Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions

Transcription

1 Nils Berglund EPFL, Seminar of Probability and Stochastic Processes Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions Nils Berglund MAPMO, Université d Orléans 11 April 2016 with Christian Kuehn (TU Munich)

2 Plan Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 1. Motivation and main result 2. Introduction to regularity structures 3. Extension to FitzHugh Nagumo

3 1. FitzHugh Nagumo SDE Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 du t = [u t u 3 t + v t ] dt + σ dw t dv t = ε[a u t bv t ] dt u t : membrane potential of neuron v t : gating variable (proportion of open ion channels) v u t ε = 0.1 b = 0 a = σ = 0.03 u t

4 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 FitzHugh Nagumo SPDE t u = u + u u 3 + v + ξ t v = a 1 u + a 2 v u = u(t, x) R, v = v(t, x) R (or R n ), (t, x) D = R + T d, d = 2, 3 ξ(t, x) Gaussian space-time white noise: E [ ξ(t, x)ξ(s, y) ] = δ(t s)δ(x y) ξ: distribution defined by ξ, ϕ = W ϕ, {W h } h L 2 (D), E[W h W h ] = h, h (Link to simulation)

5 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Main result Mollified noise: ξ ε = ϱ ε ξ where ϱ ε (t, x) = 1 ϱ ( t, x ) ε d+2 ε 2 ε with ϱ compactly supported, integral 1 Theorem [NB & C. Kuehn, Elec J Probab 21 (18):1-48 (2016)] There exists a choice of renormalisation constant C(ε), lim ε 0 C(ε) =, such that t u ε = u ε + [1 + C(ε)]u ε (u ε ) 3 + v ε + ξ ε t v ε = a 1 u ε + a 2 v ε admits a sequence of local solutions (u ε, v ε ), converging in probability to a limit (u, v) as ε 0.

6 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Main result Mollified noise: ξ ε = ϱ ε ξ where ϱ ε (t, x) = 1 ϱ ( t, x ) ε d+2 ε 2 ε with ϱ compactly supported, integral 1 Theorem [NB & C. Kuehn, Elec J Probab 21 (18):1-48 (2016)] There exists a choice of renormalisation constant C(ε), lim ε 0 C(ε) =, such that t u ε = u ε + [1 + C(ε)]u ε (u ε ) 3 + v ε + ξ ε t v ε = a 1 u ε + a 2 v ε admits a sequence of local solutions (u ε, v ε ), converging in probability to a limit (u, v) as ε 0. Local solution means up to a random possible explosion time Initial conditions should be in appropriate Hölder spaces C(ε) log(ε 1 ) for d = 2 and C(ε) ε 1 for d = 3 Similar results for more general cubic nonlinearity and v R n

7 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 2. Mild solutions of SPDE t u = u + F (u) + ξ u(0, x) = u 0 (x)

8 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 2. Mild solutions of SPDE t u = u + F (u) + ξ u(0, x) = u 0 (x) Construction of mild solution via Duhamel formula: t u = u u(t, x) = G(t, x y)u 0 (y) dy =: (e t u 0 )(x) where G(t, x): heat kernel (compatible with bc)

9 2. Mild solutions of SPDE Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 t u = u + F (u) + ξ u(0, x) = u 0 (x) Construction of mild solution via Duhamel formula: t u = u u(t, x) = G(t, x y)u 0 (y) dy =: (e t u 0 )(x) where G(t, x): heat kernel (compatible with bc) t u = u + f u(t, x) = (e t u 0 )(x) + Notation: u = Gu 0 + G f t 0 e (t s) f (s, )(x) ds

10 2. Mild solutions of SPDE Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 t u = u + F (u) + ξ u(0, x) = u 0 (x) Construction of mild solution via Duhamel formula: t u = u u(t, x) = G(t, x y)u 0 (y) dy =: (e t u 0 )(x) where G(t, x): heat kernel (compatible with bc) t u = u + f u(t, x) = (e t u 0 )(x) + Notation: u = Gu 0 + G f t 0 e (t s) f (s, )(x) ds t u = u + ξ u = Gu 0 + G ξ (stochastic convolution)

11 2. Mild solutions of SPDE Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 t u = u + F (u) + ξ u(0, x) = u 0 (x) Construction of mild solution via Duhamel formula: t u = u u(t, x) = G(t, x y)u 0 (y) dy =: (e t u 0 )(x) where G(t, x): heat kernel (compatible with bc) t u = u + f u(t, x) = (e t u 0 )(x) + Notation: u = Gu 0 + G f t 0 e (t s) f (s, )(x) ds t u = u + ξ u = Gu 0 + G ξ (stochastic convolution) t u = u + ξ + F (u) u = Gu 0 + G [ξ + F (u)] Aim: use Banach s fixed-point theorem but which function space?

