.244,-0.942) -- (5.242,-0.928) --
          (5.240,-0.913) -- (5.238,-0.899) -- (5.236,-0.884) --
          (5.235,-0.870) -- (5.233,-0.855) -- (5.231,-0.841) --
          (5.230,-0.826) -- (5.229,-0.812) -- (5.227,-0.797) --
          (5.226,-0.783) -- (5.225,-0.768) -- (5.224,-0.754) --
          (5.223,-0.739) -- (5.222,-0.725) -- (5.221,-0.711) --
          (5.220,-0.696) -- (5.219,-0.681) -- (5.218,-0.667) --
          (5.217,-0.652) -- (5.217,-0.638) -- (5.216,-0.623) --
          (5.215,-0.609) -- (5.215,-0.594) -- (5.214,-0.580) --
          (5.214,-0.565) -- (5.213,-0.551) -- (5.213,-0.536) --
          (5.212,-0.522) -- (5.212,-0.507) -- (5.211,-0.493) --
          (5.211,-0.478) -- (5.211,-0.464) -- (5.210,-0.449) --
          (5.210,-0.435) -- (5.210,-0.420) -- (5.209,-0.406) --
          (5.209,-0.391) -- (5.209,-0.377) -- (5.209,-0.362) --
          (5.208,-0.348) -- (5.208,-0.333) -- (5.208,-0.319) --
          (5.208,-0.304) -- (5.208,-0.290) -- (5.208,-0.275) --
          (5.207,-0.261) -- (5.207,-0.246) -- (5.207,-0.232) --
          (5.207,-0.217) -- (5.207,-0.203) -- (5.207,-0.188) --
          (5.207,-0.174) -- (5.207,-0.159) -- (5.207,-0.145) --
          (5.206,-0.130) -- (5.206,-0.116) -- (5.206,-0.101) --
          (5.206,-0.087) -- (5.206,-0.072) -- (5.206,-0.058) --
          (5.206,-0.043) -- (5.206,-0.029) -- (5.206,-0.014) -- (5.206,0.000)
          -- (5.206,0.015) -- (5.206,0.029) -- (5.206,0.044) -- (5.206,0.058)
          -- (5.206,0.073) -- (5.206,0.087) -- (5.206,0.102) -- (5.206,0.116)
          -- (5.206,0.131) -- (5.207,0.145) -- (5.207,0.160) -- (5.207,0.174)
          -- (5.207,0.189) -- (5.207,0.203) -- (5.207,0.217) -- (5.207,0.232)
          -- (5.207,0.246) -- (5.207,0.261) -- (5.208,0.275) -- (5.208,0.290)
          -- (5.208,0.304) -- (5.208,0.319) -- (5.208,0.333) -- (5.208,0.348)
          -- (5.209,0.362) -- (5.209,0.377) -- (5.209,0.392) -- (5.209,0.406)
          -- (5.210,0.421) -- (5.210,0.435) -- (5.210,0.450) -- (5.211,0.464)
          -- (5.211,0.479) -- (5.211,0.493) -- (5.212,0.508) -- (5.212,0.522)
          -- (5.213,0.537) -- (5.213,0.551) -- (5.214,0.566) -- (5.214,0.580)
          -- (5.215,0.595) -- (5.215,0.609) -- (5.216,0.624) -- (5.217,0.638)
          -- (5.217,0.653) -- (5.218,0.667) -- (5.219,0.682) -- (5.220,0.696)
          -- (5.221,0.711) -- (5.222,0.725) -- (5.223,0.740) -- (5.224,0.754)
          -- (5.225,0.769) -- (5.226,0.783) -- (5.227,0.798) -- (5.229,0.812)
          -- (5.230,0.827) -- (5.231,0.841) -- (5.233,0.856) -- (5.235,0.870)
          -- (5.236,0.885) -- (5.238,0.899) -- (5.240,0.914) -- (5.242,0.928)
          -- (5.244,0.943) -- (5.247,0.957) -- (5.249,0.971) -- (5.251,0.986)
          -- (5.254,1.000) -- (5.257,1.015) -- (5.260,1.029) -- (5.263,1.044)
          -- (5.266,1.058) -- (5.269,1.073) -- (5.273,1.087) -- (5.276,1.102)
          -- (5.280,1.117) -- (5.284,1.131) -- (5.288,1.145) -- (5.292,1.160)
          -- (5.297,1.175) -- (5.301,1.189) -- (5.306,1.204) -- (5.311,1.218)
          -- (5.316,1.233) -- (5.321,1.247) -- (5.327,1.262) -- (5.332,1.276)
          -- (5.338,1.291) -- (5.344,1.305) -- (5.349,1.320) -- (5.355,1.334)
          -- (5.362,1.349) -- (5.368,1.363) -- (5.374,1.378) -- (5.380,1.392)
          -- (5.386,1.407) -- (5.393,1.421) -- (5.399,1.436) -- (5.405,1.450)
          -- (5.411,1.465) -- (5.417,1.479) -- (5.423,1.494) -- (5.429,1.508)
          -- (5.434,1.523) -- (5.440,1.537) -- (5.445,1.552) -- (5.449,1.566)
          -- (5.454,1.581) -- (5.457,1.595) -- (5.461,1.610) -- (5.464,1.624)
          -- (5.467,1.639) -- (5.469,1.653) -- (5.470,1.668) -- (5.471,1.682)
          -- (5.472,1.697) -- (5.472,1.711) -- (5.471,1.725) -- (5.470,1.740)
          -- (5.468,1.754) -- (5.465,1.769) -- (5.462,1.783) -- (5.458,1.798)
          -- (5.453,1.812) -- (5.448,1.827) -- (5.443,1.841) -- (5.437,1.856)
          -- (5.430,1.871) -- (5.423,1.885) -- (5.415,1.899) -- (5.408,1.914)
          -- (5.399,1.929) -- (5.391,1.943) -- (5.382,1.958) -- (5.373,1.972)
          -- (5.364,1.987) -- (5.355,2.001) -- (5.346,2.016) -- (5.337,2.030)
          -- (5.328,2.045) -- (5.319,2.059) -- (5.311,2.074) -- (5.302,2.088)
          -- (5.294,2.103) -- (5.287,2.117) -- (5.279,2.132) -- (5.272,2.146)
          -- (5.265,2.161) -- (5.259,2.175) -- (5.253,2.190) -- (5.247,2.204)
          -- (5.242,2.219) -- (5.237,2.233) -- (5.233,2.248) -- (5.229,2.262)
          -- (5.225,2.277) -- (5.222,2.291) -- (5.219,2.306) -- (5.216,2.320)
          -- (5.214,2.335) -- (5.212,2.349) -- (5.210,2.364) -- (5.209,2.378)
          -- (5.207,2.393) -- (5.206,2.407) -- (5.205,2.422) -- (5.204,2.436)
          -- (5.203,2.450) -- (5.203,2.465) -- (5.202,2.479) -- (5.202,2.494)
          -- (5.201,2.508) -- (5.201,2.523) -- (5.201,2.538) -- (5.201,2.552)
          -- (5.201,2.567) -- cycle;

