Nils Berglund nils.berglund@univ-orleans.fr http://www.univ-orleans.fr/mapmo/membres/berglund/ SPA 2017 Invited Session: Regularity structures A thermodynamic characterization of some regularity structures near the subcriticality threshold Nils Berglund MAPMO, Université d Orléans, France Moskva / Moscow, July 24 2017 with Christian Kuehn (TU Munich)
Fractional Allen Cahn-type SPDEs Thermodynamic characterization of regularity structures July 24, 2017 1/11 t u = ( ) ρ/2 u + F (u) + ξ
Fractional Allen Cahn-type SPDEs Thermodynamic characterization of regularity structures July 24, 2017 1/11 t u = ( ) ρ/2 u + F (u) + ξ u = u(t, x) R, (t, x) R + T d, d 1 ( ) ρ/2 =: ρ/2 : Fractional Laplacian F polynomial of degree N ξ space-time white noise: E[ξ(t, x)ξ(y, s)] = δ(x y)δ(t s) ξ, ϕ = W ϕ N (0, ϕ 2 L 2 ), E[W ϕ W ϕ ] = ϕ, ϕ
Fractional Allen Cahn-type SPDEs Thermodynamic characterization of regularity structures July 24, 2017 1/11 t u = ( ) ρ/2 u + F (u) + ξ u = u(t, x) R, (t, x) R + T d, d 1 ( ) ρ/2 =: ρ/2 : Fractional Laplacian F polynomial of degree N ξ space-time white noise: E[ξ(t, x)ξ(y, s)] = δ(x y)δ(t s) ξ, ϕ = W ϕ N (0, ϕ 2 L 2 ), E[W ϕ W ϕ ] = ϕ, ϕ Motivations: Better understanding of local subcriticality FitzHugh Nagumo equation [B, Kuehn, EJP 2016] t u = u + u u 3 + v + ξ t v = a 1 u + a 2 v
Fractional Allen Cahn-type SPDEs Thermodynamic characterization of regularity structures July 24, 2017 1/11 t u = ( ) ρ/2 u + F (u) + ξ u = u(t, x) R, (t, x) R + T d, d 1 ( ) ρ/2 =: ρ/2 : Fractional Laplacian F polynomial of degree N ξ space-time white noise: E[ξ(t, x)ξ(y, s)] = δ(x y)δ(t s) ξ, ϕ = W ϕ N (0, ϕ 2 L 2 ), E[W ϕ W ϕ ] = ϕ, ϕ Motivations: Better understanding of local subcriticality FitzHugh Nagumo equation [B, Kuehn, EJP 2016] t u = u + u u 3 + v + ξ t v = δ v + a 1 u + a 2 v
Fractional Allen Cahn-type SPDEs Thermodynamic characterization of regularity structures July 24, 2017 1/11 t u = ( ) ρ/2 u + F (u) + ξ u = u(t, x) R, (t, x) R + T d, d 1 ( ) ρ/2 =: ρ/2 : Fractional Laplacian F polynomial of degree N ξ space-time white noise: E[ξ(t, x)ξ(y, s)] = δ(x y)δ(t s) ξ, ϕ = W ϕ N (0, ϕ 2 L 2 ), E[W ϕ W ϕ ] = ϕ, ϕ Motivations: Better understanding of local subcriticality FitzHugh Nagumo equation [B, Kuehn, EJP 2016] t u = u + u u 3 + v + ξ t v = ρ/2 v + a 1 u + a 2 v
Case ρ = 2, N = 3 Thermodynamic characterization of regularity structures July 24, 2017 2/11 Φ 4 d model: tu = u u 3 + ξ
Case ρ = 2, N = 3 Thermodynamic characterization of regularity structures July 24, 2017 2/11 Φ 4 d model: tu = u u 3 + ξ Mollified noise: ξ ε = ϱ ε ξ with ϱ ε (t, x) = 1 ϱ ( t, x ) ε d+2 ε 2 ε where ϱ compact support, integral 1 Theorem d {2, 3}. choice of renormalisation const C(ε), lim ε 0 C(ε) =, t u ε = u ε + C(ε)u ε (u ε ) 3 + ξ ε admits sequence u ε of local solutions, converging in probability to a limit u as ε 0.
