Structures de regularité. de FitzHugh Nagumo
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- Lenard Horatio Morris
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1 Nils Berglund Strasbourg, Séminaire Calcul Stochastique Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo Nils Berglund MAPMO, Université d Orléans 13 novembre 2015 avec Christian Kuehn (TU Vienne)
2 Plan Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18 1. Motivation and main result 2. Introduction to regularity structures 3. Extension to FitzHugh Nagumo
3 FitzHugh Nagumo SDE Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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, preprint 2015, arxiv/ ] 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
6 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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, preprint 2015, arxiv/ ] 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 Mild solutions of SPDE Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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?
8 Mild solutions of SPDE Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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?
9 Mild solutions of SPDE Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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?
10 Mild solutions of SPDE Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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?
11 Mild solutions of SPDE Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 Hölder spaces Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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: x y t s 1/2 + d i=1 x i y i Parabolic scaling: 1 x y δ η( δ ) 1 η( t s δ d+2 δ 2, x y δ )
13 Hölder spaces Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 C 3/2 s G ξ C 1/2 s 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 and Debussche 2003] Why not use mollified noise? Limit ε 0 does not exist
15 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 C 3/2 s G ξ C 1/2 s 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 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) Renormalisation: modification of the lift Ψ Aternative approaches for d = 3: [Catellier & Chouk], [Kupiainen] Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18
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] Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18
18 Regularity structures Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18
19 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 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18
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 Notation: f = f f 1 X + f 2 X 2 Regularity structure: Generalised Taylor basis whose basis elements can also be singular distributions Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18
22 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. Γτ τ β<α T β τ T α 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 Polynomial reg. structure on R d : X k = X k X k d d T k, k = d i=1 k i Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18
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. Γτ τ β<α T β τ T α 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 Polynomial reg. structure on R d : X k = X k X k d d T k, k = d i=1 k i Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18
24 Regularity structure for t u = u u 3 + ξ Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18 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 structure for t u = u u 3 + ξ Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18 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 Fixed-point equation for t u = u u 3 + ξ Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18 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 is distribution describing U near z Γ z z G describes translations: Π z = Π z Γ z z Spaces{ 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)
27 Fixed-point equation for t u = u u 3 + ξ Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18 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 ) Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 RKf = G Rf and the following diagrams commute: (u 0, Z ε S ) U (u 0, Z ε S ) U R R R R (u 0, ξ ε ) u ε (u 0, ξ ε ) u ε S S where Kf = If + polynomial term + nonlocal term
29 Canonical model Z ε = (Π ε, Γ ε ) Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 RKf = G Rf and the following diagrams commute: (u 0, Z ε S ) U (u 0, Z ε S ) U R R R R (u 0, ξ ε ) u ε (u 0, ξ ε ) u ε S S where Kf = If + polynomial term + nonlocal term
30 Canonical model Z ε = (Π ε, Γ ε ) Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 RKf = G Rf and the following diagrams commute: (u 0, Z ε S ) U (u 0, Z ε S ) U R R R R (u 0, ξ ε ) u ε (u 0, ξ ε ) u ε S S where Kf = If + polynomial term + nonlocal term
31 Canonical model Z ε = (Π ε, Γ ε ) Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 Why do we need to renormalise? Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18 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 + G ε (z z 1 ) 2 dz 1 in 2nd Wiener chaos, bdd Renormalised model: ( Π ε z)(z) = (Π z )(z) C 1 (ε) C 1(ε)
33 Why do we need to renormalise? Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18 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 + G ε (z z 1 ) 2 dz 1 in 2nd Wiener chaos, bdd Renormalised model: ( Π ε z)(z) = (Π z )(z) C 1 (ε) C 1(ε)
34 Why do we need to renormalise? Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18 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 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /18 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 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 Eτ 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 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 τ s<0 τ s 0 Conclusion
37 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 c s<0 τ τ + τ c s 0 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 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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
39 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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
40 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 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 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 Conclusion
42 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 Conclusion
43 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 Conclusion
44 Structures de regularité et renormalisation d EDPS de FitzHugh Nagumo 13 novembre /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 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, preprint arxiv/
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