          \\fill[violet!35] (5.201,-2.189) -- (5.201,-2.175) --
          (5.202,-2.160) -- (5.202,-2.146) -- (5.202,-2.131) --
          (5.202,-2.117) -- (5.203,-2.103) -- (5.203,-2.088) --
          (5.203,-2.073) -- (5.204,-2.059) -- (5.204,-2.044) --
          (5.204,-2.030) -- (5.205,-2.015) -- (5.205,-2.001) --
          (5.206,-1.986) -- (5.206,-1.972) -- (5.207,-1.957) --
          (5.207,-1.943) -- (5.208,-1.928) -- (5.208,-1.914) --
          (5.209,-1.899) -- (5.209,-1.885) -- (5.210,-1.871) --
          (5.211,-1.856) -- (5.211,-1.841) -- (5.212,-1.827) --
          (5.212,-1.812) -- (5.213,-1.798) -- (5.213,-1.783) --
          (5.214,-1.769) -- (5.214,-1.754) -- (5.215,-1.740) --
          (5.215,-1.725) -- (5.216,-1.711) -- (5.216,-1.696) --
          (5.217,-1.682) -- (5.217,-1.667) -- (5.218,-1.653) --
          (5.218,-1.638) -- (5.218,-1.624) -- (5.218,-1.609) --
          (5.219,-1.595) -- (5.219,-1.580) -- (5.219,-1.566) --
          (5.219,-1.551) -- (5.219,-1.537) -- (5.219,-1.522) --
          (5.219,-1.508) -- (5.219,-1.493) -- (5.219,-1.479) --
          (5.219,-1.464) -- (5.219,-1.450) -- (5.219,-1.435) --
          (5.219,-1.421) -- (5.219,-1.406) -- (5.219,-1.392) --
          (5.218,-1.377) -- (5.218,-1.363) -- (5.218,-1.348) --
          (5.218,-1.334) -- (5.217,-1.319) -- (5.217,-1.305) --
          (5.217,-1.290) -- (5.217,-1.276) -- (5.216,-1.261) --
          (5.216,-1.247) -- (5.216,-1.232) -- (5.215,-1.218) --
          (5.215,-1.203) -- (5.215,-1.189) -- (5.214,-1.174) --
          (5.214,-1.160) -- (5.214,-1.145) -- (5.213,-1.131) --
          (5.213,-1.116) -- (5.213,-1.102) -- (5.212,-1.087) --
          (5.212,-1.073) -- (5.212,-1.058) -- (5.211,-1.044) --
          (5.211,-1.029) -- (5.211,-1.015) -- (5.210,-1.000) --
          (5.210,-0.986) -- (5.210,-0.971) -- (5.210,-0.957) --
          (5.209,-0.942) -- (5.209,-0.928) -- (5.209,-0.913) --
          (5.209,-0.899) -- (5.208,-0.884) -- (5.208,-0.870) --
          (5.208,-0.855) -- (5.208,-0.841) -- (5.207,-0.826) --
          (5.207,-0.812) -- (5.207,-0.797) -- (5.207,-0.783) --
          (5.207,-0.768) -- (5.207,-0.754) -- (5.206,-0.739) --
          (5.206,-0.725) -- (5.206,-0.711) -- (5.206,-0.696) --
          (5.206,-0.681) -- (5.206,-0.667) -- (5.206,-0.652) --
          (5.205,-0.638) -- (5.205,-0.623) -- (5.205,-0.609) --
          (5.205,-0.594) -- (5.205,-0.580) -- (5.205,-0.565) --
          (5.205,-0.551) -- (5.205,-0.536) -- (5.205,-0.522) --
          (5.205,-0.507) -- (5.204,-0.493) -- (5.204,-0.478) --
          (5.204,-0.464) -- (5.204,-0.449) -- (5.204,-0.435) --
          (5.204,-0.420) -- (5.204,-0.406) -- (5.204,-0.391) --
          (5.204,-0.377) -- (5.204,-0.362) -- (5.204,-0.348) --
          (5.204,-0.333) -- (5.204,-0.319) -- (5.204,-0.304) --
          (5.204,-0.290) -- (5.204,-0.275) -- (5.204,-0.261) --
          (5.204,-0.246) -- (5.204,-0.232) -- (5.204,-0.217) --
          (5.204,-0.203) -- (5.204,-0.188) -- (5.204,-0.174) --
          (5.204,-0.159) -- (5.204,-0.145) -- (5.204,-0.130) --
          (5.204,-0.116) -- (5.204,-0.101) -- (5.204,-0.087) --
          (5.204,-0.072) -- (5.204,-0.058) -- (5.204,-0.043) --
          (5.204,-0.029) -- (5.204,-0.014) -- (5.204,0.000) -- (5.204,0.015)
          -- (5.204,0.029) -- (5.205,0.044) -- (5.205,0.058) -- (5.205,0.073)
          -- (5.205,0.087) -- (5.205,0.102) -- (5.205,0.116) -- (5.205,0.131)
          -- (5.205,0.145) -- (5.205,0.160) -- (5.206,0.174) -- (5.206,0.189)
          -- (5.206,0.203) -- (5.206,0.217) -- (5.206,0.232) -- (5.206,0.246)
          -- (5.207,0.261) -- (5.207,0.275) -- (5.207,0.290) -- (5.207,0.304)
          -- (5.208,0.319) -- (5.208,0.333) -- (5.208,0.348) -- (5.208,0.362)
          -- (5.209,0.377) -- (5.209,0.392) -- (5.210,0.406) -- (5.210,0.421)
          -- (5.210,0.435) -- (5.211,0.450) -- (5.211,0.464) -- (5.212,0.479)
          -- (5.212,0.493) -- (5.213,0.508) -- (5.214,0.522) -- (5.214,0.537)
          -- (5.215,0.551) -- (5.216,0.566) -- (5.217,0.580) -- (5.217,0.595)
          -- (5.218,0.609) -- (5.219,0.624) -- (5.220,0.638) -- (5.221,0.653)
          -- (5.222,0.667) -- (5.224,0.682) -- (5.225,0.696) -- (5.226,0.711)
          -- (5.228,0.725) -- (5.230,0.740) -- (5.231,0.754) -- (5.233,0.769)
          -- (5.235,0.783) -- (5.237,0.798) -- (5.239,0.812) -- (5.241,0.827)
          -- (5.244,0.841) -- (5.246,0.856) -- (5.249,0.870) -- (5.252,0.885)
          -- (5.255,0.899) -- (5.259,0.914) -- (5.262,0.928) -- (5.266,0.943)
          -- (5.270,0.957) -- (5.274,0.971) -- (5.278,0.986) -- (5.283,1.000)
          -- (5.288,1.015) -- (5.293,1.029) -- (5.298,1.044) -- (5.304,1.058)
          -- (5.310,1.073) -- (5.317,1.087) -- (5.323,1.102) -- (5.331,1.117)
          -- (5.338,1.131) -- (5.346,1.145) -- (5.354,1.160) -- (5.362,1.175)
          -- (5.371,1.189) -- (5.380,1.204) -- (5.390,1.218) -- (5.400,1.233)
          -- (5.410,1.247) -- (5.421,1.262) -- (5.432,1.276) -- (5.444,1.291)
          -- (5.455,1.305) -- (5.467,1.320) -- (5.480,1.334) -- (5.492,1.349)
          -- (5.505,1.363) -- (5.518,1.378) -- (5.532,1.392) -- (5.545,1.407)
          -- (5.559,1.421) -- (5.572,1.436) -- (5.586,1.450) -- (5.599,1.465)
          -- (5.612,1.479) -- (5.626,1.494) -- (5.638,1.508) -- (5.651,1.523)
          -- (5.663,1.537) -- (5.675,1.552) -- (5.686,1.566) -- (5.696,1.581)
          -- (5.706,1.595) -- (5.715,1.610) -- (5.723,1.624) -- (5.730,1.639)
          -- (5.736,1.653) -- (5.742,1.668) -- (5.745,1.682) -- (5.748,1.697)
          -- (5.750,1.711) -- (5.750,1.725) -- (5.749,1.740) -- (5.746,1.754)
          -- (5.743,1.769) -- (5.737,1.783) -- (5.731,1.798) -- (5.723,1.812)
          -- (5.714,1.827) -- (5.703,1.841) -- (5.692,1.856) -- (5.679,1.871)
          -- (5.665,1.885) -- (5.651,1.899) -- (5.635,1.914) -- (5.618,1.929)
          -- (5.601,1.943) -- (5.584,1.958) -- (5.565,1.972) -- (5.547,1.987)
          -- (5.528,2.001) -- (5.509,2.016) -- (5.491,2.030) -- (5.472,2.045)
          -- (5.453,2.059) -- (5.435,2.074) -- (5.418,2.088) -- (5.401,2.103)
          -- (5.384,2.117) -- (5.368,2.132) -- (5.353,2.146) -- (5.339,2.161)
          -- (5.325,2.175) -- (5.312,2.190) -- (5.301,2.204) -- (5.290,2.219)
          -- (5.279,2.233) -- (5.270,2.248) -- (5.261,2.262) -- (5.254,2.277)
          -- (5.247,2.291) -- (5.240,2.306) -- (5.235,2.320) -- (5.230,2.335)
          -- (5.225,2.349) -- (5.221,2.364) -- (5.218,2.378) -- (5.215,2.393)
          -- (5.213,2.407) -- (5.210,2.422) -- (5.209,2.436) -- (5.207,2.450)
          -- (5.206,2.465) -- (5.205,2.479) -- (5.204,2.494) -- (5.203,2.508)
          -- (5.202,2.523) -- (5.202,2.538) -- (5.201,2.552) -- (5.201,2.567)
          -- cycle;

          \\node[violet!80!black,font=\\scriptsize,align=left,anchor=west] at
          (5.35,2.5) {law of $\\Phi_T$\\\\given $\\Phi_t$};

          \\node[black!55,font=\\scriptsize,anchor=west] at (5.35,-2.05)
          {$\\nu_T$};

          \\draw[green!50!black,thick] (0,0) -- (5.200,1.696);

          \\draw[green!50!black,thick] (0,0) -- (5.200,-1.696);

          \\draw[blue!70!black] (0.000,0.000) -- (0.026,0.052) --
          (0.052,-0.003) -- (0.078,-0.078) -- (0.104,0.054) -- (0.130,0.003)
          -- (0.156,0.095) -- (0.182,0.065) -- (0.208,0.143) -- (0.234,0.043)
          -- (0.260,-0.159) -- (0.286,-0.279) -- (0.312,-0.321) --
          (0.338,-0.294) -- (0.364,-0.485) -- (0.390,-0.450) --
          (0.416,-0.566) -- (0.442,-0.666) -- (0.468,-0.617) --
          (0.494,-0.525) -- (0.520,-0.546) -- (0.546,-0.503) --
          (0.572,-0.556) -- (0.598,-0.540) -- (0.624,-0.439) --
          (0.650,-0.311) -- (0.676,-0.312) -- (0.702,-0.235) --
          (0.728,-0.169) -- (0.754,-0.096) -- (0.780,-0.010) -- (0.806,0.001)
          -- (0.832,-0.046) -- (0.858,-0.072) -- (0.884,0.106) --
          (0.910,-0.018) -- (0.936,-0.085) -- (0.962,-0.096) -- (0.988,0.075)
          -- (1.014,0.055) -- (1.040,0.177) -- (1.066,0.154) -- (1.092,0.201)
          -- (1.118,0.136) -- (1.144,0.133) -- (1.170,0.295) -- (1.196,0.372)
          -- (1.222,0.453) -- (1.248,0.525) -- (1.274,0.487) -- (1.300,0.247)
          -- (1.326,0.189) -- (1.352,0.394) -- (1.378,0.319) -- (1.404,0.399)
          -- (1.430,0.401) -- (1.456,0.520) -- (1.482,0.469) -- (1.508,0.426)
          -- (1.534,0.412) -- (1.560,0.527) -- (1.586,0.520) -- (1.612,0.397)
          -- (1.638,0.371) -- (1.664,0.308) -- (1.690,0.267) -- (1.716,0.292)
          -- (1.742,0.220) -- (1.768,0.445) -- (1.794,0.450) -- (1.820,0.398)
          -- (1.846,0.468) -- (1.872,0.437) -- (1.898,0.329) -- (1.924,0.439)
          -- (1.950,0.540) -- (1.976,0.496) -- (2.002,0.486) -- (2.028,0.467)
          -- (2.054,0.339) -- (2.080,0.440) -- (2.106,0.418) -- (2.132,0.445)
          -- (2.158,0.414) -- (2.184,0.369) -- (2.210,0.403) -- (2.236,0.411)
          -- (2.262,0.392) -- (2.288,0.253) -- (2.314,0.343) -- (2.340,0.392)
          -- (2.366,0.353) -- (2.392,0.407) -- (2.418,0.428) -- (2.444,0.592)
          -- (2.470,0.609) -- (2.496,0.608) -- (2.522,0.698) -- (2.548,0.824)
          -- (2.574,0.981) -- (2.600,1.087);