Case ρ = 2, N = 3 Thermodynamic characterization of regularity structures July 24, 2017 2/11 Φ 4 d model: tu = u u 3 + ξ Mollified noise: ξ ε = ϱ ε ξ with ϱ ε (t, x) = 1 ϱ ( t, x ) ε d+2 ε 2 ε where ϱ compact support, integral 1 Theorem d {2, 3}. choice of renormalisation const C(ε), lim ε 0 C(ε) =, t u ε = u ε + C(ε)u ε (u ε ) 3 + ξ ε admits sequence u ε of local solutions, converging in probability to a limit u as ε 0. d = 2: [Da Prato & Debussche 2004] C(ε) = C 1 log(ε 1 ) d = 3: [Hairer 2014], also [Catellier & Chouk], [Kupiainen] C(ε) = C 1 ε 1 + C 2 log(ε 1 ) + C 3
Thermodynamic characterization of regularity structures July 24, 2017 3/11 Mild solutions and Hölder spaces ( t )u = h u = G h ( t )u = F (u) + ξ u = G ξ + G F (u) Which function space?
Thermodynamic characterization of regularity structures July 24, 2017 3/11 Mild solutions and Hölder spaces ( t )u = h u = G h ( t )u = F (u) + ξ u = G ξ + G F (u) Which function space? Hölder spaces C α for f : I R, with I R compact interval: 0 < α < 1: f (x) f (y) C x y α x y α > 1: f C α and f C α 1 f (x) f (y) C x y α α < 0: f distribution, f, ηx δ Cδ α with ηx(y) δ = 1 x y δ η( δ )
Mild solutions and Hölder spaces Thermodynamic characterization of regularity structures July 24, 2017 3/11 ( t )u = h u = G h ( t )u = F (u) + ξ u = G ξ + G F (u) Which function space? Hölder spaces C α for f : I R, with I R compact interval: 0 < α < 1: f (x) f (y) C x y α x y α > 1: f C α and f C α 1 f (x) f (y) C x y α α < 0: f distribution, f, ηx δ Cδ α with ηx(y) δ = 1 x y δ η( δ ) Parabolic scaling C α s : x y t s 1/2 + d i=1 x i y i Facts: 1. α / Z, f Cs α G f Cs α+2 2. ξ Cs α a.s. α < d+2 2 (Schauder)
Mild solutions and Hölder spaces Thermodynamic characterization of regularity structures July 24, 2017 3/11 ( t )u = h u = G h ( t )u = F (u) + ξ u = G ξ + G F (u) Which function space? Hölder spaces C α for f : I R, with I R compact interval: 0 < α < 1: f (x) f (y) C x y α x y α > 1: f C α and f C α 1 f (x) f (y) C x y α α < 0: f distribution, f, ηx δ Cδ α with ηx(y) δ = 1 x y δ η( δ ) Parabolic scaling C α s : x y t s 1/2 + d i=1 x i y i Facts: 1. α / Z, f Cs α G f Cs α+2 (Schauder) 2. ξ Cs α a.s. α < d+2 2 Consequence: G ξ Cs α a.s. α < 2 d 2 0 for d 2
Thermodynamic characterization of regularity structures July 24, 2017 4/11 Regularity structures [Hairer, Inventiones Math. 198, 269 504, 2014]: M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) u ε S M
Thermodynamic characterization of regularity structures July 24, 2017 4/11 Regularity structures [Hairer, Inventiones Math. 198, 269 504, 2014]: M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) S M u ε u = G (ξ ε u 3 ) U = I(Ξ U 3 ) + ϕ1 + polynomial terms U 0 = 0 U 1 = I(Ξ) + ϕ1 +... U1 3 = I(Ξ) 3 + 3ϕI(Ξ) 2 + 3ϕ 2 I(Ξ) + ϕ 3 1 +... U 2 = I(Ξ) I(I(Ξ) 3 ) 3ϕI(I(Ξ) 2 ) 3ϕ 2 I(I(Ξ)) + ϕ1 +...