12 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Hölder spaces Definition of C α for f : I R, with I R a compact interval: 0 < α < 1: f (x) f (y) C x y α α > 1: f C α and f C α 1 α < 0: f distribution, f, η δ x Cδ α where η δ x(y) = 1 δ x y x y η( δ ) for all test functions η C α Property: f C α, 0 < α < 1 f C α 1 where f, η = f, η Remark: f C 1+α f (x) f (y) C x y 1+α. See e.g f (x) = x + x 3/2

13 Hölder spaces Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Definition of C α for f : I R, with I R a compact interval: 0 < α < 1: f (x) f (y) C x y α α > 1: f C α and f C α 1 α < 0: f distribution, f, η δ x Cδ α where η δ x(y) = 1 δ x y x y η( δ ) for all test functions η C α Property: f C α, 0 < α < 1 f C α 1 where f, η = f, η Remark: f C 1+α f (x) f (y) C x y 1+α. See e.g f (x) = x + x 3/2 Case of the heat kernel: ( t )u = f u = G f Parabolic scaling C α s : x y t s 1/2 + d i=1 x i y i Parabolic scaling Cs α : 1 x y δ η( δ ) 1 η( t s δ d+2 δ 2, x y δ )

14 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Schauder estimates and fixed-point equation Schauder estimate α / Z, f C α s G f C α+2 s Fact: in dimension d, space-time white noise ξ C α s a.s. α < d+2 2

15 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Schauder estimates and fixed-point equation Schauder estimate α / Z, f C α s G f C α+2 s Fact: in dimension d, space-time white noise ξ C α s a.s. α < d+2 2 Fixed-point equation: u = Gu 0 + G [ξ + F (u)] d = 1: ξ Cs 3/2 G ξ Cs 1/2 F (u) defined d = 3: ξ C 5/2 s G ξ C 1/2 s F (u) not defined d = 2: ξ C 2 s G ξ C 0 s F (u) not defined Boundary case, can be treated with Besov spaces [Da Prato & Debussche 2003] Why not use mollified noise? Limit ε 0 does not exist

16 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Regularity structures Basic idea of Martin Hairer [Inventiones Math. 198, , 2014]: Lift mollified fixed-point equation u = Gu 0 + G [ξ ε + F (u)] to a larger space called a Regularity structure M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) u ε S M u ε = S(u 0, ξ ε ): classical solution of mollified equation U = S(u 0, Z ε ): solution map in regularity structure S and R are continuous (in suitable topology)

17 Regularity structures Basic idea of Martin Hairer [Inventiones Math. 198, , 2014]: Lift mollified fixed-point equation u = Gu 0 + G [ξ ε + F (u)] to a larger space called a Regularity structure S M (u 0, MZ ε ) U M MΨ R M (u 0, ξ ε ) û ε S M u ε = S(u 0, ξ ε ): classical solution of mollified equation U = S(u 0, Z ε ): solution map in regularity structure S and R are continuous (in suitable topology) Renormalisation: modification of the lift Ψ Aternative approaches for d = 3: [Catellier & Chouk 13], [Kupiainen 15] Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18

18 Regularity structures Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18

19 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Basic idea: Generalised Taylor series f : I R, 0 < α < 1 f C 2+α f C 2 and f C α Associate with f the triple (f, f, f ) When does a triple (f 0, f 1, f 2 ) represent a function f C 2+α?

20 Basic idea: Generalised Taylor series f : I R, 0 < α < 1 f C 2+α f C 2 and f C α Associate with f the triple (f, f, f ) When does a triple (f 0, f 1, f 2 ) represent a function f C 2+α? When there is a constant C such that for all x, y I f 0 (y) f 0 (x) (y x)f 1 (x) 1 2 (y x)2 f 2 (x) C x y 2+α f 1 (y) f 1 (x) (y x)f 2 (x) C x y 1+α f 2 (y) f 2 (x)) C x y α Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18

21 Basic idea: Generalised Taylor series f : I R, 0 < α < 1 f C 2+α f C 2 and f C α Associate with f the triple (f, f, f ) When does a triple (f 0, f 1, f 2 ) represent a function f C 2+α? When there is a constant C such that for all x, y I f 0 (y) f 0 (x) (y x)f 1 (x) 1 2 (y x)2 f 2 (x) C x y 2+α f 1 (y) f 1 (x) (y x)f 2 (x) C x y 1+α f 2 (y) f 2 (x)) C x y α Notation: f = f f 1 X + f 2 X 2 Regularity structure: Generalised Taylor basis whose basis elements can also be singular distributions Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18

22 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Definition of a regularity structure Definition [M. Hairer, Inventiones Math 2014] A Regularity structure is a triple (A, T, G) where 1. Index set: A R, bdd below, locally finite, 0 A 2. Model space: T = T α, each T α Banach space, T 0 = span(1) R α A 3. Structure group: G group of linear maps Γ : T T such that Γτ τ T β τ T α β<α and Γ1 = 1 Γ G.