          \\draw[blue!70!black,opacity=0.25] (2.600,1.087) -- (2.626,1.077)
          -- (2.652,1.194) -- (2.678,1.455) -- (2.704,1.268) -- (2.730,1.271)
          -- (2.756,1.348) -- (2.782,1.094) -- (2.808,1.134) -- (2.834,1.136)
          -- (2.860,0.982) -- (2.886,0.917) -- (2.912,0.852) -- (2.938,0.895)
          -- (2.964,0.793) -- (2.990,0.674) -- (3.016,0.649) -- (3.042,0.717)
          -- (3.068,0.780) -- (3.094,0.856) -- (3.120,0.767) -- (3.146,0.805)
          -- (3.172,0.823) -- (3.198,0.903) -- (3.224,0.760) -- (3.250,0.584)
          -- (3.276,0.328) -- (3.302,0.445) -- (3.328,0.310) -- (3.354,0.421)
          -- (3.380,0.438) -- (3.406,0.517) -- (3.432,0.590) -- (3.458,0.569)
          -- (3.484,0.617) -- (3.510,0.532) -- (3.536,0.453) -- (3.562,0.701)
          -- (3.588,0.741) -- (3.614,0.597) -- (3.640,0.628) -- (3.666,0.574)
          -- (3.692,0.559) -- (3.718,0.687) -- (3.744,0.815) -- (3.770,0.764)
          -- (3.796,0.764) -- (3.822,0.918) -- (3.848,0.992) -- (3.874,1.007)
          -- (3.900,1.014) -- (3.926,1.181) -- (3.952,1.153) -- (3.978,1.240)
          -- (4.004,1.300) -- (4.030,1.212) -- (4.056,1.288) -- (4.082,1.261)
          -- (4.108,1.270) -- (4.134,1.155) -- (4.160,1.175) -- (4.186,1.179)
          -- (4.212,1.284) -- (4.238,1.272) -- (4.264,1.217) -- (4.290,1.415)
          -- (4.316,1.399) -- (4.342,1.364) -- (4.368,1.211) -- (4.394,1.255)
          -- (4.420,1.462) -- (4.446,1.538) -- (4.472,1.683) -- (4.498,1.560)
          -- (4.524,1.688) -- (4.550,1.707) -- (4.576,1.626) -- (4.602,1.650)
          -- (4.628,1.704) -- (4.654,1.697) -- (4.680,1.724) -- (4.706,1.914)
          -- (4.732,1.978) -- (4.758,1.902) -- (4.784,1.676) -- (4.810,1.807)
          -- (4.836,1.690) -- (4.862,1.631) -- (4.888,1.859) -- (4.914,1.835)
          -- (4.940,1.764) -- (4.966,1.685) -- (4.992,1.628) -- (5.018,1.803)
          -- (5.044,1.865) -- (5.070,1.911) -- (5.096,1.875) -- (5.122,1.694)
          -- (5.148,1.755) -- (5.174,1.861) -- (5.200,1.696);

          \\fill[violet] (5.200,1.696) circle (1.6pt); \\fill[violet]
          (5.200,-1.696) circle (1.6pt);

          \\fill[green!50!black] (2.600,0.848) circle (1.4pt);
          \\fill[green!50!black] (2.600,-0.848) circle (1.4pt);

          \\draw[red!70!black,-<gtr>,thin,opacity=0.97] (2.600,0.848) --
          (2.600,1.087);

          \\draw[red!70!black,-<gtr>,thin,opacity=0.38] (2.600,-0.848) --
          (2.600,1.087);

          \\draw[red!80!black,-<gtr>,very thick] (2.600,0.746) --
          (2.600,1.087);

          \\fill[blue!70!black] (2.600,1.087) circle (1.8pt);

          \\node[blue!70!black,anchor=south east] at (2.600,1.087)
          {$\\Phi_t\\,$};

          \\node[violet!80!black,font=\\scriptsize,anchor=north west] at
          (5.200,1.696) {$y_+$};

          \\node[violet!80!black,font=\\scriptsize,anchor=south west] at
          (5.200,-1.696) {$y_-$};

          \\node[red!80!black,font=\\scriptsize,anchor=west,align=left] at
          (2.756,-0.348) {$(T-t)\\,\\omega_t$\\\\$=\\mathbb E[\\Phi_t-\\tfrac
          tT \\Phi_T\\,\|\\,\\Phi_t]$};

          \\node[font=\\footnotesize] at (2.60,2.75) {(a) the observer at
          resolution $t$};

          \\end{scope}

          \\begin{scope}[xshift=10.6cm,yshift=-2.3cm]

          \\draw[black!45] (-2.25,0) -- (2.25,0);

          \\draw[black!45] (-2.25,0) -- (-2.25,2.95);

          \\foreach \\v/\\l in {0/{10^{-4}},1.24/{10^{-2}},2.48/{1}}{\\draw[black!45]
          (-2.25,\\v) -- (-2.33,\\v) node[left,black!70,font=\\scriptsize]{$\\l$};}

          \\foreach \\u in {-4,-2,0,2,4}{\\draw[black!45] (\\u*0.52,0) --
          (\\u*0.52,-0.08) node[below,black!70,font=\\scriptsize]{$\\u$};}

          \\draw[black,dashed] (-2.184,0.000) -- (-2.158,0.000) --
          (-2.132,0.000) -- (-2.106,0.024) -- (-2.080,0.078) --
          (-2.054,0.132) -- (-2.028,0.185) -- (-2.002,0.237) --
          (-1.976,0.288) -- (-1.950,0.339) -- (-1.924,0.389) --
          (-1.898,0.439) -- (-1.872,0.488) -- (-1.846,0.536) --
          (-1.820,0.583) -- (-1.794,0.630) -- (-1.768,0.676) --
          (-1.742,0.722) -- (-1.716,0.766) -- (-1.690,0.811) --
          (-1.664,0.854) -- (-1.638,0.897) -- (-1.612,0.939) --
          (-1.586,0.980) -- (-1.560,1.021) -- (-1.534,1.061) --
          (-1.508,1.100) -- (-1.482,1.139) -- (-1.456,1.177) --
          (-1.430,1.214) -- (-1.404,1.251) -- (-1.378,1.287) --
          (-1.352,1.322) -- (-1.326,1.357) -- (-1.300,1.391) --
          (-1.274,1.424) -- (-1.248,1.457) -- (-1.222,1.489) --
          (-1.196,1.520) -- (-1.170,1.551) -- (-1.144,1.581) --
          (-1.118,1.610) -- (-1.092,1.639) -- (-1.066,1.667) --
          (-1.040,1.694) -- (-1.014,1.721) -- (-0.988,1.747) --
          (-0.962,1.772) -- (-0.936,1.796) -- (-0.910,1.820) --
          (-0.884,1.843) -- (-0.858,1.866) -- (-0.832,1.888) --
          (-0.806,1.909) -- (-0.780,1.930) -- (-0.754,1.950) --
          (-0.728,1.969) -- (-0.702,1.987) -- (-0.676,2.005) --
          (-0.650,2.022) -- (-0.624,2.039) -- (-0.598,2.055) --
          (-0.572,2.070) -- (-0.546,2.084) -- (-0.520,2.098) --
          (-0.494,2.111) -- (-0.468,2.124) -- (-0.442,2.135) --
          (-0.416,2.146) -- (-0.390,2.157) -- (-0.364,2.167) --
          (-0.338,2.176) -- (-0.312,2.184) -- (-0.286,2.192) --
          (-0.260,2.199) -- (-0.234,2.205) -- (-0.208,2.211) --
          (-0.182,2.216) -- (-0.156,2.220) -- (-0.130,2.224) --
          (-0.104,2.227) -- (-0.078,2.230) -- (-0.052,2.231) --
          (-0.026,2.232) -- (0.000,2.233) -- (0.026,2.232) -- (0.052,2.231)
          -- (0.078,2.230) -- (0.104,2.227) -- (0.130,2.224) -- (0.156,2.220)
          -- (0.182,2.216) -- (0.208,2.211) -- (0.234,2.205) -- (0.260,2.199)
          -- (0.286,2.192) -- (0.312,2.184) -- (0.338,2.176) -- (0.364,2.167)
          -- (0.390,2.157) -- (0.416,2.146) -- (0.442,2.135) -- (0.468,2.124)
          -- (0.494,2.111) -- (0.520,2.098) -- (0.546,2.084) -- (0.572,2.070)
          -- (0.598,2.055) -- (0.624,2.039) -- (0.650,2.022) -- (0.676,2.005)
          -- (0.702,1.987) -- (0.728,1.969) -- (0.754,1.950) -- (0.780,1.930)
          -- (0.806,1.909) -- (0.832,1.888) -- (0.858,1.866) -- (0.884,1.843)
          -- (0.910,1.820) -- (0.936,1.796) -- (0.962,1.772) -- (0.988,1.747)
          -- (1.014,1.721) -- (1.040,1.694) -- (1.066,1.667) -- (1.092,1.639)
          -- (1.118,1.610) -- (1.144,1.581) -- (1.170,1.551) -- (1.196,1.520)
          -- (1.222,1.489) -- (1.248,1.457) -- (1.274,1.424) -- (1.300,1.391)
          -- (1.326,1.357) -- (1.352,1.322) -- (1.378,1.287) -- (1.404,1.251)
          -- (1.430,1.214) -- (1.456,1.177) -- (1.482,1.139) -- (1.508,1.100)
          -- (1.534,1.061) -- (1.560,1.021) -- (1.586,0.980) -- (1.612,0.939)
          -- (1.638,0.897) -- (1.664,0.854) -- (1.690,0.811) -- (1.716,0.766)
          -- (1.742,0.722) -- (1.768,0.676) -- (1.794,0.630) -- (1.820,0.583)
          -- (1.846,0.536) -- (1.872,0.488) -- (1.898,0.439) -- (1.924,0.389)
          -- (1.950,0.339) -- (1.976,0.288) -- (2.002,0.237) -- (2.028,0.185)
          -- (2.054,0.132) -- (2.080,0.078) -- (2.106,0.024) -- (2.132,0.000)
          -- (2.158,0.000) -- (2.184,0.000);