Thermodynamic characterization of regularity structures July 24, 2017 4/11 Regularity structures [Hairer, Inventiones Math. 198, 269 504, 2014]: M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) S M u ε u = G (ξ ε u 3 ) U = I(Ξ U 3 ) + ϕ1 + polynomial terms U 0 = 0 U 1 = I(Ξ) + ϕ1 +... U1 3 = I(Ξ) 3 + 3ϕI(Ξ) 2 + 3ϕ 2 I(Ξ) + ϕ 3 1 +... U 2 = I(Ξ) I(I(Ξ) 3 ) 3ϕI(I(Ξ) 2 ) 3ϕ 2 I(I(Ξ)) + ϕ1 +...
Thermodynamic characterization of regularity structures July 24, 2017 4/11 Regularity structures [Hairer, Inventiones Math. 198, 269 504, 2014]: M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) S M u ε u = G (ξ ε u 3 ) U = I(Ξ U 3 ) + ϕ1 + polynomial terms U 0 = 0 U 1 = I(Ξ) + ϕ1 +... U1 3 = I(Ξ) 3 + 3ϕI(Ξ) 2 + 3ϕ 2 I(Ξ) + ϕ 3 1 +... U 2 = I(Ξ) I(I(Ξ) 3 ) 3ϕI(I(Ξ) 2 ) 3ϕ 2 I(I(Ξ)) + ϕ1 +... =: 3ϕ 3ϕ 2 + ϕ1 +...
Regularity structures [Hairer, Inventiones Math. 198, 269 504, 2014]: M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) S M u ε u = G (ξ ε u 3 ) U = I(Ξ U 3 ) + ϕ1 + polynomial terms U 0 = 0 U 1 = I(Ξ) + ϕ1 +... U1 3 = I(Ξ) 3 + 3ϕI(Ξ) 2 + 3ϕ 2 I(Ξ) + ϕ 3 1 +... U 2 = I(Ξ) I(I(Ξ) 3 ) 3ϕI(I(Ξ) 2 ) 3ϕ 2 I(I(Ξ)) + ϕ1 +... =: 3ϕ 3ϕ 2 + ϕ1 +... 1 d 2 5 3d 2 4 d 3 d 2 0 Locally subcritical: Hölder exponents are bdd below Thermodynamic characterization of regularity structures July 24, 2017 4/11
Thermodynamic characterization of regularity structures July 24, 2017 5/11 Case of fractional Laplacian 0 < ρ < 2, ρ/2 := ( ) ρ/2 generator of a Lévy process with kernel G ρ (t, x) = 1 (2π) d e i x ξ e t ξ ρ dξ R d
Thermodynamic characterization of regularity structures July 24, 2017 5/11 Case of fractional Laplacian 0 < ρ < 2, ρ/2 := ( ) ρ/2 generator of a Lévy process with kernel G ρ (t, x) = 1 (2π) d e i x ξ e t ξ ρ dξ R d Easy to check: 1. s = (ρ, 1,..., 1) G ρ regularising of order ρ: f C α s, α + ρ / Z G ρ f C α+ρ s
Thermodynamic characterization of regularity structures July 24, 2017 5/11 Case of fractional Laplacian 0 < ρ < 2, ρ/2 := ( ) ρ/2 generator of a Lévy process with kernel G ρ (t, x) = 1 (2π) d e i x ξ e t ξ ρ dξ R d Easy to check: 1. s = (ρ, 1,..., 1) G ρ regularising of order ρ: f C α s, α + ρ / Z G ρ f C α+ρ s 2. ξ C α s a.s. α < ρ+d 2
Thermodynamic characterization of regularity structures July 24, 2017 5/11 Case of fractional Laplacian 0 < ρ < 2, ρ/2 := ( ) ρ/2 generator of a Lévy process with kernel G ρ (t, x) = 1 (2π) d e i x ξ e t ξ ρ dξ R d Easy to check: 1. s = (ρ, 1,..., 1) G ρ regularising of order ρ: f C α s, α + ρ / Z G ρ f C α+ρ s 2. ξ C α s a.s. α < ρ+d 2 3. t u = ρ/2 u + F (u) degree N + ξ loc. subcritical ρ > ρ c = d N 1 N+1
Thermodynamic characterization of regularity structures July 24, 2017 5/11 Case of fractional Laplacian 0 < ρ < 2, ρ/2 := ( ) ρ/2 generator of a Lévy process with kernel G ρ (t, x) = 1 (2π) d e i x ξ e t ξ ρ dξ R d Easy to check: 1. s = (ρ, 1,..., 1) G ρ regularising of order ρ: f C α s, α + ρ / Z G ρ f C α+ρ s 2. ξ C α s a.s. α < ρ+d 2 3. t u = ρ/2 u + F (u) degree N + ξ loc. subcritical ρ > ρ c = d N 1 N+1 Idea: Fixed point equ U = I(Ξ + F (U)) + ϕ1 +... Idea: Ξ s = ρ+d 2 κ =: α 0 Idea: I(Ξ) N s = N(α 0 + ρ) = N 2 (ρ d) Nκ Idea: I(Ξ) N s > Ξ s ρ > ρ c Idea: then induction on fixed point application
