Sharp estimates on metastable lifetimes for one- and two-dimensional Allen Cahn SPDEs

Nils Berglund nils.berglund@univ-orleans.fr http://www.univ-orleans.fr/mapmo/membres/berglund/ COSMOS Workshop, École des Ponts ParisTech Sharp estimates on metastable lifetimes for one- and two-dimensional Allen Cahn SPDEs Nils Berglund MAPMO, Université d Orléans Paris, 2 February 2016 Joint works with Barbara Gentz (Bielefeld), and with Giacomo Di Gesu (Paris) and Hendrik Weber (Warwick)

Metastability for finite-dimensional SDEs dx t = V (x t ) dt + 2ε dw t Mont Puget V R d R confining potential τ x y = inf{t > 0 x t B ε (y)} first-hitting time of small ball B ε (y), when starting in x Arrhenius law (1889): E[τy x [V (z) V (x)]/ε ] e uminy x Col de Sugiton z Eyring Kramers law (1935, 1940): Eigenvalues of Hessian of V at minimum x: 0 < ν 1 ν 2 ν d Eigenvalues of Hessian of V at saddle z: λ 1 < 0 < λ 2 λ d E[τy x ] = 2π λ 2...λ d λ 1 ν 1...ν d e [V (z) V (x)]/ε [1 + Oε(1)] Calanque de Sugiton y Arrhenius law: proved by Freidlin, Wentzell (1979) using large deviations Eyring Kramers law: Bovier, Eckhoff, Gayrard, Klein (2004) using potential theory, Helffer, Klein, Nier (2004) using Witten aplacian,... Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 1/17(19)

Potential-theoretic proof Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 2/17(19) Magic formula: for A, B R d disjoint sets, E ν A,B [τ B ] = 1 cap A (B) B c h A,B(y) e V (y)/ε dy Equilibrium measure: ν A,B proba measure concentrated on A Committor function: h A,B (y) = P y {τ A < τ B } Capacity: cap A (B) = ε (A B) c h A,B(y) 2 e V (y)/ε dy Property: cap A (B) = ε inf h H 1,h A =1,h B =0 (A B) h(y) 2 e V (y)/ε dy c Proving Eyring Kramers formula: A and B small sets around x and y Variational arguments: cap A (B) ε λ1 2πε (2πε) d 1 λ 2...λ d (2πε) d V (z)/ε e aplace asymptotics: B c h A,B (y) e V (y)/ε dy ν 1...ν d Use Harnack inequalities to show that E ν A,B [τ B ] E x [τ B ] V (x)/ε e

Deterministic Allen Cahn PDE Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 3/17(19) [Chafee & Infante 74, Allen & Cahn 75] x [0, ] u(x, t) R t u(x, t) = xx u(x, t) + f (u(x, t)) Either periodic or zero-flux Neumann boundary conditions In this talk: f (u) = u u 3 (results more general) Energy function: V [u] = [1 0 2 u (x) 2 1 2 u(x)2 + 1 4 u(x)4 ] dx min Scaling limit of particle system with potential V (y) = N2 2 2 N i=1 (y i+1 y i ) 2 + N i=1 [ 1 2 y i 2 + 1 4 y i 4] Stationary solutions: u 0 (x) = u 0(x) + u 0 (x) 3 critical points of V

Stationary solutions Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 4/17(19) u 0 (x) = f (u 0(x)) = u 0 (x) + u 0 (x) 3 u ± (x) ±1 u 0 (x) 0 Nonconstant solutions satisfying b.c. (expressible in terms of Jacobi elliptic fcts) H = 1 2 (u ) 2 + 1 2 u2 1 4 u4 1 1 u H = 1 4 u

Stability of stationary solutions u 0 (x) = u 0(x) + u 0 (x) 3 Variational eq around u 0 : t v t (x) = v t (x) + [1 3u 0 (x) 2 ]v t (x) Sturm iouville spectrum of RHS determines stability of u 0 u ± ±1: always stable (global minima of V ) u 0 0: always unstable, eigenvalues λ k = ( βkπ )2 1 (Neumann b.c.: β = 1, periodic b.c.: β = 2) 0 Neumann b.c.: Number of positive eigenvalues (= unstable directions) Transition state 1 2 3 1 2 3 4 0 π 2π 3π 4π 1 2 3 0 Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 5/17(19)

Coarsening dynamics Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 6/17(19) [Carr & Pego 89, Chen 04] (ink to simulation)