          \\draw[blue!70!black,thick] (-2.158,0.000) -- (-2.106,0.000) --
          (-2.054,0.000) -- (-2.002,0.081) -- (-1.950,0.049) --
          (-1.898,0.187) -- (-1.846,0.272) -- (-1.794,0.366) --
          (-1.742,0.457) -- (-1.690,0.555) -- (-1.638,0.646) --
          (-1.586,0.756) -- (-1.534,0.822) -- (-1.482,0.893) --
          (-1.430,0.997) -- (-1.378,1.066) -- (-1.326,1.129) --
          (-1.274,1.209) -- (-1.222,1.266) -- (-1.170,1.330) --
          (-1.118,1.401) -- (-1.066,1.462) -- (-1.014,1.514) --
          (-0.962,1.572) -- (-0.910,1.620) -- (-0.858,1.673) --
          (-0.806,1.724) -- (-0.754,1.766) -- (-0.702,1.802) --
          (-0.650,1.847) -- (-0.598,1.881) -- (-0.546,1.920) --
          (-0.494,1.950) -- (-0.442,1.985) -- (-0.390,2.014) --
          (-0.338,2.046) -- (-0.286,2.076) -- (-0.234,2.109) --
          (-0.182,2.144) -- (-0.130,2.187) -- (-0.078,2.246) --
          (-0.026,2.668) -- (0.026,2.668) -- (0.078,2.246) -- (0.130,2.188)
          -- (0.182,2.144) -- (0.234,2.105) -- (0.286,2.077) -- (0.338,2.043)
          -- (0.390,2.018) -- (0.442,1.983) -- (0.494,1.952) -- (0.546,1.921)
          -- (0.598,1.884) -- (0.650,1.848) -- (0.702,1.803) -- (0.754,1.763)
          -- (0.806,1.719) -- (0.858,1.672) -- (0.910,1.622) -- (0.962,1.576)
          -- (1.014,1.518) -- (1.066,1.464) -- (1.118,1.408) -- (1.170,1.330)
          -- (1.222,1.276) -- (1.274,1.199) -- (1.326,1.133) -- (1.378,1.061)
          -- (1.430,0.982) -- (1.482,0.915) -- (1.534,0.833) -- (1.586,0.731)
          -- (1.638,0.672) -- (1.690,0.581) -- (1.742,0.469) -- (1.794,0.399)
          -- (1.846,0.247) -- (1.898,0.151) -- (1.950,0.060) -- (2.002,0.000)
          -- (2.054,0.000) -- (2.106,0.000) -- (2.158,0.000);

          \\node[font=\\scriptsize,anchor=west,align=left] at (0.75,2.75)
          {--\\,--\\, $N(0,\\mathcal R_t)$: law of\\\\\\phantom{--\\,--\\,}
          $(T-t)^{-1}(\\Phi_t-\\tfrac tT \\Phi_T)$};

          \\node[blue!70!black,font=\\scriptsize,anchor=west] at (0.75,2.05)
          {law of $\\omega_t$};

          \\node[font=\\footnotesize] at (0,5.05) {(b) densities, logarithmic
          scale};

          \\end{scope}

          \\end{tikzpicture}

          \\end{document}
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      </with>

      <small|<with|color|dark grey|<math|T=1>, <math|t=1/2>, <math|\<nu\><rsub|T><around|(|\<mathd\>y|)>\<propto\>\<mathe\><rsup|-2*y<rsup|4>+6*y<rsup|2>>N<around|(|0,1|)><around|(|\<mathd\>y|)>> (grey). Green: bridges to the endpoints <math|y<rsub|\<pm\>>>; violet: posterior. The peak at <math|0> in (b): a fold of <math|\<Phi\><rsub|t>\<mapsto\>\<omega\><rsub|t>> between the wells.>>

      General field: <math|<math-up|Cov><around|(|\<omega\><rsub|t>|)>=\<cal-R\><rsub|t>-S<rsub|t><rsup|-1>*J<rsub|t>*\<bbb-E\><around|[|<math-up|Cov><rsub|t>|]>*J<rsub|t>*S<rsub|t><rsup|-1>=K*J<rsub|t>+J<rsub|t>*\<bbb-E\><around|[|\<mathD\><rsup|2>V<rsub|t><around|(|\<Phi\><rsub|t>|)>|]>*J<rsub|t>>:
      the free bridge noise <strong|minus what the observer cannot resolve>;
      Nelson's mass <math|m<rsup|2>+\<bbb-E\>*V<rprime|''>>, at resolution
      <math|t>, and finite.
    </with>
  </hidden>|<\hidden>
    <mytitle|The noise is sub-Gaussian>

    <\with|font-size|0.9>
      <math|\<omega\><rsub|t>> is minus the score of the law of
      <math|\<Phi\><rsub|t>> (density <math|p<rsub|t>>), seen through <math|J<rsub|t>>: <math|\<omega\><rsub|t>=-J<rsub|t>*\<mathD\>log
      p<rsub|t><around|(|\<Phi\><rsub|t>|)>>. Given <math|\<Phi\><rsub|T>>,
      the interacting field is still the Gaussian bridge:

      <\equation*>
        \<Phi\><rsub|t>=J<rsub|t>*\<Phi\><rsub|T>+\<zeta\><rsub|t>,<space|1em>\<zeta\><rsub|t>\<sim\>N<around*|(|0,Q<rsub|t>*<around|(|1-J<rsub|t>|)>|)><space|1em>\<Rightarrow\><space|1em>\<omega\><rsub|t>=\<bbb-E\><around*|[|S<rsub|t><rsup|-1>\<zeta\><rsub|t><mid|\|>\<Phi\><rsub|t>|]>.
      </equation*>

      <\proposition*>
        <math|\<omega\><rsub|t>> is dominated in convex order by
        <math|N<around|(|0,\<cal-R\><rsub|t>|)>>, with
        <math|\<cal-R\><rsub|t>=S<rsub|t><rsup|-1>-Q<rsub|T><rsup|-1>> the
        <strong|Wetterich regulator>; in particular

        <\equation*>
          \<bbb-E\>*\<mathe\><rsup|u<around|\<langle\>|h,\<omega\><rsub|t>|\<rangle\>>>\<leqslant\>exp
          <around*|(|<frac|u<rsup|2>|2>*<around|\<langle\>|h,\<cal-R\><rsub|t>*h|\<rangle\>>|)>,
        </equation*>

        uniformly in the interaction, with no convexity (conditional Jensen).
      </proposition*>

      <\small>
        <\itemize>
          <item>Heat profile, finite spatial cutoff: <math|\<cal-R\><rsub|t>=A/<around|(|\<mathe\><rsup|A/t<rsup|2>>-1|)>\<leqslant\>t<rsup|2>>
          if <math|T=\<infty\>>; <math|\<cal-R\><rsub|t>\<leqslant\><frac|4|3>*t<rsup|2>> if <math|T\<geqslant\>2*t>.

          <item>With <math|\<mathd\>F<rsub|s>=\<Upsilon\><rsub|s>*\<mathd\>W<rsub|s>> (Lecture<nbsp>4) and Itô:
          <math|\<bbb-E\><big|int><rsub|0><rsup|t><around|\<\|\|\>|\<Upsilon\><rsub|s><rsup|\<ast\>>*J<rsub|s>*h|\<\|\|\>><rsup|2>*\<mathd\>s\<leqslant\>2*<around|(|1+\<mathe\><rsup|-1>|)>*t<rsup|2><around|\<\|\|\>|h|\<\|\|\>><rsup|2>>:
          a smoothed derivative of the force; no terminal force needed.

          <item><math|\<Phi\><rsup|4><rsub|3>>: the noise in
          <math|A*\<Phi\><rsub|t>=A*Q<rsub|t>*F<rsub|t>+\<omega\><rsub|t>> is
          controlled a priori; what remains is coercivity of the renormalised
          force.
        </itemize>
      </small>
    </with>
  </hidden>|<\hidden>
    <mytitle|And the force?>

    <\with|font-size|0.9>
      <math|F<rsub|t>=-\<mathD\>V<rsub|t><around|(|\<Phi\><rsub|t>|)>=-\<mathD\>\<Gamma\><rsub|t><around|(|\<psi\><rsub|t>|)>>.
      Read it from the measure at resolution <math|t>,
      <math|\<nu\><rsub|t>\<propto\>\<mathe\><rsup|-V<rsub|t>>\<mu\><rsup|Q<rsub|t>>>.