Thermodynamic characterization of regularity structures July 24, 2017 6/11 Model space U = I(Ξ + F (U)) + ϕ1 +... U F : symbols representing solution U F F U F : symbols representing equation (i.e. Ξ + F (U))
Thermodynamic characterization of regularity structures July 24, 2017 6/11 Model space U = I(Ξ + F (U)) + ϕ1 +... U F : symbols representing solution U F F U F : symbols representing equation (i.e. Ξ + F (U)) given by induction W 0 = U 0 = and W m = W m 1 U m 1 Um 1 N {Ξ} U m = I(W m ) {X k } with AB := {ττ : τ A, τ B} Then U F = U m, F F = m U m ) m 0 m 0(W
Thermodynamic characterization of regularity structures July 24, 2017 6/11 Model space U = I(Ξ + F (U)) + ϕ1 +... U F : symbols representing solution U F F U F : symbols representing equation (i.e. Ξ + F (U)) given by induction W 0 = U 0 = and W m = W m 1 U m 1 Um 1 N {Ξ} U m = I(W m ) {X k } with AB := {ττ : τ A, τ B} Then U F = U m, F F = m U m ) m 0 m 0(W Questions: Let A F = { τ s : τ F F } 1. Estimate h F = #(A F R ) (number of negative Hölder exponents) 2. Estimate c F = #{τ F F : τ s < 0} (number of singular symbols)
Thermodynamic characterization of regularity structures July 24, 2017 7/11 Number of negative Hölder exponents h F = #(A F R ) Theorem 1 ρ + d 1 h F 1 + N + 1 ρ ρ c (ρ + d)dn N + 1 1 ρ ρ c
Thermodynamic characterization of regularity structures July 24, 2017 7/11 Number of negative Hölder exponents h F = #(A F R ) Theorem 1 ρ + d 1 h F 1 + N + 1 ρ ρ c Proof: τ s = ρ+d 2 p(τ) + q(τ)ρ + k s(τ) O(κ) p = #Ξ, q = #I, 0 k s = polynomial exp. D 0 (U) = {(p(τ), q(τ)): τ U} N 2 0 (ρ + d)dn N + 1 1 ρ ρ c
Thermodynamic characterization of regularity structures July 24, 2017 7/11 Number of negative Hölder exponents h F = #(A F R ) Theorem 1 ρ + d 1 h F 1 + N + 1 ρ ρ c (ρ + d)dn N + 1 1 ρ ρ c D 0 (W 3 ) for N = d = 3 ρ = 2 I ρ = 21 11 Ξ ρ = 9 5 ρ = ρ c = 3 2
Thermodynamic characterization of regularity structures July 24, 2017 7/11 Number of negative Hölder exponents h F = #(A F R ) Theorem 1 ρ + d 1 h F 1 + N + 1 ρ ρ c Proof: τ s = ρ+d 2 p(τ) + q(τ)ρ + k s(τ) O(κ) p = #Ξ, q = #I, 0 k s = polynomial exp. (ρ + d)dn N + 1 1 ρ ρ c D 0 (U) = {(p(τ), q(τ)): τ U} N 2 0 (p, q ) D 0 (U n ) = convex env. of nd 0 (U) N 2 0 lim m D 0 (U m ) = truncated cone τ s < 0 p = 1 + N 1 N q & τ triangle h F = number of lattice points in triangle q = (ρ+d)n (N+1)(ρ ρ + O(κ) c) q p q = ρ+d 2 p + qρ = 0 N N 1 (p 1)
Thermodynamic characterization of regularity structures July 24, 2017 8/11 Number of symbols c F = #{τ F F : τ s < 0} Theorem 2 C N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) c F C + N (ρ ρ c) 3/2 e β Nd/(ρ ρ c)
Thermodynamic characterization of regularity structures July 24, 2017 8/11 Number of symbols c F = #{τ F F : τ s < 0} Theorem 2 C N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) c F C + N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) Case N = 2: τ trees of degree 3, p leaves, q edges d i := # vertices of degree i d 1 + d 2 + d 3 = q + 1 d 1 + 2d 2 + 3d 3 = 2q d 1 = p + 1 {deg =1} I Ξ