23 Definition of a regularity structure Definition [M. Hairer, Inventiones Math 2014] A Regularity structure is a triple (A, T, G) where 1. Index set: A R, bdd below, locally finite, 0 A 2. Model space: T = α A T α, each T α Banach space, T 0 = span(1) R 3. Structure group: G group of linear maps Γ : T T such that and Γ1 = 1 Γ G. Polynomial regularity structure on R: A = N 0 T k R, T k = span(x k ) Γ h (X k ) = (X h) k h R Γτ τ β<α T β τ T α Polynomial reg. structure on R d : X k = X k X k d d T k, k = d i=1 k i Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18

24 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Regularity structure for t u = u u 3 + ξ New symbols: Ξ, representing ξ, Hölder exponent Ξ s = α 0 = d+2 2 κ New symbols: I(τ), representing G f, Hölder exponent I(τ) s = τ s + 2 New symbols: τσ, Hölder exponent τσ s = τ s + σ s

25 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Regularity structure for t u = u u 3 + ξ New symbols: Ξ, representing ξ, Hölder exponent Ξ s = α 0 = d+2 2 κ New symbols: I(τ), representing G f, Hölder exponent I(τ) s = τ s + 2 New symbols: τσ, Hölder exponent τσ s = τ s + σ s τ Symbol τ s d = 3 d = 2 Ξ Ξ α κ 2 κ I(Ξ) 3 3α κ 0 3κ I(Ξ) 2 2α κ 0 2κ I(I(Ξ) 3 )I(Ξ) 2 5α κ 2 5κ I(Ξ) α κ 0 κ I(I(Ξ) 3 )I(Ξ) 4α κ 2 4κ I(I(Ξ) 2 )I(Ξ) 2 4α κ 2 4κ I(Ξ) 2 X i X i 2α κ 1 2κ I(I(Ξ) 3 1 ) 3α κ 2 3κ

26 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Fixed-point equation for t u = u u 3 + ξ u = G [ξ ε u 3 ] + Gu 0 U = I(Ξ U 3 ) + ϕ U 0 = 0 U 1 = + ϕ1 U 2 = + ϕ1 3ϕ +...

27 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Fixed-point equation for t u = u u 3 + ξ u = G [ξ ε u 3 ] + Gu 0 U = I(Ξ U 3 ) + ϕ U 0 = 0 U 1 = To prove convergence, we need + ϕ1 U 2 = + ϕ1 3ϕ +... A model (Π, Γ): z R d+1, Π z τ is distribution describing τ near z Γ z z G describes translations: Π z = Π z Γ z z Spaces{ of modelled distributions D γ = f : R d+1 } T β : f (z) Γ z z f ( z) β z z γ β s β<γ equipped with a seminorm The Reconstruction theorem: provides a unique map R : D γ C α s (α = inf A) s.t. Rf Π z f (z), η δ s,z δ γ (constructed using wavelets)

28 Canonical model Z ε = (Π ε, Γ ε ) Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Defined inductively by (Π ε zξ)( z) = ξ ε ( z) (Π ε zx k )( z) = ( z z) k (Π ε zτσ)( z) = (Π ε zτ)( z)(π ε zσ)( z) (Π ε zi(τ))( z) = G( z z )(Π ε zτ)(z ) dz polynomial term

29 Canonical model Z ε = (Π ε, Γ ε ) Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Defined inductively by (Π ε zξ)( z) = ξ ε ( z) (Π ε zx k )( z) = ( z z) k (Π ε zτσ)( z) = (Π ε zτ)( z)(π ε zσ)( z) (Π ε zi(τ))( z) = G( z z )(Π ε zτ)(z ) dz polynomial term

30 Canonical model Z ε = (Π ε, Γ ε ) Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Defined inductively by (Π ε zξ)( z) = ξ ε ( z) (Π ε zx k )( z) = ( z z) k (Π ε zτσ)( z) = (Π ε zτ)( z)(π ε zσ)( z) (Π ε zi(τ))( z) = G( z z )(Π ε zτ)(z ) dz polynomial term

