Stochastic Quantization Equations

Stochastic quantization equations

Hao Shen (Columbia University)

February 23, 2016

Page 1/25 1 2 I Ornstein-Uhlenbeck (d=0): S(X )= 2 X dX = X dt + dB t t t 4 1 2 1 4 d I model: S()= 2 ( (x)) + 4 (x) d x ´ r @ = 3 + ⇠ t 1 2 2 I Sine-Gordon (d=2): S()= 2 ( (x)) + cos((x))d x ´ r 1 @t u = u + sin(u)+⇠ 2

Stochastic quantization Stochastic quantization: consider a Euclidean quantum ﬁeld theory measure as a stationary distribution of a stochastic process.

exp( S()) /Z @ = S()/ + ⇠ D ) t where ⇠ is space-time white noise E[⇠(x, t)⇠(¯x, t¯)] = (d)(x x¯)(t t¯).

Page 2/25 Stochastic quantization Stochastic quantization: consider a Euclidean quantum ﬁeld theory measure as a stationary distribution of a stochastic process.

exp( S()) /Z @ = S()/ + ⇠ D ) t where ⇠ is space-time white noise E[⇠(x, t)⇠(¯x, t¯)] = (d)(x x¯)(t t¯). 1 2 I Ornstein-Uhlenbeck (d=0): S(X )= 2 X dX = X dt + dB t t t 4 1 2 1 4 d I model: S()= 2 ( (x)) + 4 (x) d x ´ r @ = 3 + ⇠ t 1 2 2 I Sine-Gordon (d=2): S()= 2 ( (x)) + cos((x))d x ´ r 1 @t u = u + sin(u)+⇠ 2

Page 2/25 I The solution to the linear equation (when d 2) is a.s. a distribution – so 3 and sin(u) are meaningless! I Let ⇠ be smooth noise and ⇠ ⇠ " " ! 1 @t u" = u" + sin(u")+⇠" 2 Then u does not converge to any nontrivial limit as " 0. " !

Stochastic quantization: questions & diﬃculties

I Field theory: construct the measure exp( S()) /Z D I Stochastic PDE: study well-posedness.

@ = 3 + ⇠ t 1 @t u = u + sin(u)+⇠ 2

Page 3/25 Stochastic quantization: questions & diﬃculties

I Field theory: construct the measure exp( S()) /Z D I Stochastic PDE: study well-posedness.

@ = 3 + ⇠ t 1 @t u = u + sin(u)+⇠ 2

I The solution to the linear equation (when d 2) is a.s. a distribution – so 3 and sin(u) are meaningless! I Let ⇠ be smooth noise and ⇠ ⇠ " " ! 1 @t u" = u" + sin(u")+⇠" 2 Then u does not converge to any nontrivial limit as " 0. " !

Page 3/25 I Alternative theories: Dirichlet forms for 4 in 2D (Albeverio & Rockner ’91); Paracontrolled distribution method by Gubinelli et al, for 4 in 3D (Catellier & Chouk ’13); Renormalization group method for 4 in 3D (Kupiainen ’14).

Two solution theories

@ = 3 + ⇠ t 1 @t u = u + sin(u)+⇠ 2

I Da Prato - Debussche method: 4 I in 2D (Da Prato & Debussche ’03), 2 I sine-Gordon with < 4⇡ (Hairer & S. ’14).

I Hairer’s regularity structure theory: 4 I in 3D (Hairer ’13), 2 16⇡ I sine-Gordon with for 4⇡ < (Hairer & S. ’14) (expect to  3 apply to all 2 < 8⇡;inprogress).

Page 4/25 Two solution theories

@ = 3 + ⇠ t 1 @t u = u + sin(u)+⇠ 2

I Da Prato - Debussche method: 4 I in 2D (Da Prato & Debussche ’03), 2 I sine-Gordon with < 4⇡ (Hairer & S. ’14).

I Hairer’s regularity structure theory: 4 I in 3D (Hairer ’13), 2 16⇡ I sine-Gordon with for 4⇡ < (Hairer & S. ’14) (expect to  3 apply to all 2 < 8⇡;inprogress). I Alternative theories: Dirichlet forms for 4 in 2D (Albeverio & Rockner ’91); Paracontrolled distribution method by Gubinelli et al, for 4 in 3D (Catellier & Chouk ’13); Renormalization group method for 4 in 3D (Kupiainen ’14).

Page 4/25 Outline

I Sine-Gordon equation 2 I < 4⇡: Apply Da Prato - Debussche method 2 I 4⇡ < 16⇡/3: Apply regularity structure  I 4 equation I Equation from gauge theory

Page 5/25 I If exp(i") had nontrivial limit (actually not!) then 1 @t v = v + f (v) 2

Da Prato - Debussche method

1 @t u" = u" + sin(u")+⇠" 2 Write u" =" + v" where 1 @t " = " + ⇠" 2

Then v" satisﬁes 1 @t v" = v" + sin(") cos(v")+cos(") sin(v") 2

New random input: exp(i")=cos(")+i sin(").

