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 Φ model: S(φ)= 2 ( φ(x)) + 4 φ(x) d x ´ r @ φ =∆φ φ3 + ⇠ t − 1 2 2 I Sine-Gordon (d=2): S(φ)= 2β ( φ(x)) + cos(φ(x))d x ´ r 1 @t u = ∆u + sin(βu)+⇠ 2 Stochastic quantization Stochastic quantization: consider a Euclidean quantum field theory measure as a stationary distribution of a stochastic process. exp( S(φ)) φ/Z @ φ = δS(φ)/δφ + ⇠ − D ) t − where ⇠ is space-time white noise E[⇠(x, t)⇠(¯x, t¯)] = δ(d)(x x¯)δ(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(φ)) φ/Z @ φ = δS(φ)/δφ + ⇠ − D ) t − where ⇠ is space-time white noise E[⇠(x, t)⇠(¯x, t¯)] = δ(d)(x x¯)δ(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 Φ model: S(φ)= 2 ( φ(x)) + 4 φ(x) d x ´ r @ φ =∆φ φ3 + ⇠ t − 1 2 2 I Sine-Gordon (d=2): S(φ)= 2β ( φ(x)) + cos(φ(x))d x ´ r 1 @t u = ∆u + sin(βu)+⇠ 2 Page 2/25 I The solution to the linear equation (when d 2) is a.s. a ≥ distribution – so φ3 and sin(βu) are meaningless! I Let ⇠ be smooth noise and ⇠ ⇠ " " ! 1 @t u" = ∆u" + sin(βu")+⇠" 2 Then u does not converge to any nontrivial limit as " 0. " ! Stochastic quantization: questions & difficulties I Field theory: construct the measure exp( S(φ)) φ/Z − D I Stochastic PDE: study well-posedness. @ φ =∆φ φ3 + ⇠ t − 1 @t u = ∆u + sin(βu)+⇠ 2 Page 3/25 Stochastic quantization: questions & difficulties I Field theory: construct the measure exp( S(φ)) φ/Z − D I Stochastic PDE: study well-posedness. @ φ =∆φ φ3 + ⇠ t − 1 @t u = ∆u + sin(βu)+⇠ 2 I The solution to the linear equation (when d 2) is a.s. a ≥ distribution – so φ3 and sin(βu) are meaningless! I Let ⇠ be smooth noise and ⇠ ⇠ " " ! 1 @t u" = ∆u" + sin(βu")+⇠" 2 Then u does not converge to any nontrivial limit as " 0. " ! Page 3/25 I Alternative theories: Dirichlet forms for Φ4 in 2D (Albeverio & Rockner ’91); Paracontrolled distribution method by Gubinelli et al, for Φ4 in 3D (Catellier & Chouk ’13); Renormalization group method for Φ4 in 3D (Kupiainen ’14). Two solution theories @ φ =∆φ φ3 + ⇠ t − 1 @t u = ∆u + sin(βu)+⇠ 2 I Da Prato - Debussche method: 4 I Φ in 2D (Da Prato & Debussche ’03), 2 I sine-Gordon with β < 4⇡ (Hairer & S. ’14). I Hairer’s regularity structure theory: 4 I Φ in 3D (Hairer ’13), 2 16⇡ I sine-Gordon with for 4⇡ β < (Hairer & S. ’14) (expect to 3 apply to all β2 < 8⇡;inprogress). Page 4/25 Two solution theories @ φ =∆φ φ3 + ⇠ t − 1 @t u = ∆u + sin(βu)+⇠ 2 I Da Prato - Debussche method: 4 I Φ in 2D (Da Prato & Debussche ’03), 2 I sine-Gordon with β < 4⇡ (Hairer & S. ’14). I Hairer’s regularity structure theory: 4 I Φ in 3D (Hairer ’13), 2 16⇡ I sine-Gordon with for 4⇡ β < (Hairer & S. ’14) (expect to 3 apply to all β2 < 8⇡;inprogress). I Alternative theories: Dirichlet forms for Φ4 in 2D (Albeverio & Rockner ’91); Paracontrolled distribution method by Gubinelli et al, for Φ4 in 3D (Catellier & Chouk ’13); Renormalization group method for Φ4 in 3D (Kupiainen ’14). Page 4/25 Outline I Sine-Gordon equation 2 I β < 4⇡: Apply Da Prato - Debussche method 2 I 4⇡ β < 16⇡/3: Apply regularity structure I Φ4 equation I Equation from gauge theory Page 5/25 