31 Canonical model Z ε = (Π ε, Γ ε ) Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Defined inductively by (Π ε zξ)( z) = ξ ε ( z) (Π ε zx k )( z) = ( z z) k (Π ε zτσ)( z) = (Π ε zτ)( z)(π ε zσ)( z) (Π ε zi(τ))( z) = G( z z )(Π ε zτ)(z ) dz polynomial term Then K s.t. RKf = G Rf and the following diagrams commute: D γ K D γ+2 (u 0, Z ε S ) U R R R R Cs α Cs α +2 (u 0, ξ ε ) u ε G S where α = inf A and Kf = If + polynomial term + nonlocal term

32 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Why do we need to renormalise? Let G ε = G ϱ ε where ϱ ε is the mollifier (Π ε z )(z) = (G ξ ε )(z) = (G ε ξ)(z) = G ε (z z 1 )ξ(z 1 ) dz 1 belongs to first Wiener chaos, limit ε 0 well-defined

33 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Why do we need to renormalise? Let G ε = G ϱ ε where ϱ ε is the mollifier (Π ε z )(z) = (G ξ ε )(z) = (G ε ξ)(z) = G ε (z z 1 )ξ(z 1 ) dz 1 belongs to first Wiener chaos, limit ε 0 well-defined (Π ε z )(z) = (G ξ ε )(z) 2 = G ε (z z 1 )G ε (z z 2 )ξ(z 1 )ξ(z 2 ) dz 1 dz 2 diverges as ε 0

34 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Why do we need to renormalise? Let G ε = G ϱ ε where ϱ ε is the mollifier (Π ε z )(z) = (G ξ ε )(z) = (G ε ξ)(z) = G ε (z z 1 )ξ(z 1 ) dz 1 belongs to first Wiener chaos, limit ε 0 well-defined (Π ε z )(z) = (G ξ ε )(z) 2 = G ε (z z 1 )G ε (z z 2 )ξ(z 1 )ξ(z 2 ) dz 1 dz 2 diverges as ε 0 Wick product: ξ(z 1 ) ξ(z 2 ) = ξ(z 1 )ξ(z 2 ) δ(z 1 z 2 ) (Π ε z )(z) = G ε (z z 1 )G ε (z z 2 )ξ(z 1 ) ξ(z 2 ) dz 1 dz 2 in 2nd Wiener chaos, bdd Renormalised model: ( Π ε z )(z) = (Π ε z )(z) C 1 (ε) + G ε (z z 1 ) 2 dz 1 C 1(ε)

35 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 3. The case of the FitzHugh Nagumo equations Fixed-point equation Lifted version u(t, x) = G [ξ ε + u u 3 + v](t, x) + Gu 0 (t, x) v(t, x) = t 0 u(s, x) e (t s)a 2 a 1 ds + e ta 2 v 0 U = I[Ξ + U U 3 + V ] + Gu 0 V = EU + Qv 0 where E is an integration map which is not regularising in space New symbols E(I(Ξ)) =, etc... We expect U, and thus also V to be α-hölder for α < 1 2 Thus I(U U 3 + V ) should be well-defined The standard theory has to be extended, because E does not correspond to a smooth kernel

36 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Details on implementing E Problems: Fixed-point equation requires diagonal identity (Π t,x τ)(t, x) = 0 Usual definition of K would contain Taylor series J (z)τ = X k D k G(z z)(π z τ)(d z) k! k s<α N f (z) = X k D k G(z z)(rf Π z f (z))(d z) k! k s<γ

37 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Details on implementing E Problems: Fixed-point equation requires diagonal identity (Π t,x τ)(t, x) = 0 Usual definition of K would contain Taylor series J (z)τ = X k D k G(z z)(π z τ)(d z) k! k s<α N f (z) = X k D k G(z z)(rf Π z f (z))(d z) k! Solution: k s<γ Define ΠEτ only if 2 < τ s < 0 (otherwise Eτ = 0) J (z)τ = 0 Define K only for f = τ s<0 c τ τ + τ s 0 c τ (t, x)τ =: f + f + can take Rf = Π t,x f (t, x) and thus N f = 0 for these f Time-convolution with Q lifted to (K Q f )(t, x) = c τ Eτ+ Q(t s)c τ (s, x) ds τ =: (Ef +Qf + )(t, x) τ s<0 τ s 0 Conclusion

38 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Fixed-point equation Consider t u = u + F (u, v) + ξ with F a polynomial of degree 3 If (U, V ) satisfies fixed-point equation U = I[Ξ + F (U, V )] + Gu 0 + polynomial term V = EU + QU + + Qv 0 then (RU, RV ) is solution, provided RF (U, V ) = F (RU, RV )

39 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Fixed-point equation Consider t u = u + F (u, v) + ξ with F a polynomial of degree 3 If (U, V ) satisfies fixed-point equation U = I[Ξ + F (U, V )] + Gu 0 + polynomial term V = EU + QU + + Qv 0 then (RU, RV ) is solution, provided RF (U, V ) = F (RU, RV ) Fixed point is of the form U = + ϕ1 + [ a 1 + a 2 + a 3 + a 4 ] + [ b1 + b 2 + b 3 ] +... V = + ψ1 + [ â 1 + â 2 + â 3 + â 4 ] + [ˆb1 + ˆb 2 + ˆb 3 ] +...