Stochastic partial differential equations Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 7/17(19) u t (x) = u t (x) + f (u t (x)) + 2ε Ẅtx ( xx, f (u) = u u 3 ) Ẅ tx : space time white noise (formal derivative of Brownian sheet) Construction of mild solution via Duhamel formula: u t = u t u t = e t u 0 where e t cos( kπx ) = e (kπ/)2t cos( kπx ) u t = u t + 2ε Ẅtx u t = e t u 0 + 2ε w t (x) = k t (t s) 0 e (kπ/)2 dw s (k) u t = u t + 2ε Ẅ tx + f (u t ) t 0 e (t s) Ẇ x (ds) = w t(x) cos( kπx ) Hs C α s, α < 1 2 u t = e t u 0 + 2ε w t + 0 e (t s) f (u s ) ds =: F t [u] Existence and a.s. uniqueness [Faris & Jona-asinio 1982] t

Stochastic partial differential equations Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 8/17(19) u t (x) = u t (x) + f (u t (x)) + 2ε Ẅ tx ( xx, f (u) = u u 3 ) Fourier variables: u t (x) = 1 z k (t) e i πkx/ k= dz k = λ k z k dt 1 z k1 z k2 z k3 dt + 2ε dw (k) t k 1 +k 2 +k 3 =k Itō SDE, dw (k) t : independent Wiener processes λ k = 1 + (πk/) 2 Energy functional: V [u] = 1 λ k z k 2 + 1 z k1 z k2 z k3 z k4 2 k= 4 k 1 +k 2 +k 3 +k 4 =0 dz t = V (z t ) dt + 2ε dw t

Coarsening dynamics with noise Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 9/17(19) (ink to simulation)

Eyring Kramers law for 1D SPDEs: heuristics Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 10/17(19) u t (x) = u t (x) + f (u t (x)) + 2ε Ẅ tx ( xx, f (u) = u u 3 ) Initial condition: u in near u 1 Target: u + 1, τ + = inf{t > 0 u t u + < ρ} with eigenvalues ν k Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.) u 0 (x) 0 if βπ with ev λ k =( βkπ u ts (x) = u 1 (x) β-kink stationary sol. if > βπ with ev λ k [Faris & Jona-asinio 82]: large-deviation principle Arrhenius law: E u in (V [uts] V [u ])/ε [τ + ] e [Maier & Stein 01]: formal computation; for Neumann b.c. E u in [τ + ] = 2π Rigorous proof? 1 λ 0 ν 0 k=1 λ k (V [uts] V [u ])/ε ν k e What happens when π as then λ 1 0? )2 1

Eyring Kramers law for 1D SPDEs: heuristics Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 10/17(19) u t (x) = u t (x) + f (u t (x)) + 2ε Ẅ tx ( xx, f (u) = u u 3 ) Initial condition: u in near u 1 Target: u + 1, τ + = inf{t > 0 u t u + < ρ} with eigenvalues ν k Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.) u 0 (x) 0 if βπ with ev λ k =( βkπ u ts (x) = u 1 (x) β-kink stationary sol. if > βπ with ev λ k [Faris & Jona-asinio 82]: large-deviation principle Arrhenius law: E u in (V [uts] V [u ])/ε [τ + ] e [Maier & Stein 01]: formal computation; for Neumann b.c. E u in [τ + ] 2π Rigorous proof? 1 λ 0 ν 0 k=1 λ k (V [uts] V [u ])/ε ν k e What happens when π as then λ 1 0? )2 1

Eyring Kramers law for 1D SPDEs: heuristics Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 10/17(19) u t (x) = u t (x) + f (u t (x)) + 2ε Ẅ tx ( xx, f (u) = u u 3 ) Initial condition: u in near u 1 Target: u + 1, τ + = inf{t > 0 u t u + < ρ} with eigenvalues ν k Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.) u 0 (x) 0 if βπ with ev λ k =( βkπ u ts (x) = u 1 (x) β-kink stationary sol. if > βπ with ev λ k [Faris & Jona-asinio 82]: large-deviation principle Arrhenius law: E u in (V [uts] V [u ])/ε [τ + ] e [Maier & Stein 01]: formal computation; for Neumann b.c. E u in [τ + ] 2π Rigorous proof? 1 λ 0 ν 0 k=1 λ k (V [uts] V [u ])/ε ν k e What happens when βπ as then λ 1 0? )2 1