Page 6/25 Da Prato - Debussche method

1 @t u" = u" + sin(u")+⇠" 2 Write u" =" + v" where 1 @t " = " + ⇠" 2

Then v" satisﬁes 1 @t v" = v" + sin(") cos(v")+cos(") sin(v") 2

New random input: exp(i")=cos(")+i sin(").

I If exp(i") had nontrivial limit (actually not!) then 1 @t v = v + f (v) 2

Page 6/25 PDE argument Let f be a smooth function, and C with > 1, 2 1 @t v = v + f (v) 2 Let K =(@ 1 ) 1 be the heat kernel. Then: t 2 : v K ( f (v)) M 7! ⇤ deﬁnes a map from C 1 to C 1 itself: ↵ (↵,) I Young’s Thm: g C , h C , ↵ + >0 gh C min 2 2 ) 2 f (v) C (> 1) 2 1 (Example of ↵ + <0: BdB with B C 2 Brownian motion) 2 I Schauder’s estimate: “heat kernel gives two more regularities”

v C +2 C 1 M 2 ✓

Page 7/25 Da Prato - Debussche method

I Back to our equation 1 @t v✏ = v✏ + sin(") cos(v")+cos(") sin(v") 2 Q: Does exp(i ) converge to a limit in C with > 1? "

I Kolmogorov: To show random processes F F C , " ! 2

p p E F",' h z i . p p ' ( E F" F ,'z 0 z h i ! z for p 1. 8 I

Page 8/25 1 E "(z)"(z0) 2⇡ log( z z0 + ") 2 1⇠ | | E "(z) log " ( ⇥ ⇠⇤2⇡ !1 2 ⇥ ⇤ i" def /(4⇡) i" Renormalise: e " = " e

2 2 ¯ E "(z) "(z0) ( z z0 + ") 2⇡ indicates " C 2⇡ ⇠ | | ! 2 h i

Da Prato - Debussche method: second moment Question ei" ? in C with > 1 ! 2 i (z) I Want: E ' (z)e " dz . 2 h ´ i I Using characteristic function of Gaussian i"(z) i"(z ) E e e 0 2 2 E ("(z) "(z0)) ⇥= e 2 ⇤ ⇥ ⇤

Page 9/25 Da Prato - Debussche method: second moment Question ei" ? in C with > 1 ! 2 i (z) I Want: E ' (z)e " dz . 2 h ´ i I Using characteristic function of Gaussian i"(z) i"(z ) E e e 0 2 2 E ("(z) "(z0)) ⇥= e 2 ⇤ ⇥ ⇤ 1 E "(z)"(z0) 2⇡ log( z z0 + ") 2 1⇠ | | E "(z) log " ( ⇥ ⇠⇤2⇡ !1 2 ⇥ ⇤ i" def /(4⇡) i" Renormalise: e " = " e

2 2 ¯ E "(z) "(z0) ( z z0 + ") 2⇡ indicates " C 2⇡ ⇠ | | ! 2 h i Page 9/25 Da Prato - Debussche method: higher moments

+ + +

z z 2/(2⇡) | 0| z z 2/(2⇡) | 0|

Page 10/25 Da Prato - Debussche method: higher moments

+ + . + + + + z z 2/(2⇡) | 0| z z 2/(2⇡) | 0| 1 Conclusion: @t v = 2 v + Im( ) cos(v)+Re( ) sin(v) is locally well-posed for 2 < 4⇡. Solutions of

1 2/(4⇡) @t u" = u" + " sin(u")+⇠" 2 converge to a limit u. (Recall u =+v) Page 11/25 A Stochastic ODE example:

dXt = f (Xt ) dBt

1 I For B Brownian motion, dB C (R+) with < ;the 2 2 argument breaks down - one needs extra information to deﬁne the product f (Xt )dBt . I Extra information can be given by rough path theory.

Theory of regularity structure and 2 4⇡ If C with 1, 2  1 @t v = v + f (v) 2 “Young’s theorem - Schauder’s estimate” argument breaks down.

Page 12/25 Theory of regularity structure and 2 4⇡ If C with 1, 2  1 @t v = v + f (v) 2 “Young’s theorem - Schauder’s estimate” argument breaks down.

A Stochastic ODE example:

dXt = f (Xt ) dBt

1 I For B Brownian motion, dB C (R+) with < ;the 2 2 argument breaks down - one needs extra information to deﬁne the product f (Xt )dBt . I Extra information can be given by rough path theory.