      <\proposition*>
        If <math|V<rsub|T>> is even and <math|\<Gamma\><rsub|t>> is convex,

        <\equation*>
          \<bbb-E\>*\<mathe\><rsup|u<around|\<langle\>|h,F<rsub|t>|\<rangle\>>>\<leqslant\>exp
          <around*|(|<frac|u<rsup|2>|2>*<around|\<langle\>|h,S<rsub|t><rsup|-1>*h|\<rangle\>>|)>.
        </equation*>
      </proposition*>

      <\enumerate>
        <item><strong|Always:> <math|\<mathD\><rsup|2>V<rsub|t>\<leqslant\>S<rsub|t><rsup|-1>>
        (<math|<math-up|Cov><rsub|t>\<geqslant\>0>), then Cameron\UMartin:

        <\equation*>
          \<bbb-E\>*\<mathe\><rsup|u<around|\<langle\>|h,F<rsub|t>|\<rangle\>>>\<leqslant\>\<mathe\><rsup|<frac|u<rsup|2>|2>*<around|\<langle\>|h,<around|(|S<rsub|t><rsup|-1>-Q<rsub|t><rsup|-1>|)>*h|\<rangle\>>>*<space|0.17em>\<bbb-E\>*\<mathe\><rsup|u<around|\<langle\>|Q<rsub|t><rsup|-1>*h,\<Phi\><rsub|t>|\<rangle\>>>.
        </equation*>

        <item><strong|Convexity:> <math|\<Gamma\><rsub|t>> convex
        <math|\<Rightarrow\>> <math|V<rsub|t>> convex
        <math|\<Rightarrow\>> (Prékopa) <math|\<Phi\><rsub|t>> no wider than
        the free field: <math|\<bbb-E\>*\<mathe\><rsup|<around|\<langle\>|k,\<Phi\><rsub|t>|\<rangle\>>>\<leqslant\>\<mathe\><rsup|<frac|1|2>*<around|\<langle\>|k,Q<rsub|t>*k|\<rangle\>>>>.
      </enumerate>

      <\small>
        <\itemize>
          <item>Without convexity: <math|\<mathe\><rsup|<frac|u<rsup|2>|2>*<around|\<langle\>|h,\<cal-R\><rsub|t>*h|\<rangle\>>>*\<bbb-E\><rsub|\<nu\><rsub|T>>\<mathe\><rsup|u<around|\<langle\>|K*h,\<Phi\><rsub|T>|\<rangle\>>>>:
          the noise plus the terminal field seen through <math|K>.

          <item><math|\<mathD\>\<Gamma\><rsup|<math-up|EAA>><rsub|t><around|(|\<psi\><rsub|t>|)>=\<omega\><rsub|t>>,
          sub-Gaussian for free. <math|\<Gamma\><rsup|<math-up|EAA>><rsub|t>>
          convex <math|\<Leftrightarrow\>> <math|\<nu\><rsub|t>> log-concave:
          not enough (negative mass: <math|<math-up|Var>F<rsub|t>\<sim\>1/\<varepsilon\>>).
        </itemize>
      </small>
    </with>
  </hidden>|<\hidden>
    <mytitle|How rough is the noise?>

    Spectral densities, heat profile, <math|T=\<infty\>> (<math|X<rsub|t>>: the free field, <math|V=0>),
    <math|A=<around|\||k|\|><rsup|2>+m<rsup|2>>:

    <\equation*>
      <tabular*|<tformat|<cwith|1|-1|1|-1|cell-lsep|0.8em>|<cwith|1|-1|1|-1|cell-rsep|0.8em>|<table|<row|<cell|<text|white
      noise>>|<cell|<text|GFF>>|<cell|<text|free noise
      >A*X<rsub|t>>|<cell|<text|bound >\<cal-R\><rsub|t>>>|<row|<cell|1>|<cell|A<rsup|-1>>|<cell|A*\<mathe\><rsup|-A/t<rsup|2>>>|<cell|<frac|A|\<mathe\><rsup|A/t<rsup|2>>-1>>>>>>
    </equation*>

    <\small>
      <\itemize>
        <item><math|\<cal-R\><rsub|t>\<leqslant\>t<rsup|2>*\<mathe\><rsup|-A/2*t<rsup|2>>>:
        at resolution <math|t>, <math|\<omega\><rsub|t>\<lesssim\>t\<times\>>
        white noise smoothed at scale <math|1/t>, so
        <math|\<omega\><rsub|t><around|(|x|)>=O<around|(|t<rsup|1+d/2>|)>>.

        <item>At scale <math|1/t> this is the size of regularity
        <math|-1-d/2> (<math|-5/2> in <math|d=3>): one derivative rougher
        than white noise, like the free noise <math|A<rsup|1/2>*\<xi\>>,
        <strong|whatever the interaction>.

        <item>Not a limit distribution: on a fixed <math|h> only
        <math|<around|\<langle\>|h,\<omega\><rsub|t>|\<rangle\>>=O<around|(|t|)>>, while the free noise stays bounded.

        <item>On resolved modes (<math|<around|\||k|\|>\<ll\>t>) the bound exceeds the free noise
        by <math|t<rsup|2>/A>; it is attained by a pinning potential
        <math|\<kappa\><around|\<\|\|\>|\<varphi\>|\<\|\|\>><rsup|2>/2>, <math|\<kappa\>\<to\>\<infty\>>.

        <item><strong|Pathwise> (convex domination of the norm, Doob): a.s.
        for <math|t\<geqslant\>1> (<math|t\<leqslant\>T/2> if <math|T\<less\>\<infty\>>), at finite spatial cutoff, with a random constant <math|<wide|C|~>>,

        <\equation*>
          sup<rsub|x><around|(|1+<around|\||x|\|>|)><rsup|-\<delta\>><around|\||\<omega\><rsub|t><around|(|x|)>|\|>\<leqslant\><wide|C|~>*t<rsup|1+d/2><sqrt|log
          <around|(|2+t|)>>.
        </equation*>
      </itemize>
    </small>
  </hidden>|<\hidden>
    <mytitle|The noise as a stochastic observable>

    <\itemize>
      <item><math|<around|(|\<omega\><rsub|t>|)><rsub|t\<leqslant\>T>> is a
      <strong|martingale>, with terminal value Nelson's Euclidean noise
      <math|K*\<Phi\><rsub|T>+\<mathD\>V<rsub|T><around|(|\<Phi\><rsub|T>|)>>.

      <item>At each resolution a <strong|smooth random field>; every
      <math|<around|\<langle\>|h,\<omega\><rsub|t>|\<rangle\>>> is
      <strong|sub-Gaussian> with variance
      <math|\<leqslant\><around|\<langle\>|h,\<cal-R\><rsub|t>*h|\<rangle\>>>,
      uniformly in the interaction and the cutoff.

      <item>Paired with functionals of the coarse field it is the
      <strong|transpose of the smoothed derivative>
      <math|\<mathD\><rsub|J<rsub|t>h>> (DS at every scale).

      <item>Expected <strong|local> in the sense of Lecture<nbsp>3: germs
      <math|A*\<Phi\><rsub|t><around|(|x|)>+n<rsub|t><around|(|\<Phi\><rsub|t><around|(|x|)>|)>>,
      <math|n<rsub|t>> a renormalised cube; local iff the force is. For
      <math|\<Phi\><rsup|4><rsub|3>>: local but not closed (no terminal representative).

      

      <item>For <math|\<Phi\><rsup|4><rsub|3>>,
      <math|A*\<varphi\>+V<rprime|'><around|(|\<varphi\>|)>> is not expected
      to be a random variable (Nelson; Albeverio\ULiang\UZegarlinski), but
      its conditional expectations are, consistently:
      <math|\<omega\><rsub|s>=\<bbb-E\><rsub|s>*\<omega\><rsub|t>>.
    </itemize>

    <\myframed>
      The Euclidean field equation of <math|\<Phi\><rsup|4><rsub|3>> has a
      well-defined sub-Gaussian noise: an <strong|observable without a
      terminal representative>.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Infinite volume>

    \;

    On <math|\<bbb-T\><rsup|d><rsub|L>>, the free energy <math|V<rsub|0><around|(|0|)>> and
    <math|V<rsub|t>> are <strong|extensive> (<math|\<sim\>L<rsup|d>>). Three
    <strong|intensive> objects survive:

    <\enumerate-roman>
      <item><with|font-series|bold|the force.> For
      <math|V<rsub|T>=<big|int>v<rsub|T><around|(|\<varphi\>|)>>, on
      <math|\<bbb-R\><rsup|d>>:

      <\equation*>
        \<Phi\><rsub|t>=<big|int><rsub|0><rsup|t><wide|Q|\<dot\>><rsub|s>*F<rsub|s><space|0.17em>\<mathd\>s+<big|int><rsub|0><rsup|t><wide|Q|\<dot\>><rsub|s><rsup|1/2>\<mathd\>W<rsub|s>,<space|2em>F<rsub|t><around|(|x|)>=-\<bbb-E\><rsub|t><around*|[|v<rsub|T><rprime|'><around|(|\<Phi\><rsub|T><around|(|x|)>|)>|]>.
      </equation*>

      Everything is local; infinite-volume limits should solve this.
      Uniqueness of solutions <math|\<leftrightarrow\>> uniqueness of the
      theory; non-uniqueness expected at phase transitions
      (<math|\<Phi\><rsup|4><rsub|2>>, deep double well: Glimm\UJaffe\USpencer 1975).

      <item><with|font-series|bold|the Burgers flow> of the force (Lecture<nbsp>4);

      <item><with|font-series|bold|differences of free energies>
      <math|\<cal-W\><rsub|T><rsup|<around|(|L|)>><around|(|f|)>=-log<big|int>\<mathe\><rsup|-f>\<mathd\>\<nu\><rsub|T><rsup|<around|(|L|)>>>, <math|f>
      local: bounded as <math|L\<to\>\<infty\>>; basis of the infinite-volume
      analysis of <math|\<Phi\><rsup|4><rsub|2>> and <math|exp
      <around|(|\<Phi\>|)><rsub|2>>.
    </enumerate-roman>

    Estimates must be <strong|local> (weighted norms, volume-independent
    constants): the analogue of \Pcoming down from infinity\Q for the parabolic dynamics.
    <small|<with|color|dark grey|Barashkov\UGubinelli, PMP 2023>>
  </hidden>|<\hidden>
    <mytitle|Differences of free energies>

    <\with|font-size|0.9>

      Perturb the optimal drift <math|u<rsup|\<ast\>>> of the Boué\UDupuis formula (Lecture<nbsp>4), <math|X<rsup|u<rsup|\<ast\>>>=\<Phi\>>: <math|u=u<rsup|\<ast\>>+w>,
      <math|X<rsup|u>=\<Phi\>+I<around|(|w|)>>,
      <math|I<rsub|t><around|(|w|)>=<big|int><rsub|0><rsup|t><wide|Q|\<dot\>><rsub|s><rsup|1/2>*w<rsub|s>\<mathd\>s>.
      Euler\ULagrange gives the cross term