Eyring Kramers law for 1D SPDEs: main result Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 11/17(19) Theorem: Neumann b.c. [B & Gentz, Elec J Proba 2013] If < π c with c > 0, then E u in [τ + ] = 2π 1 λ k e (V [uts] V [u ])/ε [1 + O(ε 1/2 log ε 3/2 )] λ 0 ν 0 k=1 ν k If > π + c, then same formula with extra factor 1 2 (since 2 saddles) and λ k instead of λ k If π c π, then E u in [τ + ] = 2π λ 1 + 3ε/2 λ k λ 0 ν 0 ν 1 k=2 ν k e (V [uts] V [u ])/ε Ψ + (λ 1 / [1 + R(ε)] 3ε/2) where Ψ + explicit, involves Bessel function K 1/4, lim α Ψ + (α) = 1 If π π + c, then same formula, with another function Ψ, involving Bessel functions I ±1/4, lim α Ψ (α) = 2

Eyring Kramers law for 1D SPDEs: comments Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 12/17(19) Periodic b.c.: similar result [B & Gentz, Elec J Proba 2013] For > 2π: extra factor ε because saddle is a whole curve Proof: relies on spectral Galerkin approximation Details Similar results by F. Barret [Annales IHP, 2015] using different method (away from bifurcation points) For Neumann b.c. and < π: spectral determinant in prefactor is explicitly computable (Euler product formulas) 1 λ 0 ν 0 λ k = 1 k=1 ν k 2 ( kπ )2 1 = 1 k=1 ( kπ )2 + 2 2 1 ( kπ )2 k=1 2 + ( = kπ )2 Similar expression for periodic b.c. and < 2π sin() 2 sinh( 2) For larger, techniques for Feynman path integrals allow to compute the spectral determinants in prefactors [Maier & Stein]

Eyring Kramers law for 1D SPDEs: comments Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 12/17(19) Periodic b.c.: similar result [B & Gentz, Elec J Proba 2013] For > 2π: extra factor ε because saddle is a whole curve Proof: relies on spectral Galerkin approximation Details Similar results by F. Barret [Annales IHP, 2015] using different method (away from bifurcation points) For Neumann b.c. and < π: spectral determinant in prefactor is explicitly computable (Euler product formulas) 1 λ 0 ν 0 λ k = 1 k=1 ν k 2 ( kπ )2 1 = 1 k=1 ( kπ )2 + 2 2 k=1 1 ( kπ )2 Similar expression for periodic b.c. and < 2π 1 + 2( kπ )2 = sin() 2 sinh( 2) For larger, techniques for Feynman path integrals allow to compute the spectral determinants in prefactors [Maier & Stein]

Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 13/17(19) The two-dimensional case (ink to simulation)

Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 14/17(19) The two-dimensional case arge-deviation principle: [Hairer & Weber, Ann. Fac. Sc. Toulouse, 2015] Naive computation of prefactor fails: log k (N 2 ) 1 ( k π )2 1 + 2( k π )2 log(1 32 k (N 2 ) k 2 π ) 2 k (N 2 ) 32 k 2 π 2 32 π 2 1 r dr r 2 = In fact, the equation needs to be renormalised Theorem: [Da Prato & Debussche 2003] et ξ δ be a mollification on scale δ of white noise. Then t u = u + [1 + εc(δ)]u u 3 + 2εξ δ with C(δ) log(δ 1 ) admits local solution converging as δ 0. (Global version: [Mourrat & Weber 2015])

Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 15/17(19) Renormalisation Problem: For d = 2, stoch. convolution t 0 e (t s) Ẇ x (ds) is a distribution δ-mollification should be equivalent to Galerkin approx. k N = δ 1 : w N (x, t) = a k (t) 1 Ωk x ei a k N t k = 0 e µ k(t s) dw s (k) µ k = (Ω k ) 2 Ω = βπ/ lim t t 0 e( 1l)(t s) Ẇ x (ds) = φ N is a Gaussian free field, s.t. 2 C N = 2 Eφ 2 N = E φ N 2 1 = 2 2(µ k +1) = Tr(P N[ +1l] 1 ) 2 log(n) k N Wick powers φ 2 N = φ2 N C N φ 3 N = φ3 N 3C Nφ N φ 4 N = φ4 N 6C Nφ 2 N + 3C 2 N have zero mean and uniformly bounded variance

Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 16/17(19) Computation of the prefactor Work in progress with Giacomo Di Gesu and Hendrik Weber Consider for simplicity < βπ transition state in 0 Galerkin-truncated renormalised potential V N = 1 2 T 2[u N (x)2 u N (x) 2 ] dx + 1 4 T 2 u N(x) 4 dx Using Nelson estimate: cap A (B) λ0 ε 2π Symmetry argument: 2πε λ k 0< k N h A,B(z) e VN(z)/ε dz = 1 B c 2 e VN(z)/ε dz = 1 2 Z N(ε) Z N (ε) 2 2πε ν k e VN(,0)/ε where V N (, 0) = 1 4 2 + 3 2 2 C N ε k N Prefactor proportionnal to (since ν k = λ k + 3) 0< k N λ k λ k +3 e3/λ k converges since log[ λ k λ k +3 e3/λ k ] = O( 1 k 4 )

Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 17/17(19) Outlook Dim d = 3: more difficult because 2 renormalisation constants needed Potentials with more than two wells: essentially understood Potentials invariant under symmetry group: done for SDEs [5,6] Mainly open: non-gradient systems But see poster by Manon Baudel References: For this talk: [1,2,3]; Overview: [4] 1. N. B. and Barbara Gentz, Anomalous behavior of the Kramers rate at bifurcations in classical field theories, J. Phys. A: Math. Theor. 42, 052001 (2009) 2., The Eyring Kramers law for potentials with nonquadratic saddles, Markov Processes Relat. Fields 16, 549 598 (2010) 3., Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers law and beyond, Electronic J. Probability 18, (24):1 58 (2013) 4. N. B., Kramers law: Validity, derivations and generalisations, Markov Processes Relat. Fields 19, 459 490 (2013) 5. N. B. and Sébastien Dutercq, The Eyring Kramers law for Markovian jump processes with symmetries, J. Theoretical Probability, First Online (2015) 6., Interface dynamics of a metastable mass-conserving spatially extended diffusion, J. Statist. Phys. 162, 334 370 (2016)

Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 18/17(19) Sketch of proof: Spectral Galerkin approximation u (d) t (x) = 1 d k= d z k(t) e i πkx/ dz t = V (z t ) dt + 2ε dw t Theorem [Blömker & Jentzen 13] For all γ (0, 1 2 ) there exists an a.s. finite r.v. Z Ω R + s.t. ω Ω sup u t (ω) u (d) t (ω) < Z(ω)d γ d N 0 t T Proposition (using potential theory) ε 0 > 0 ε < ε 0 d 0 (ε) < d d 0 ν d proba measure on B r (u ) Br (u ) Ev 0 [τ (d) + ]ν d (dv 0 ) = C(d, ε) e H(d)/ε [1 + R(ε)] Proposition (using large deviations and lots of other stuff) H 0 = V [u ts ] V [u ]. η > 0 ε 0, T 1, H 1 ε < ε 0 d 0 < s.t. d d 0 sup E v 0 [τ+] 2 T1 2 e 2(H0+η)/ε, sup v 0 B r (u ) sup d d 0 v 0 B r (u ) E v 0 [(τ (d) + ) 2 ] T 2 1 e 2H 1/ε

Main step of the proof Set T Kr = C(, ε) e H 0/ε et B = B ρ (u + ) and define nested sets B B B + at -distance δ Ω K,d = { sup t [0,KT Kr ] (d) P(Ω c K,d ) P{Z > δd γ } + Ev 0 [τ (d) B ] KT Kr On Ω K,d one has τ (d) B + E v (d) E v (d) 0 [τ (d) B + 0 [τ (d) B + τ B τ (d) B v t v (d) t δ, τ (d) B KT Kr } Cauchy Schwarz. 1 {ΩK,d }] E v 0 [τ B 1 {ΩK,d }] E v (d) ] E v (d) 0 [τ (d) B + lim sup P(Ω c K,d ) = M(ε) d K 0 [τ (d) 1 {Ω c K,d }] E v 0 [τ B ] E v (d) B 1 {ΩK,d }] 0 [τ (d) B ] + E v 0 [τ B 1 {Ω c K,d }] Integrate against ν d and use Cauchy Schwarz to bound error terms: E v 0 [τ B 1 {Ω c K,d }] E v 0 [τ 2 B ]P(Ω c K,d ), take d and finally K large Back Sharp estimates on metastable lifetimes for 1D and 2D Allen Cahn SPDEs 2 February 2016 19/17(19)