Discussion of Hugh Thomas' talk

A probabilistic perspective on ol ' theory

im Tom KLOSE Amphituhedron DayNikolas TAPIA 8 April 2022

I

• STOCHASTIC QUANT ISAT 1 ON goes back to [ PARISI & WU,1981] i

consider Euclidean QFT measure as stationary measure of a stochastic process

* static picture : old f) i = exp f- 54 ) ) d¥ ( formal !)

• functional derivative* Langevin dynamics :

'¥4 = - 81¥ t 3.

10=0/4 , xD ,t E fictitious time space

- time white noise

x space - timeEL3A ,

a) 54yd] = 8ft - s) f'd 'Cx -y)

• scold := f I ↳ fowl ' t £ local ' d '" x1124

* off = DOI - ol'

t 3 ④ equation on IR+ × Id, loco, -1=0)

2

• Fixed - point argument for : to =P * C- f't 3)

* of'"=O → of

"'= P * 3 = :P

- 1 - K

- K-3+2 - k

NB : 3e C-'¥

as. ⇒ P e C a. s .(Sikander )

* ol '" = P * C - P'

t 3)

ill - posed : C'

xcpacf.gl to f.ge Gap is well- def IFF atp > 0.

• Cure : consider abstract FP problem [ Hairy] :

OI = pf - I ' t )

-2k 1-3k y -k

Iterate ! -1- K O n

- Eft ) = - t 9411 - 2 - 29th t CD9H ,X7

3• ITE e S

'

( IR"' ) w/ * ITE i= 3

,I smoothed STWN

* TIE := P * IT't

* IT'II :-. IT4 1TE

Ex.: IT

'=P * IT

'= P * IT' I

'

=P * I E regularised at scale e

• Problem in regularity structures : counter terms to ensure ex.

③

^ aof it :-&:P. .

ft - D) be = - ok + 3. t§g, God k Id* [ Brunel

,Chandra

, Cheryrev , Hairer]with C

,led := ELTIEA

-

I 101]A

Negative twisted antipodes. " Extraction - contraction procedure"

F [ Bruised, Haireritambohti) & (Chandra , Hairer ]

4• Explicitly for the ol ' eq .

( Berglund & Bruised) :

dtif, = DOT, - TO} t 3, t ↳ let t GCE ) Toe

* Cited = ELTIEA-ti lol) for some Ti w I degt;) a 0

Examples:

• A-

= - t 4 I . - 4 I • I •

• Each of the shaded trees gives a Gum At vacuum Feynman diagram !( IT acts multiplicatively on forests )

• Eft' lol)

-9Let's look at this one in some

more detail !

5

Eft' lol) = ELITE ,lol )' ] = Eff L . . . ? 3×06 Idea .

. . - ded)+ use Nick 's thin !

=

2t 4 t 4 t [ 0 's ]

q

Contest

•

•I • 5-1

> -

2 cohesion rule 8'

•a • *"

•

•t

6are. These computations become unwieldy very quickly !

• We need to exploit cancellations for of :-. fig toe to exist !

In the spirit of the original motivation :

2 Can we use AMPLITUHEDRON - like structures to organise

our Feynman diagram computations ?

( RS allow to treat locally subcortical = superrenomalisabk stochastic PDES,

i. e. # It : deg G) a 03 so .

Ever more important when one wants to study critical SPDES ! )

THANK YOU !