40 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Fixed-point equation Consider t u = u + F (u, v) + ξ with F a polynomial of degree 3 If (U, V ) satisfies fixed-point equation U = I[Ξ + F (U, V )] + Gu 0 + polynomial term V = EU + QU + + Qv 0 then (RU, RV ) is solution, provided RF (U, V ) = F (RU, RV ) Fixed point is of the form U = + ϕ1 + [ a 1 + a 2 + a 3 + a 4 ] + [ b1 + b 2 + b 3 ] +... V = + ψ1 + [ â 1 + â 2 + â 3 + â 4 ] + [ˆb1 + ˆb 2 + ˆb 3 ] +... Prove existence of fixed point in (modification of) D γ with γ = 1 + κ Extend from small interval [0, T ] up to first exit from large ball Deal with renormalisation procedure Conclusion

41 Renormalisation Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Renormalisation group: group of linear maps M : T T Associated model: Π M z s.t. Π M τ = ΠMτ where Π z = ΠΓ fz Allen Cahn eq.: M = e C 1L 1 C 2 L 2 with L 1 : 1, L 2 : 1 FHN eq.: the same group suffices because Q is smoothing

42 Renormalisation Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Renormalisation group: group of linear maps M : T T Associated model: Π M z s.t. Π M τ = ΠMτ where Π z = ΠΓ fz Allen Cahn eq.: M = e C 1L 1 C 2 L 2 with L 1 : 1, L 2 : 1 FHN eq.: the same group suffices because Q is smoothing Look for r.v. Π z τ s.t. if Π (ε) z E Π z τ, η λ z 2 λ 2 τ s+κ = (Π (ε) z ) Mε then κ, θ > 0 s.t. E Π z τ (ε) Π z τ, ηz λ 2 ε 2θ λ 2 τ s+κ Then ( Π (ε) (ε) z, Γ z ) converges to limiting model, with explicit L p bounds

43 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Renormalisation Renormalisation group: group of linear maps M : T T Associated model: Π M z s.t. Π M τ = ΠMτ where Π z = ΠΓ fz Allen Cahn eq.: M = e C 1L 1 C 2 L 2 with L 1 : 1, L 2 : 1 FHN eq.: the same group suffices because Q is smoothing Look for r.v. Π z τ s.t. if Π (ε) z E Π z τ, η λ z 2 λ 2 τ s+κ = (Π (ε) z ) Mε then κ, θ > 0 s.t. E Π z τ (ε) Π z τ, ηz λ 2 ε 2θ λ 2 τ s+κ Then ( Π (ε) (ε) z, Γ z ) converges to limiting model, with explicit L p bounds Renormalised equations have nonlinearity F s.t. F (MU, MV ) = MF (U, V ) + terms of Hölder exponent > 0 FHN eq. with cubic nonlinearity F = α 1 u + α 2 v + β 1 u 2 + β 2 uv + β 3 v 2 + γ 1 u 3 + γ 2 u 2 v + γ 3 uv 2 + γ 4 v 3 F (u, v) = F (u, v) c 0 (ε) c 1 (ε)u c 2 (ε)v with the c i (ε) depending on C 1, C 2, provided either d = 2 or γ 2 = 0

44 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs 11 April /18 Concluding remarks Models with t u of order u 4 + v 4 and t v of order u 2 + v should be renormalisable Current approach does not work when singular part (t, x)-dependent Global existence: recent progress by J.-C. Mourrat and H. Weber on 2D and 3D Allen Cahn More quantitative results? References Martin Hairer, A theory of regularity structures, Invent. Math. 198 (2), pp (2014) Martin Hairer, Introduction to Regularity Structures, lecture notes (2013) Ajay Chandra, Hendrik Weber, Stochastic PDEs, regularity structures, and interacting particle systems, preprint arxiv/ N. B., Christian Kuehn, Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions, Elec J Probab 21 (18):1-48 (2016) N. B., Mayonnaise et élections américaines, Dossier Pour la Science (2016)