      <\equation*>
        \<bbb-E\><big|int><around|\<langle\>|u<rsup|\<ast\>>,w|\<rangle\>>=-\<bbb-E\><around|\<langle\>|\<mathD\>V<rsub|T><around|(|\<Phi\><rsub|T>|)>,I<rsub|T><around|(|w|)>|\<rangle\>>.
      </equation*>

      <\proposition*>
        <dueto|Relative variational principle><with|font-shape|right|(formal)> For <math|f> bounded below,

        <\equation*>
          -log <big|int>\<mathe\><rsup|-f>*\<mathd\>\<nu\><rsub|T>=inf<rsub|w>
          \<bbb-E\><around*|[|f<around|(|\<Phi\><rsub|T>+I<rsub|T><around|(|w|)>|)>+B<rsub|V<rsub|T>><around|(|\<Phi\><rsub|T>,I<rsub|T><around|(|w|)>|)>+<frac|1|2>*<big|int><rsub|0><rsup|T><around|\<\|\|\>|w|\<\|\|\>><rsup|2>|]>,
        </equation*>

        over adapted <math|w> with <math|\<bbb-E\><big|int><around|\<\|\|\>|w<rsub|s>|\<\|\|\>><rsup|2>\<mathd\>s\<less\>\<infty\>>: the two extensive infima cancel exactly.
      </proposition*>

      <math|B<rsub|V><around|(|\<phi\>,h|)>=V<around|(|\<phi\>+h|)>-V<around|(|\<phi\>|)>-<around|\<langle\>|\<mathD\>V<around|(|\<phi\>|)>,h|\<rangle\>>>:
      the <strong|first-order Taylor remainder>.

      <\itemize>
        <item>Minimiser: <math|\<Phi\><rsup|f>=\<Phi\>+I<around|(|w<rsup|\<ast\>>|)>> solves
        the FBSDE with terminal potential <math|V<rsub|T>+f>, a perturbation of
        <math|\<Phi\>>:

        <\equation*>
          F<rsup|f><rsub|s>-F<rsub|s>=-\<bbb-E\><rsub|s><around|[|\<mathD\>f<around|(|\<Phi\><rsup|f><rsub|T>|)>+\<mathD\>V<rsub|T><around|(|\<Phi\><rsup|f><rsub|T>|)>-\<mathD\>V<rsub|T><around|(|\<Phi\><rsub|T>|)>|]>.
        </equation*>
      </itemize>
    </with>
  </hidden>|<\hidden>
    <mytitle|Infinite-volume measures>

    Local interaction <math|V<rsub|T>=<big|int>v<rsub|T><around|(|\<varphi\><around|(|x|)>|)>*\<mathd\>x>:

    <\equation*>
      B<rsub|V<rsub|T>><around|(|\<phi\>,h|)>=<big|int>b<rsub|T><around|(|\<phi\><around|(|x|)>,h<around|(|x|)>|)>*\<mathd\>x,<space|2em>b<rsub|T><around|(|a,h|)>=v<rsub|T><around|(|a+h|)>-v<rsub|T><around|(|a|)>-v<rsub|T><rprime|'><around|(|a|)>*h.
    </equation*>

    <\itemize>
      <item><math|b<rsub|T><around|(|a,0|)>=0>: only the region where the
      perturbation acts counts. Every term is <strong|intensive>: no free
      energy, only <math|\<Phi\>> and its perturbation.

      <item><math|f> local <math|\<Rightarrow\>> the optimal <math|w> should
      decay away from its support, at the rate of decay of correlations.

      <item>Several solutions of the FBSDE (phase transitions)
      <math|\<Rightarrow\>> several measures, each with its own relative variational principle.

      <item><math|\<Phi\><rsup|4><rsub|3>>: the remainder term carries the
      counterterms; the change of variables of Barashkov\UGubinelli (Duke 2020) is still
      needed.
    </itemize>

    <\myframed>
      <strong|Formal statement> (at fixed UV cutoff <math|T>). An infinite-volume measure is the law of
      <math|\<Phi\><rsub|T>> for a solution of the infinite-volume FBSDE
      satisfying the relative variational principle for all local <math|f>.
    </myframed>
  </hidden>|<\hidden>
    <vspace*|2.5cm>

    \;

    <cqft-part-title|Appendices: additional material>

    <vspace|0.3cm>

    <with|color|dark grey|Lecture<nbsp>5>

    \;
  </hidden>|<\hidden>
    <mytitle|A semigroup of renormalisation maps>

    \;

    <math|\<frak-R\><rsub|S><around|(|V|)>=-log
    <around|(|\<mu\><rsup|S>\<ast\>\<mathe\><rsup|-V>|)>>. Since
    <math|\<mu\><rsup|S>\<ast\>\<mu\><rsup|S<rprime|'>>=\<mu\><rsup|S+S<rprime|'>>>,

    <\equation*>
      \<frak-R\><rsub|S>\<circ\>\<frak-R\><rsub|S<rprime|'>>=\<frak-R\><rsub|S+S<rprime|'>>=\<frak-R\><rsub|S<rprime|'>>\<circ\>\<frak-R\><rsub|S>,<space|2em>V<rsub|t>=\<frak-R\><rsub|S<rsub|t>><around|(|V<rsub|T>|)>:
    </equation*>

    an orbit of an <strong|abelian> semigroup indexed by the cone of positive
    covariances, nonlinear on <math|V>, linear on <math|\<mathe\><rsup|-V>>.
    Generators:

    <\equation*>
      \<cal-P\><rsub|B><around|(|V|)>=<tfrac|1|2><math-up|Tr><around|(|B*\<mathD\><rsup|2>*V|)>-<tfrac|1|2><around|\<langle\>|\<mathD\>V,B*\<mathD\>V|\<rangle\>>,<space|2em>\<partial\><rsub|t>*V<rsub|t>=-\<cal-P\><rsub|<wide|Q|\<dot\>><rsub|t>><around|(|V<rsub|t>|)>.
    </equation*>

    <\proposition*>
      <dueto|Flatness><with|font-shape|right|(formal)>
      <math|<around|[|\<cal-P\><rsub|B>,\<cal-P\><rsub|B<rprime|'>>|]>=0>.
      <math|V<rsub|t>> depends on the profile only through
      <math|S<rsub|t>>; <math|V<rsub|0><around|(|0|)>> and
      <math|\<nu\><rsub|T>> only through <math|Q<rsub|T>>.
    </proposition*>

    Different profiles with the same endpoints are different
    <strong|coordinate systems> on the same RG trajectory: they change
    <math|\<Phi\>> and intermediate observables, not the terminal theory.
    <strong|No holonomy.>
  </hidden>|<\hidden>
    <mytitle|Perturbation theory and counterterms>

    <\with|font-size|0.9>
      \;

      Gaussian convolution <strong|lowers the Wick ordering>:
      <math|V<rsub|t>=<big|int><around|[|<frac|\<lambda\>|12>:\<phi\><rsup|4>:<rsub|t>+\<gamma\><rsub|T>:\<phi\><rsup|2>:<rsub|t>|]>+O<around|(|\<lambda\><rsup|2>|)>>
      (<math|\<gamma\><rsub|T>=O<around|(|\<lambda\><rsup|2>|)>>, kept for later use;
      <math|:\<phi\><rsup|p>:<rsub|t>=H<rsub|p><around|(|\<phi\>;c<rsub|t>|)>>, <math|c<rsub|t>=Q<rsub|t><around|(|x,x|)>>):
      dual to <math|<big|int>H<rsub|4>*<around|(|X<rsub|t>;c<rsub|t>|)>> being an observable (Lecture<nbsp>3). At second order, <math|S=S<rsub|t>>:

      <\proposition*>
        <with|font-shape|right|(formal)> To second order in
        <math|\<lambda\>>:

        <\equation*>
          V<rsub|t>=<big|int><around*|[|<frac|\<lambda\>|12>:\<phi\><rsup|4>:<rsub|t>+\<gamma\><rsub|T>:\<phi\><rsup|2>:<rsub|t>|]>-<frac|\<lambda\><rsup|2>|288>*<big|sum><rsub|k=1><rsup|4><binom|4|k><rsup|2>k!<big|iint>S*<around|(|x,y|)><rsup|k><space|0.17em>:\<phi\><rsup|4-k>:<rsub|t><around|(|x|)><space|0.17em>:\<phi\><rsup|4-k>:<rsub|t><around|(|y|)>+O<around|(|\<lambda\><rsup|3>|)>.
        </equation*>
      </proposition*>

      <math|T\<to\>\<infty\>> at fixed <math|t>, <math|d=3>,
      <math|G<around|(|z|)>\<sim\><around|(|4*\<pi\><around|\||z|\|>|)><rsup|-1>> (the kernel of <math|A<rsup|-1>>):

      <\itemize>
        <item><math|k=1,2>: finite, nonlocal.

        <item><math|k=3> (<strong|sunset>):
        <math|S<rsup|3>\<sim\><around|\||z|\|><rsup|-3>>; local divergence
        <math|-<frac|\<lambda\><rsup|2>|3><around|(|<big|int>S<rsup|3>|)>*<big|int>:\<phi\><rsup|2>:<rsub|t>>
        (+ finite), cancelled by <math|\<gamma\><rsub|T>=<frac|\<lambda\><rsup|2>|3>*<big|int>Q<rsub|T><rsup|3>=<frac|\<lambda\><rsup|2>|48*\<pi\><rsup|2>>*log
        T+O<around|(|1|)>>.