Page 12/25 Stochastic ODE example

For smooth function f

dXt = f (Xt ) dBt

I X locally “looks like” Brownian motion

X X = g (B B )+sth. smoother t t0 t0 · t t0 I If X is as above, then so is f (X ),andfurthermore t 0 f (Xs ) dBs is also as above. ´ I Only need to deﬁne one product BdB.

Page 13/25 Theory of regularity structure and 2 4⇡

1 dXt = f (Xt ) dBt @t v = v + f (v) 2 1/2 " 1 " dB C C 2 2 The solutions, at small scale, behave like

t X B = dBs analogous v K ⇠ ˆ0 ⇠ ⇤

where K =(@ 1 ) 1. t 2 I Only need to deﬁne one product (K ). · ⇤ I Awholetheory(Theoryofregularitystructuresrecently developed by Martin Hairer) behind this “analogy”.

Page 14/25 I Need to analyze p-th moment of (K ) for arbitrary p. ⇤ I This renormalization amounts to change of equation 1 @t v✏ = v✏ + Im( ✏) cos(v✏) C✏ cos(v✏) cos0(v✏) 2 ⇣ ⌘ + Re( ) sin(v ) C sin(v ) sin0(v ) ✏ ✏ ✏ ✏ ✏ ⇣ ⌘ The two red terms cancel.

Regularity structure and 2 4⇡:momentsof (K ) ⇤ I First moment:

2 E (z) K(z w) ¯ (w)dw = K(z w) z w 2⇡ dw ˆR2+1 ˆR2+1 | | h i For 2 4⇡ non-integrable singularity at z w. ⇡ I Renormalization: deﬁne the product to be

def 2 (K ) = lim ✏(K ¯ ✏) C" (C" = K(z) z 2⇡ dz) ⇤ ✏ 0 ⇤ ˆ | | ! h i

Page 15/25 Regularity structure and 2 4⇡:momentsof (K ) ⇤ I First moment:

2 E (z) K(z w) ¯ (w)dw = K(z w) z w 2⇡ dw ˆR2+1 ˆR2+1 | | h i For 2 4⇡ non-integrable singularity at z w. ⇡ I Renormalization: deﬁne the product to be

def 2 (K ) = lim ✏(K ¯ ✏) C" (C" = K(z) z 2⇡ dz) ⇤ ✏ 0 ⇤ ˆ | | ! h i I Need to analyze p-th moment of (K ) for arbitrary p. ⇤ I This renormalization amounts to change of equation 1 @t v✏ = v✏ + Im( ✏) cos(v✏) C✏ cos(v✏) cos0(v✏) 2 ⇣ ⌘ + Re( ) sin(v ) C sin(v ) sin0(v ) ✏ ✏ ✏ ✏ ✏ ⇣ ⌘ The two red terms cancel. Page 15/25 Larger values of 2

I At 2 = 4⇡, C 1 -need K 2 · / I At 2 = 16⇡/3, C 4 3 -need K( K ) 2 · · / I At 2 = 6⇡, C 3 2 -need K( K( K )) 2 · · · I ...... Inﬁnite thresholds: 16⇡ 8(n 1) 0 < 4⇡< < 6⇡<...< ⇡<... 8⇡ 3 n ! Problem: show well-posedness for all 2 < 8⇡ (in progress). I Sine-Gordon ﬁeld theory: Kosterlitz-Thouless phase transition at 8⇡ (Frohlich & Spencer ’81)

Page 16/25 Outline

I Sine-Gordon equation 2 I < 4⇡: Apply Da Prato - Debussche method 2 I 4⇡< < 16⇡/3: Apply regularity structure

I 4 equation I Equation from gauge theory

Page 17/25 This method breaks down in 3D, where : 2 : C 1 (>0) 0 2

4:DaPrato-Debusschemethod

@ = 3 + ⇠ t Write = 0 + v, @t 0 =0 + ⇠ @ v =v (3 + 32v + 3 v 2 + v 3) t 0 0 0 Two steps in d = 2: I Renormalize: 2 3 2 3 I Replace , by Wick powers : :, : : C (>0). 0 0 0 0 2 I Nice fact: equivalence of moments.

I Solve v by ﬁxed point argument (say, in space v C 1) 2

Page 18/25 4:DaPrato-Debusschemethod

I Solve v by ﬁxed point argument (say, in space v C 1) 2 This method breaks down in 3D, where : 2 : C 1 (>0) 0 2

Page 18/25 4 For equation in 3D, write = 0 + 1 + R where

@t 0 =0 + ⇠

@ = 3 t 1 1 0 with ansatz 2 1 R K ( ) C (>0) ⇠ ⇤ 0 2 Need to deﬁne 2 (K (2)) -anotherrenormalization. 0 · ⇤ 0

4 in 3D: regularity structures

Regularity structure theory: I ﬁxed point argument in space of functions/distributions with delicate descriptions of local behavior, i.e. locally behave like Brownian path or K etc. ⇤ I only need to deﬁne a few objects (often through renormalization procedure).