Thermodynamic characterization of regularity structures July 24, 2017 8/11 Number of symbols c F = #{τ F F : τ s < 0} Theorem 2 C N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) c F C + N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) Case N = 2: τ trees of degree 3, p leaves, q edges d i := # vertices of degree i d 1 + d 2 + d 3 = q + 1 d 1 + 2d 2 + 3d 3 = 2q d 1 = p + 1 {deg =1} q = 2n binary tree with q + 1 vertices q = 2n + 1 binary tree with q + 2 vertices minus one edge I Ξ
Number of symbols c F = #{τ F F : τ s < 0} Theorem 2 C N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) c F C + N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) Case N = 2: τ trees of degree 3, p leaves, q edges d i := # vertices of degree i d 1 + d 2 + d 3 = q + 1 d 1 + 2d 2 + 3d 3 = 2q d 1 = p + 1 {deg =1} q = 2n binary tree with q + 1 vertices q = 2n + 1 binary tree with q + 2 vertices minus one edge One has to count trees up to homeomorphism (1/0.4072... )n Wedderburn Etherington numbers W n c [Otter 1948] n 3/2 Thermodynamic characterization of regularity structures July 24, 2017 8/11 I Ξ
Thermodynamic characterization of regularity structures July 24, 2017 9/11 Statistical properties Ω = {τ F F : τ s < 0}, P uniform measure Properties of random variables X : Ω R when ρ ρ c?
Thermodynamic characterization of regularity structures July 24, 2017 9/11 Statistical properties Ω = {τ F F : τ s < 0}, P uniform measure Properties of random variables X : Ω R when ρ ρ c? Case X = Q = #I: E[Q/q ] = 1 + O(ρ ρ c ) et Var[Q/q ] = O((ρ ρ c ) 2 ) lim ρ ρ c (ρ ρ c ) log P{Q/q x} = β N d(1 x) x [0, 1] P{Q / NN} e γ/(ρ ρc)
Thermodynamic characterization of regularity structures July 24, 2017 9/11 Statistical properties Ω = {τ F F : τ s < 0}, P uniform measure Properties of random variables X : Ω R when ρ ρ c? Case X = Q = #I: E[Q/q ] = 1 + O(ρ ρ c ) et Var[Q/q ] = O((ρ ρ c ) 2 ) lim ρ ρ c (ρ ρ c ) log P{Q/q x} = β N d(1 x) x [0, 1] P{Q / NN} e γ/(ρ ρc) Other interesting random variables: Number P of Ξ: function of Q Hölder exponent: concentrated in 0 Degree distribution: close to ( N 1 N, 0,..., 0, 1 N ) Height and diameter [Broutin & Flajolet]: of order 1/ ρ ρ c N = d = 3, ρ {1.8, 1.75, 1.7, 1.76, 1.6, 1.59}
Thermodynamic characterization of regularity structures July 24, 2017 10/11 Renormalisation t u ε = ρ/2 u ε + C(ε)u ε (u ε ) 3 + ξ ε
Thermodynamic characterization of regularity structures July 24, 2017 10/11 Renormalisation t u ε = ρ/2 u ε + C(ε)u ε (u ε ) 3 + ξ ε BPHZ theory [Bruned, Hairer & Zambotti 2016, Chandra & Hairer 2016]: We expect C(ε) ε τ s (ε 0 = log(ε 1 )) τ F F : τ s<0
Thermodynamic characterization of regularity structures July 24, 2017 10/11 Renormalisation t u ε = ρ/2 u ε + C(ε)u ε (u ε ) 3 + ξ ε BPHZ theory [Bruned, Hairer & Zambotti 2016, Chandra & Hairer 2016]: We expect C(ε) ε τ s (ε 0 = log(ε 1 )) τ F F : τ s<0 = c F E[ε τ s ]