        <item><math|k=4>: vacuum energy <math|\<sim\>T>; cancels in
        <math|\<nu\><rsub|T>>.
      </itemize>

      Renormalisation <math|=> choosing <math|V<rsub|T>> so that
      <math|V<rsub|t>> at <strong|fixed> resolution converges.
      <math|\<nu\><rsub|t>\<ll\>\<mu\><rsup|Q<rsub|t>>> expected for
      <math|t\<less\>\<infty\>>; the terminal <math|\<Phi\><rsup|4><rsub|3>>
      is singular w.r.t. the GFF.
    </with>
  </hidden>|<\hidden>
    <mytitle|Second order in graphs>

    \;

    \;

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      \\node[vtx] (a) at (0,0) {}; \\node[vtx] (b) at (1.3,0) {};

      \\draw[prop] (a) to[bend left=35] (b); \\draw[prop] (a) to[bend
      right=35] (b);

      \\foreach \\t in {135,225}{\\draw[thick] (a)--++(\\t:.5);}

      \\foreach \\t in {45,315}{\\draw[thick] (b)--++(\\t:.5);}

      \\node at (.65,-1.05) {$:\\!\\varphi^4\\!:$};\\node[font=\\scriptsize]
      at (.65,-1.5) {bubble, finite in $d=3$};

      \\end{scope}

      % sunset

      \\begin{scope}[xshift=6.4cm]

      \\node[vtx] (a) at (0,0) {}; \\node[vtx] (b) at (1.3,0) {};

      \\draw[prop] (a) to[bend left=45] (b); \\draw[prop] (a)--(b);
      \\draw[prop] (a) to[bend right=45] (b);

      \\draw[thick] (a)--++(180:.5); \\draw[thick] (b)--++(0:.5);

      \\node at (.65,-1.05) {$:\\!\\varphi^2\\!:$};\\node[font=\\scriptsize]
      at (.65,-1.5) {sunset, $\\sim\\lambda^2\\log T$};

      \\end{scope}

      % vacuum

      \\begin{scope}[xshift=9.4cm]

      \\node[vtx] (a) at (0,0) {}; \\node[vtx] (b) at (1.3,0) {};

      \\draw[prop] (a) to[bend left=60] (b); \\draw[prop] (a) to[bend
      left=20] (b); \\draw[prop] (a) to[bend right=20] (b); \\draw[prop] (a)
      to[bend right=60] (b);

      \\node at (.65,-1.05) {$1$};\\node[font=\\scriptsize] at (.65,-1.5)
      {vacuum, a constant};

      \\end{scope}

      \\end{tikzpicture}

      \\end{document}

      \;
    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    <small|<with|color|dark grey|<math|k> lines <math|S> between two
    Wick-ordered vertices. <math|k=1,\<ldots\>,4> from left to right; Wick ordering removes
    self-loops. Only the sunset needs a further counterterm,
    <math|\<gamma\><rsub|T>>.>>
  </hidden>|<\hidden>
    <mytitle|Stochastic localisation: what and why>

    Eldan 2013: write a measure as a mixture of simpler ones, by a
    measure-valued martingale. Here

    <\equation*>
      \<mu\><rsub|t><rsup|\<phi\>><around|(|\<mathd\>y|)>\<propto\>\<mathe\><rsup|<around|\<langle\>|S<rsub|t><rsup|-1>\<phi\>,y|\<rangle\>>-<frac|1|2><around|\<langle\>|y,\<cal-R\><rsub|t>*y|\<rangle\>>>*\<nu\><rsub|T><around|(|\<mathd\>y|)>,<space|2em>\<Phi\><rsub|t>=J<rsub|t>\<Phi\><rsub|T>+\<zeta\><rsub|t>:
    </equation*>

    <math|\<Phi\><rsub|t>> is a noisy <strong|observation> of
    <math|\<Phi\><rsub|T>>, with precision <math|\<cal-R\><rsub|t>> gained,
    and <math|\<mu\><rsub|t>> the <strong|posterior>. Eldan's
    identity-control case: <math|\<cal-R\><rsub|t>=t\<cdot\>1>.

    <\itemize>
      <item><strong|Martingale:> Itô splits variance and entropy into
      revealed and hidden parts.

      <item><strong|Simpler pieces:> <math|\<nu\><rsub|T>> log-concave
      <math|\<Rightarrow\>> <math|\<mu\><rsub|t>> is
      <math|\<cal-R\><rsub|t>>-uniformly log-concave.

      <item><strong|One quantity:> the loss is governed by
      <math|Cov<rsub|t>>, with explicit dynamics. An inequality reduces to:
      <em|the localisation is not too fast>.

      <item><strong|Algorithms:> sampling
      (El<nbsp>Alaoui\UMontanari\USellke), denoising diffusions (Montanari);
      <math|\<omega\><rsub|t>> is minus the score, seen through
      <math|J<rsub|t>>.
    </itemize>
  </hidden>|<\hidden>
    <mytitle|Entropy along the localisation>

    <\with|font-size|0.9>
      <math|\<rho\>\<geqslant\>0>, <math|\<bbb-E\><rsub|\<nu\><rsub|T>>*\<rho\>=1>,
      <math|M<rsub|t>=\<bbb-E\><rsub|t>*\<rho\><around|(|\<Phi\><rsub|T>|)>=G<rsub|t><around|(|\<Phi\><rsub|t>|)>>,
      <math|G<rsub|t><around|(|\<phi\>|)>=<big|int>\<rho\>*\<mathd\>\<mu\><rsub|t><rsup|\<phi\>>>.

      <\itemize>
        <item><math|M<rsub|0>=1>, <math|M<rsub|T>=\<rho\><around|(|\<Phi\><rsub|T>|)>>,
        and <math|\<mathd\>M<rsub|t>=<around|\<langle\>|\<mathD\>G<rsub|t><around|(|\<Phi\><rsub|t>|)>,<wide|Q|\<dot\>><rsub|t><rsup|1/2>*\<mathd\>W<rsub|t>|\<rangle\>>>.

        <item>Itô for <math|M*log M>:

        <\equation*>
          Ent<rsub|\<nu\><rsub|T>><around|(|\<rho\>|)>=<frac|1|2>*\<bbb-E\><big|int><rsub|0><rsup|T><frac|\<mathd\><around|\<langle\>|M|\<rangle\>><rsub|t>|M<rsub|t>>=2*\<bbb-E\><big|int><rsub|0><rsup|T><around*|\<\|\|\>|<wide|Q|\<dot\>><rsub|t><rsup|1/2>*\<mathD\><sqrt|G<rsub|t>><around|(|\<Phi\><rsub|t>|)>|\<\|\|\>><rsup|2>*\<mathd\>t.
        </equation*>

        <item>Stopping at <math|t>: <strong|revealed + hidden> entropy,

        <\equation*>
          Ent<rsub|\<nu\><rsub|T>><around|(|\<rho\>|)>=\<bbb-E\><around|[|M<rsub|t>log M<rsub|t>|]>+\<bbb-E\>*Ent<rsub|\<mu\><rsub|t><rsup|\<Phi\><rsub|t>>><around|(|\<rho\>|)>.
        </equation*>

        <item>The increments see the conditional law:
        <math|\<mathD\>G<rsub|t><around|(|\<phi\>|)>=S<rsub|t><rsup|-1>*Cov<rsub|\<mu\><rsub|t><rsup|\<phi\>>><around|(|\<rho\>,y|)>>.

        <item><strong|Entropic stability> (Chen\UEldan; <math|<wide|\<cal-R\>|\<dot\>><rsub|r>=S<rsub|r><rsup|-1><wide|Q|\<dot\>><rsub|r>S<rsub|r><rsup|-1>>): if
        <math|<wide|\<cal-R\>|\<dot\>><rsub|r>*Cov<rsub|\<mu\><rsub|r><rsup|\<phi\>>>*<wide|\<cal-R\>|\<dot\>><rsub|r>\<leqslant\>\<alpha\><rsub|r>*<wide|\<cal-R\>|\<dot\>><rsub|r>>
        for all <math|\<phi\>> and <math|r\<leqslant\>t>, then
        <math|Ent<rsub|\<nu\><rsub|T>><around|(|\<rho\>|)>\<leqslant\>\<mathe\><rsup|<big|int><rsub|0><rsup|t>\<alpha\><rsub|r>*\<mathd\>r>*\<bbb-E\>*Ent<rsub|\<mu\><rsub|t><rsup|\<Phi\><rsub|t>>><around|(|\<rho\>|)>>:
        the low resolutions must not hide too much.
      </itemize>

      <small|<with|color|dark grey|Bauerschmidt\UBodineau\UDagallier, Probab.
      Surveys 2024: their fluctuation measure is our stochastic localisation
      (scale reversed).>>
    </with>
  </hidden>|<\hidden>
    <mytitle|Multiscale Bakry\UÉmery>

    <\with|font-size|0.9>
      <\theorem*>
        <dueto|Bauerschmidt\UBodineau><with|font-shape|right|(formal)> If for
        all <math|t,\<phi\>>, with <math|\<kappa\><rsub|t>\<in\>\<bbb-R\>>
        (possibly negative),

        <\equation*>
          <wide|Q|\<dot\>><rsub|t>*\<mathD\><rsup|2>*V<rsub|t><around|(|\<phi\>|)>*<wide|Q|\<dot\>><rsub|t>+<frac|1|2>*<wide|Q|\<ddot\>><rsub|t>\<geqslant\>\<kappa\><rsub|t>*<wide|Q|\<dot\>><rsub|t>,
        </equation*>

        then <math|Ent<rsub|\<nu\><rsub|T>><around|(|\<rho\>|)>\<leqslant\><frac|2|\<varrho\>>*\<bbb-E\><rsub|\<nu\><rsub|T>><around|\<\|\|\>|<wide|Q|\<dot\>><rsub|T><rsup|1/2>*\<mathD\><sqrt|\<rho\>>|\<\|\|\>><rsup|2>>,
        <math|<frac|1|\<varrho\>>=<big|int><rsub|0><rsup|T>\<mathe\><rsup|-2*<big|int><rsub|t><rsup|T>\<kappa\><rsub|r>*\<mathd\>r>*\<mathd\>t>.
      </theorem*>

      <\itemize>
        <item>Gradient bound at each scale + entropy formula. Gaussian check:
        <math|V<rsub|T>=0>, <math|Q<rsub|t>=t*Q<rsub|T>/T>,
        <math|\<kappa\>=0> <math|\<Rightarrow\>> Gaussian LSI.