Page 19/25 4 in 3D: regularity structures

@t 0 =0 + ⇠

@ = 3 t 1 1 0 with ansatz 2 1 R K ( ) C (>0) ⇠ ⇤ 0 2 Need to deﬁne 2 (K (2)) -anotherrenormalization. 0 · ⇤ 0

Page 19/25 Remarks

Some other results: I Global solution / solution on entire space: 4 (Mourrat & Weber) (Hairer & Matetski) I Weak universality results: KPZ (Hairer & Quastel), 4 (Hairer & Xu) (Xu & S.) I Non-Gaussian noises and central limit theorems: KPZ (Hairer & S.), 4 (Xu & S.) I Convergence from particle systems: 4 (Mourrat & Weber)

Page 20/25 Stochastic quantization of Maxwell Eq.

d = 2. Let (A , A ) be a vector ﬁeld. Let F = @ A @ A . 1 2 A 1 2 2 1 1 1 exp F 2 dx A = exp (@ A @ A )2 dx A 2 ˆ A D 2 ˆ 1 2 2 1 D ⇣ ⌘ ⇣ ⌘ The stochastic PDE: @ A = @ F + ⇠ = @2A @ @ A + ⇠ t 1 2 A 1 2 1 1 2 2 1 @ A = @ F + ⇠ = @2A @ @ A + ⇠ t 2 1 A 2 1 2 1 2 1 2

I Not parobolic.

I Gauge symmetry: For any function f (x) let A¯j = Aj + @j f . Then F ¯ = @ A¯ @ A¯ = @ A @ A = F A 1 2 2 1 1 2 2 1 A therefore A¯ satisﬁes the same equation.

Page 21/25 Stochastic quantization of Maxwell Eq.

@ A = @ F + ⇠ = @2A @ @ A + ⇠ t 1 2 A 1 2 1 1 2 2 1 @ A = @ F + ⇠ = @2A @ @ A + ⇠ t 2 1 A 2 1 2 1 2 1 2 “De Turck trick": Let B solves

@t Bj =Bj + ⇠j (j = 1, 2)

Then let t A1(t)=B1(t) @1 B(s) ds ˆ0 r· t A2(t)=B2(t) @2 B(s) ds ˆ0 r· We can check:

@ A = @ F + ⇠ ,@A = @ F + ⇠ t 1 2 B 1 t 2 1 B 2 and, F = @ A @ A = @ B @ B = F A 1 2 2 1 1 2 2 1 B Page 22/25 Nonlinear SPDE with gauge Consider quantum ﬁeld theory for vector A and complex valued 1 exp F 2 + DA 2 dx A 2 ˆ A | j | D D j=1,2 ⇣ X ⌘ The stochastic PDE: @ A = @2A @ @ A i ¯DA DA + ⇠ t 1 2 1 1 2 2 1 1 1 ⇣ ⌘ @ A = @2A @ @ A i ¯DA DA + ⇠ t 2 1 2 1 2 1 2 2 2 A A ⇣ ⌘ @t = Dj Dj +⇣ Xj A where Dj is gauge covariant derivative:

DA=@ iA (j = 1, 2) j j j

(") (") I Let (A , ) solve the smooth noise Eq, show limit " 0. ! Page 23/25 Nonlinear SPDE with gauge

@ A = @2A @ @ A i ¯DA DA + ⇠ t 1 2 1 1 2 2 1 1 1 ⇣ ⌘ @ A = @2A @ @ A i ¯DA DA + ⇠ t 2 1 2 1 2 1 2 2 2 A A ⇣ ⌘ @t = Dj Dj +⇣ Xj De Turck trick: (") @ B(") =B(") i (")DB (") (")DB(") (") + ⇠(") t j j j j j (") B(") B⇣(") (") (") (") i⌘ t B(")(s) ds (") @ = D D + i( B ) + e 0 r· ⇣ t j j r· ´ Xj Then deﬁne t (") (") (") Aj (t)=Bj (t) @j B (s) ds ˆ0 r· (") i t B(")(s) ds (") (t)=e 0 (t) ´ r·

Page 24/25 Nonlinear SPDE with gauge

@ A = @2A @ @ A i ¯DA DA + ⇠ t 1 2 1 1 2 2 1 1 1 ⇣ ⌘ @ A = @2A @ @ A i ¯DA DA + ⇠ t 2 1 2 1 2 1 2 2 2 A A ⇣ ⌘ @t = Dj Dj +⇣ Xj Expected result: Let (A("), (")) solve the smooth noise Eq. There exists a family (parametrized by time) of gauge transformations (") (") such that (A("), (")) converges to a nontrivial limit. Gt Gt

Page 25/25