Thermodynamic characterization of regularity structures July 24, 2017 10/11 Renormalisation t u ε = ρ/2 u ε + C(ε)u ε (u ε ) 3 + ξ ε BPHZ theory [Bruned, Hairer & Zambotti 2016, Chandra & Hairer 2016]: We expect C(ε) ε τ s (ε 0 = log(ε 1 )) τ F F : τ s<0 = c F E[ε τ s ] P{ τ s = h} e γ Nh/(ρ ρ c) γ N = N+1 N β N ρ+d 2 < h < 0 number of elements number of elements 5 4 3 2 1 rho = 1.8 0 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 12 10 8 6 4 2 0 homogeneity rho = 1.65 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 homogeneity number of elements number of elements 8 7 6 5 4 3 2 1 rho = 1.75 0 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 30 25 20 15 10 5 0 homogeneity rho = 1.6 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 homogeneity number of elements number of elements 12 10 8 6 4 2 rho = 1.7 0 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 45 40 35 30 25 20 15 10 5 0 homogeneity rho = 1.58 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 homogeneity
Thermodynamic characterization of regularity structures July 24, 2017 10/11 Renormalisation t u ε = ρ/2 u ε + C(ε)u ε (u ε ) 3 + ξ ε BPHZ theory [Bruned, Hairer & Zambotti 2016, Chandra & Hairer 2016]: We expect C(ε) ε τ s (ε 0 = log(ε 1 )) τ F F : τ s<0 = c F E[ε τ s ] P{ τ s = h} e γ Nh/(ρ ρ c) γ N = N+1 N β N ρ+d 2 < h < 0 number of elements number of elements 5 4 3 2 1 rho = 1.8 0 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 12 10 8 6 4 2 homogeneity rho = 1.65 number of elements number of elements 8 7 6 5 4 3 2 1 rho = 1.75 0 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 30 25 20 15 10 5 homogeneity rho = 1.6 0 0 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 homogeneity homogeneity c F log(ε 1 ) if ε > e γ N/(ρ ρ c) ( ) C(ε) ε (d ρ) c F if ε < e γ N/(ρ ρ c) e γ N/(ρ ρ c) number of elements number of elements 12 10 8 6 4 2 rho = 1.7 0 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 45 40 35 30 25 20 15 10 5 0 homogeneity rho = 1.58 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 homogeneity
Thermodynamic characterization of regularity structures July 24, 2017 11/11 References M. Hairer, A theory of regularity structures, Inv. Math. 198, 269 504 (2014) M. Hairer, Introduction to Regularity Structures, lecture notes (2013) A. Chandra, H. Weber, Stochastic PDEs, regularity structures, and interacting particle systems, Annales Mathématiques de la Faculté des Sciences de Toulouse (in press), arxiv/1508.03616 N. B., C. Kuehn, Regularity structures and renormalisation of FitzHugh Nagumo SPDEs in three space dimensions, Elec J Prob 21 (18):1 48 (2016) N. B., C. Kuehn, Model spaces of regularity structures in space-fractional SPDEs, J Statist Phys (168):331 368 (2017) Y. Bruned, M. Hairer, L. Zambotti, Algebraic renormalisation of regularity structures, arxiv/1610.08468 A. Chandra, M. Hairer, An analytic BPHZ theorem for regularity structures, arxiv/1612.08138 M. Hairer, An analyst s take on the BPHZ theorem, arxiv/1704.08634
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