        <item>Convexity of the <strong|effective> potentials, not of
        <math|V<rsub|T>>.

        <item>By <math|\<mathD\><rsup|2>*V<rsub|t>=S<rsub|t><rsup|-1>-S<rsub|t><rsup|-1>*Cov<rsub|t>*S<rsub|t><rsup|-1>>,
        for <math|<wide|Q|\<dot\>><rsub|t>> invertible the hypothesis is
        equivalent to the bound, uniform in <math|\<phi\>>,

        <\equation*>
          Cov<rsub|t>\<leqslant\>S<rsub|t>-S<rsub|t>*<wide|Q|\<dot\>><rsub|t><rsup|-1>*<around*|(|\<kappa\><rsub|t>*<wide|Q|\<dot\>><rsub|t>-<frac|1|2>*<wide|Q|\<ddot\>><rsub|t>|)>*<wide|Q|\<dot\>><rsub|t><rsup|-1>*S<rsub|t>:
        </equation*>

        at most <math|S<rsub|t>> iff <math|\<kappa\><rsub|t>*<wide|Q|\<dot\>><rsub|t>\<geqslant\><frac|1|2>*<wide|Q|\<ddot\>><rsub|t>>
        (linear profile: <math|\<kappa\><rsub|t>\<geqslant\>0>).
      </itemize>
    </with>
  </hidden>|<\hidden>
    <mytitle|Poincaré from the localisation>

    <\proposition*>
      If <math|<wide|\<cal-R\>|\<dot\>><rsub|r>*Cov<rsub|\<mu\><rsub|r><rsup|\<phi\>>>*<wide|\<cal-R\>|\<dot\>><rsub|r>\<leqslant\>\<alpha\><rsub|r>*<wide|\<cal-R\>|\<dot\>><rsub|r>>
      for all <math|\<phi\>>, <math|r\<leqslant\>t>, then

      <\equation*>
        Var<rsub|\<nu\><rsub|T>><around|(|f|)>\<leqslant\>\<mathe\><rsup|<big|int><rsub|0><rsup|t>\<alpha\>>*\<bbb-E\>*Var<rsub|\<mu\><rsub|t>><around|(|f|)>,
      </equation*>

      and, for <math|\<nu\><rsub|T>> log-concave,
      <math|Var<rsub|\<nu\><rsub|T>><around|(|f|)>\<leqslant\>\<mathe\><rsup|<big|int><rsub|0><rsup|t>\<alpha\>>*\<bbb-E\><around|\<langle\>|\<mathD\>f,\<cal-R\><rsub|t><rsup|-1>*\<mathD\>f|\<rangle\>>>.
    </proposition*>

    <strong|Proof.> <math|v<around|(|r|)>=\<bbb-E\>*Var<rsub|\<mu\><rsub|r>><around|(|f|)>>
    has <math|v<rprime|'>=-\<bbb-E\><around|\<langle\>|\<chi\>,<wide|\<cal-R\>|\<dot\>>*\<chi\>|\<rangle\>>>,
    <math|\<chi\>=Cov<around|(|f,y|)>>; Cauchy\USchwarz gives
    <math|<around|\<langle\>|\<chi\>,<wide|\<cal-R\>|\<dot\>>*\<chi\>|\<rangle\>>\<leqslant\>\<alpha\>*Var<around|(|f|)>>.
    Then Brascamp\ULieb for <math|\<mu\><rsub|t>>.

    <\itemize>
      <item>Gaussian, one mode: <math|\<mathe\><rsup|<big|int>\<alpha\>>=Q<rsub|T>/S<rsub|t>>;
      equality for linear <math|f>.

      <item><strong|KLS conjecture>: isotropic log-concave on
      <math|\<bbb-R\><rsup|n>>, <math|\<cal-R\><rsub|t>=t>:
      <math|Var<around|(|f|)>\<leqslant\>t<rsup|-1>*\<mathe\><rsup|<big|int>\<alpha\>>*\<bbb-E\><around|\||\<mathD\>f|\|><rsup|2>>.
      Keep <math|<around|\<\|\|\>|Cov<rsub|r>|\<\|\|\>>> bounded as long as
      possible (only with high probability). Eldan 2013, Chen 2021,
      Klartag\ULehec 2022, Klartag 2023: Poincaré constant
      <math|O<around|(|log n|)>>.
    </itemize>
  </hidden>|<\hidden>
    <mytitle|Two views of one martingale>

    <\with|font-size|0.9>
      <\padded-center>
        <with|font-size|0.9|<data-table|<tformat|<table|<row|<cell|these
        notes>|<cell|Bauerschmidt\UBodineau>|<cell|localisation>>|<row|<cell|<math|\<nu\><rsub|t>>>|<cell|renormalised
        measure>|<cell|law of the tilt>>|<row|<cell|<math|\<mu\><rsub|t><rsup|\<phi\>>>>|<cell|fluctuation
        measure>|<cell|tilted measure>>|<row|<cell|<math|G<rsub|t>>>|<cell|Polchinski
        semigroup>|<cell|martingale <math|<big|int>\<rho\>*\<mathd\>\<mu\><rsub|t>>>>|<row|<cell|<math|\<mathD\><rsup|2>*V<rsub|t>>>|<cell|Hessian
        of the potential>|<cell|<math|S<rsub|t><rsup|-1>-S<rsub|t><rsup|-1>Cov<rsub|t>S<rsub|t><rsup|-1>>: covariance of the tilted measure>>>>>>
      </padded-center>

      <\itemize>
        <item><strong|Bridge:> <math|\<mathD\><rsup|2>*V<rsub|t>=S<rsub|t><rsup|-1>-S<rsub|t><rsup|-1>*Cov<rsub|t>*S<rsub|t><rsup|-1>>:
        convex potentials <math|=> bounded conditional covariance.

        <item><strong|Bauerschmidt\UBodineau:> Bakry\UÉmery for the
        Kolmogorov equation, curvature <math|<wide|Q|\<dot\>>*\<mathD\><rsup|2>*V<rsub|t>*<wide|Q|\<dot\>>+<frac|1|2>*<wide|Q|\<ddot\>>>:
        gradient bound at each scale, LSI at the terminal scale, far from
        log-concavity. Uniform in <math|\<phi\>>: suits the RG (sine-Gordon,
        <math|\<beta\>\<less\>6*\<pi\>>).

        <item><strong|Localisation:> bound what is lost on
        <math|<around|[|0,t|]>>, then use a known inequality for
        <math|\<mu\><rsub|t>>. Moments of the conditional law, possibly only
        with high probability (KLS).
      </itemize>

      <small|<with|color|dark grey|Bauerschmidt\UBodineau, CPAM 2021;
      Bauerschmidt\UBodineau\UDagallier, Probab. Surveys 2024; Eldan, GAFA
      2013; Chen\UEldan, FOCS 2022.>>
    </with>
  </hidden>|<\hidden>
    <mytitle|Summary>

    \;

    <\itemize>
      <item>The <strong|Euclidean field equation>
      <math|A*\<varphi\>+V<rprime|'><around|(|\<varphi\>|)>=\<omega\>>:
      Nelson's Euclidean noise is the transpose of the derivative, fixed by
      the law of the field, not Gaussian.

      <item><strong|Field equation> at every resolution: its noise is a
      conditional expectation both of Nelson's noise and of a Gaussian bridge
      noise, hence sub-Gaussian whatever the interaction; for the forecast
      the force enters without smoothing.

      <item><strong|DS, pathwise>: an identity between stochastic
      observables, with no terminal representative needed.
    </itemize>

    \;

    <\myframed>
      <strong|Next>

      Lectures<nbsp>6\U7: a construction of <math|\<Phi\><rsup|4><rsub|3>> on
      the torus via the FBSDE.
    </myframed>
  </hidden>|<\hidden>
    <mytitle|Exercises and reading>

    <\with|font-size|0.9>
      \;

      <wide-tabular|<tformat|<cwith|1|1|1|-1|cell-valign|t>|<cwith|1|1|1|-1|cell-hyphen|t>|<cwith|1|1|1|-1|cell-lsep|0em>|<cwith|1|1|1|-1|cell-rsep|1em>|<cwith|1|1|1|1|cell-width|0.5par>|<cwith|1|1|1|1|cell-hmode|exact>|<cwith|1|1|2|2|cell-width|0.46par>|<cwith|1|1|2|2|cell-hmode|exact>|<table|<row|<\cell>
        <with|font-series|bold|Exercises> (numbering of the notes)

        <\itemize>
          <item>9.1 Doob transform; 9.2 relative variational principle; 9.3
          DS from the dynamics; 9.4 Gaussian check; 9.5 sub-Gaussian noise; 9.6 the noise of a
          two-point endpoint; 9.7 contact terms.

          <item>A.1 Commuting flows; A.2 the sunset; A.3 power counting.

          <item>B.1 Entropy along the localisation; B.2 Poincaré from the
          localisation.
        </itemize>
      </cell>|<\cell>
        <with|font-series|bold|Reading>

        <\itemize>
          <item>Nelson, <em|Remarks on Markov field equations>, 1975.

          <item>Albeverio\UYoshida, J. Funct. Anal. 2002;
          Albeverio\ULiang\UZegarlinski 2006.

          <item>Gubinelli\UHofmanová, <em|A PDE construction of the Euclidean
          <math|\<Phi\><rsup|4><rsub|3>> quantum field theory>, CMP 2021.

          <item>Bauerschmidt\UBodineau, CPAM 2021 (log-Sobolev via the
          Polchinski flow); Eldan, GAFA 2013.

          <item>Barashkov\UGubinelli, Duke 2020; PMP 2023 (infinite volume).
        </itemize>
      </cell>>>>>
    </with>
  </hidden>>
</body>

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