Regularity structures and renormalisation of FitzHughNagumo SPDEs - PowerPoint PPT Presentation

Eighth Workshop on Random Dynamical Systems Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions Nils Berglund MAPMO, Universit´ e d’Orl´ eans Bielefeld, 5 November 2015 with Christian Kuehn (TU Vienna) Nils Berglund nils.berglund@univ-orleans.fr http://www.univ-orleans.fr/mapmo/membres/berglund/

FitzHugh–Nagumo SDE d u t = [ u t − u 3 t + v t ] d t + σ d W t d v t = ε [ a − u t − bv t ] d t ⊲ u t : membrane potential of neuron ⊲ v t : gating variable (proportion of open ion channels) − u t v ε = 0 . 1 b = 0 1 a = 3 + 0 . 02 √ u t σ = 0 . 03 Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 1/15

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) Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 2/15

Main result Mollified noise: ξ ε = ̺ ε ∗ ξ � t � 1 ε 2 , x where ̺ ε ( t , x ) = ε d +2 ̺ with ̺ compactly supported, integral 1 ε Theorem [NB & C. Kuehn, preprint 2015, arXiv/1504.02953 ] 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 H¨ older 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 Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 3/15

Main result Mollified noise: ξ ε = ̺ ε ∗ ξ � t � 1 ε 2 , x where ̺ ε ( t , x ) = ε d +2 ̺ with ̺ compactly supported, integral 1 ε Theorem [NB & C. Kuehn, preprint 2015, arXiv/1504.02953 ] 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 H¨ older 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 Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 3/15

Mild solutions of SPDE ∂ t u = ∆ u + F ( u ) + ξ u (0 , x ) = u 0 ( x ) Construction of mild solution via Duhamel formula: � G ( t , x − y ) u 0 ( y ) d y = (e ∆ t u 0 )( x ) ⊲ ∂ t u = ∆ u ⇒ u ( t , x ) = where G ( t , x ): heat kernel (compatible with bc) � t u ( t , x ) = (e ∆ t u 0 )( x ) + e ∆( t − s ) f ( s , · )( x ) d s ⊲ ∂ t u = ∆ u + f ⇒ 0 Notation: u = Gu 0 + G ∗ f ⊲ ∂ 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? Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 4/15

Mild solutions of SPDE ∂ t u = ∆ u + F ( u ) + ξ u (0 , x ) = u 0 ( x ) Construction of mild solution via Duhamel formula: � G ( t , x − y ) u 0 ( y ) d y =: (e ∆ t u 0 )( x ) ⊲ ∂ t u = ∆ u ⇒ u ( t , x ) = where G ( t , x ): heat kernel (compatible with bc) � t u ( t , x ) = (e ∆ t u 0 )( x ) + e ∆( t − s ) f ( s , · )( x ) d s ⊲ ∂ t u = ∆ u + f ⇒ 0 Notation: u = Gu 0 + G ∗ f ⊲ ∂ 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? Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 4/15

Mild solutions of SPDE ∂ t u = ∆ u + F ( u ) + ξ u (0 , x ) = u 0 ( x ) Construction of mild solution via Duhamel formula: � G ( t , x − y ) u 0 ( y ) d y =: (e ∆ t u 0 )( x ) ⊲ ∂ t u = ∆ u ⇒ u ( t , x ) = where G ( t , x ): heat kernel (compatible with bc) � t u ( t , x ) = (e ∆ t u 0 )( x ) + e ∆( t − s ) f ( s , · )( x ) d s ⊲ ∂ t u = ∆ u + f ⇒ 0 Notation: u = Gu 0 + G ∗ f ⊲ ∂ 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? Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 4/15

Mild solutions of SPDE ∂ t u = ∆ u + F ( u ) + ξ u (0 , x ) = u 0 ( x ) Construction of mild solution via Duhamel formula: � G ( t , x − y ) u 0 ( y ) d y =: (e ∆ t u 0 )( x ) ⊲ ∂ t u = ∆ u ⇒ u ( t , x ) = where G ( t , x ): heat kernel (compatible with bc) � t u ( t , x ) = (e ∆ t u 0 )( x ) + e ∆( t − s ) f ( s , · )( x ) d s ⊲ ∂ t u = ∆ u + f ⇒ 0 Notation: u = Gu 0 + G ∗ f ⊲ ∂ 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? Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 4/15

Mild solutions of SPDE ∂ t u = ∆ u + F ( u ) + ξ u (0 , x ) = u 0 ( x ) Construction of mild solution via Duhamel formula: � G ( t , x − y ) u 0 ( y ) d y =: (e ∆ t u 0 )( x ) ⊲ ∂ t u = ∆ u ⇒ u ( t , x ) = where G ( t , x ): heat kernel (compatible with bc) � t u ( t , x ) = (e ∆ t u 0 )( x ) + e ∆( t − s ) f ( s , · )( x ) d s ⊲ ∂ t u = ∆ u + f ⇒ 0 Notation: u = Gu 0 + G ∗ f ⊲ ∂ 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? Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 4/15

H¨ older 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 ∀ x � = y ⊲ α > 1: f ∈ C ⌊ α ⌋ and f ′ ∈ C α − 1 ⊲ α < 0: f distribution, |� f , η δ x �| � C δ α x ( y ) = 1 δ η ( x − y where η δ δ ) for all test functions η ∈ C −⌊ α ⌋ f ′ ∈ C α − 1 where � f ′ , η � = −� f , η ′ � Property: f ∈ C α , 0 < α < 1 ⇒ 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 → | t − s | 1 / 2 + � d Parabolic scaling: | x − y | − i =1 | x i − y i | δ η ( x − y δ 2 , x − y Parabolic scaling: 1 δ d +2 η ( t − s 1 δ ) − → δ ) Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 5/15

H¨ older 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 ∀ x � = y ⊲ α > 1: f ∈ C ⌊ α ⌋ and f ′ ∈ C α − 1 ⊲ α < 0: f distribution, |� f , η δ x �| � C δ α x ( y ) = 1 δ η ( x − y where η δ δ ) for all test functions η ∈ C −⌊ α ⌋ f ′ ∈ C α − 1 where � f ′ , η � = −� f , η ′ � Property: f ∈ C α , 0 < α < 1 ⇒ 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 → | t − s | 1 / 2 + � d Parabolic scaling C α s : | x − y | − i =1 | x i − y i | δ η ( x − y δ 2 , x − y Parabolic scaling C α s : 1 δ d +2 η ( t − s 1 δ ) − → δ ) Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 5/15

Schauder estimates and fixed-point equation Schauder estimate ∈ Z , f ∈ C α G ∗ f ∈ C α +2 ⇒ α / s s s a.s. ∀ α < − d +2 Fact : in dimension d , space-time white noise ξ ∈ C α 2 Fixed-point equation: u = Gu 0 + G ∗ [ ξ + F ( u )] ⊲ d = 1: ξ ∈ C − 3 / 2 − ⇒ G ∗ ξ ∈ C 1 / 2 − ⇒ F ( u ) defined s s ⊲ d = 3: ξ ∈ C − 5 / 2 − ⇒ G ∗ ξ ∈ C − 1 / 2 − ⇒ F ( u ) not defined s s ⊲ d = 2: ξ ∈ C − 2 − ⇒ G ∗ ξ ∈ C 0 − ⇒ F ( u ) not defined s s Boundary case, can be treated with Besov spaces [Da Prato and Debussche 2003] Why not use mollified noise? Limit ε → 0 does not exist Regularity structures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions 5 November 2015 6/15