I If exp(iβΦ") had nontrivial limit (actually not!) then 1 @t v = ∆v + f (v) 2 Da Prato - Debussche method 1 @t u" = ∆u" + sin(βu")+⇠" 2 Write u" =Φ" + v" where 1 @t Φ" = ∆Φ" + ⇠" 2 Then v" satisfies 1 @t v" = ∆v" + sin(βΦ") cos(βv")+cos(βΦ") sin(βv") 2 New random input: exp(iβΦ")=cos(βΦ")+i sin(βΦ"). Page 6/25 Da Prato - Debussche method 1 @t u" = ∆u" + sin(βu")+⇠" 2 Write u" =Φ" + v" where 1 @t Φ" = ∆Φ" + ⇠" 2 Then v" satisfies 1 @t v" = ∆v" + sin(βΦ") cos(βv")+cos(βΦ") sin(βv") 2 New random input: exp(iβΦ")=cos(βΦ")+i sin(βΦ"). I If exp(iβΦ") 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 γ with γ> 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: ↵ β (↵,β) I Young’s Thm: g C , h C , ↵ + β>0 gh C min 2 2 ) 2 f (v) C γ (γ> 1) 2 − 1 δ (Example of ↵ + β<0: BdB with B C 2 − Brownian motion) 2 I Schauder’s estimate: “heat kernel gives two more regularities” v C γ+2 C 1 M 2 ✓ Page 7/25 Da Prato - Debussche method I Back to our equation 1 @t v✏ = ∆v✏ + sin(βΦ") cos(βv")+cos(βΦ") sin(βv") 2 Q: Does exp(iβΦ ) converge to a limit in C γ with γ> 1? " − γ I Kolmogorov: To show random processes F F C , " ! 2 λ p γp E F",' λ h z i . γp λ p 'λ (λ− E F" F ,'z 0 z h − i ! z for p 1. 8 ≥ I λ Page 8/25 1 E Φ"(z)Φ"(z0) 2⇡ log( z z0 + ") 2 1⇠ | − | E Φ"(z) log " ( ⇥ ⇠⇤2⇡ !1 2 ⇥ ⇤ iβΦ" def β /(4⇡) iβΦ" Renormalise: e " = "− e 2 2 ¯ β β E "(z) "(z0) ( z z0 + ")− 2⇡ indicates " C − 2⇡ ⇠ | − | ! 2 h i Da Prato - Debussche method: second moment Question eiβΦ" ? in C γ with γ> 1 ! − 2 λ iβΦ (z) γ I Want: E ' (z)e " dz . λ2 h ´ i I Using characteristic function of Gaussian iβΦ"(z) iβΦ"(z ) E e e− 0 2 β 2 E (Φ"(z) Φ"(z0)) ⇥= e− 2 − ⇤ ⇥ ⇤ Page 9/25 Da Prato - Debussche method: second moment Question eiβΦ" ? in C γ with γ> 1 ! − 2 λ iβΦ (z) γ I Want: E ' (z)e " dz . λ2 h ´ i I Using characteristic function of Gaussian iβΦ"(z) iβΦ"(z ) E e e− 0 2 β 2 E (Φ"(z) Φ"(z0)) ⇥= e− 2 − ⇤ ⇥ ⇤ 1 E Φ"(z)Φ"(z0) 2⇡ log( z z0 + ") 2 1⇠ | − | E Φ"(z) log " ( ⇥ ⇠⇤2⇡ !1 2 ⇥ ⇤ iβΦ" def β /(4⇡) iβΦ" Renormalise: e " = "− e 2 2 ¯ β β E "(z) "(z0) ( z z0 + ")− 2⇡ indicates " C − 2⇡ ⇠ | − | ! 2 h i Page 9/25 Da Prato - Debussche method: higher moments + − − + + − z z β2/(2⇡) | − 0|− z z β2/(2⇡) | − 0| Page 10/25 Da Prato - Debussche method: higher moments + − + − − − . + + + + − − z z β2/(2⇡) | − 0|− z z β2/(2⇡) | − 0| 1 Conclusion: @t v = 2 ∆v + Im( ) cos(βv)+Re( ) sin(βv) is locally well-posed for β2 < 4⇡. Solutions of 1 β2/(4⇡) @t u" = ∆u" + "− sin(βu")+⇠" 2 converge to a limit u. (Recall u =Φ+v) Page 11/25 A Stochastic ODE example: dXt = f (Xt ) dBt γ 1 I For B Brownian motion, dB C (R+) with γ< ;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 β2 4⇡ ≥ If C γ with γ 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 β2 4⇡ ≥ If C γ with γ 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 γ 1 I For B Brownian motion, dB C (R+) with γ< ;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 β2 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(βv✏) C✏ cos(βv✏) cos0(βv✏) 2 − ⇣ ⌘ + Re( ) sin(βv ) C sin(βv ) sin0(βv ) ✏ ✏ − ✏ ✏ ✏ ⇣ ⌘ The two red terms cancel.