Metastability in a class of parabolic SPDEs

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1 Metastability in a class of parabolic SPDEs Nils Berglund MAPMO, Université d Orléans CNRS, UMR 6628 et Fédération Denis Poisson Collaborators: Florent Barret, Ecole Polytechnique, Palaiseau Bastien Fernandez, CPT, CNRS, Marseille Barbara Gentz, University of Bielefeld EPF-Lausanne, 12 novembre 2010

2 Metastability in physics Examples: Supercooled liquid Supersaturated gas Wrongly magnetised ferromagnet 1

3 Metastability in physics Examples: Supercooled liquid Supersaturated gas Wrongly magnetised ferromagnet Near first-order phase transition Nucleation implies crossing energy barrier Free energy Order parameter 1-a

4 Metastability in stochastic lattice models Lattice: Λ Z d Configuration space: X = S Λ, S finite set (e.g. { 1, 1}) Hamiltonian: H : X R (e.g. Ising or lattice gas) Gibbs measure: µ β (x) = e βh(x) /Z β Dynamics: Markov chain with invariant measure µ β (e.g. Metropolis: Glauber or Kawasaki) 2

5 Metastability in stochastic lattice models Lattice: Λ Z d Configuration space: X = S Λ, S finite set (e.g. { 1, 1}) Hamiltonian: H : X R (e.g. Ising or lattice gas) Gibbs measure: µ β (x) = e βh(x) /Z β Dynamics: Markov chain with invariant measure µ β (e.g. Metropolis: Glauber or Kawasaki) Results (for β 1) on Transition time between + and or empty and full configuration Transition path Shape of critical droplet Frank den Hollander, Metastability under stochastic dynamics, Stochastic Process. Appl. 114 (2004), Enzo Olivieri and Maria Eulália Vares, Large deviations and metastability, Cambridge University Press, Cambridge, a

6 Reversible diffusion dx t = V (x t ) dt + 2ε dw t V : R d R : potential, growing at infinity W t : d-dim Brownian motion on (Ω, F, P) Reversible w.r.t. invariant measure: µ ε (dx) = e V (x)/ε Z ε dx (detailed balance) 3

7 Reversible diffusion dx t = V (x t ) dt + 2ε dw t V : R d R : potential, growing at infinity W t : d-dim Brownian motion on (Ω, F, P) Reversible w.r.t. invariant measure: Mont Puget µ ε (dx) = e V (x)/ε Z ε dx Col de Sugiton z (detailed balance) Luminy x Calanque de Sugiton y τ x y : first-hitting time of small ball B ε (y), starting in x Eyring Kramers law (Eyring 1935, Kramers 1940) Dim 1: E[τ x y ] 2π V (x) V e[v (z) V (x)]/ε (z) Dim 2: E[τ x y ] 2π λ 1 (z) det( 2 V (z)) det( 2 V (x)) e[v (z) V (x)]/ε 3-a

8 Towards a proof of Kramers law Large deviations (Wentzell & Freidlin 1969): lim ε log(e[τ y x ]) = V (z) V (x) ε 0 Analytic (Helffer, Sjöstrand 85, Miclo 95, Mathieu 95, Kolokoltsov 96,... ): low-lying spectrum of generator Potential theory/variational (Bovier, Eckhoff, Gayrard, Klein 2004): E[τy x ] = 2π det( 2 V (z)) λ 1 (z) det( 2 V (x)) e[v (z) V (x)]/ε[ 1 + O(ε 1/2 log ε 1/2 ) ] and similar asymptotics for eigenvalues of generator Witten complex (Helffer, Klein, Nier 2004): full asymptotic expansion of prefactor Distribution of τy x (Day 1983, Bovier et al 2005): lim ε 0 P{ τy x > te[τy x ] } = e t 4

9 Allen-Cahn equation t u(x, t) = xx u(x, t) + f(u(x, t)) + noise where e.g. f(u) = u u 3 x [0, L], u(x, t) R represents e.g. magnetisation Periodic b.c. Neumann b.c. x u(0, t) = x u(l, t) = 0 Noise: weak, white in space and time 5

10 Allen-Cahn equation t u(x, t) = xx u(x, t) + f(u(x, t)) + noise where e.g. f(u) = u u 3 x [0, L], u(x, t) R represents e.g. magnetisation Periodic b.c. Neumann b.c. x u(0, t) = x u(l, t) = 0 Noise: weak, white in space and time Deterministic system is gradient xx u + u u 3 = δv δu L [ 1 V [u] = 0 2 u (x) u(x) u(x)4 ] dx min 5-a

11 Stationary solutions u (x) = u(x) + u(x) 3 = d dx [ ] 6

12 Stationary solutions u (x) = u(x) + u(x) 3 = d dx [ ] u ± (x) ±1: global minima of V, stable u 0 (x) 0: unstable Neumann b.c: for k = 1, 2,..., if L > πk, u k,± (x) = ± 2m + K(m), m ) m+1 m+1 sn( kx 2k m + 1 K(m) = L 6-a

13 Stationary solutions u (x) = u(x) + u(x) 3 = d dx [ ] u ± (x) ±1: global minima of V, stable u 0 (x) 0: unstable Neumann b.c: for k = 1, 2,..., if L > πk, u k,± (x) = ± 2m + K(m), m ) m+1 m+1 sn( kx 2k m + 1 K(m) = L u + u 1,+ u 2,+ u 3,+ 0 π 2π 3π L u0 u 4,+ u 4, u 3, u 2, u 1, u 6-b

14 Stability of stationary solutions Linearisation at u(x): t ϕ = A[u]ϕ, A[u] = d2 dx u(x)2 7

15 Stability of stationary solutions Linearisation at u(x): t ϕ = A[u]ϕ, A[u] = d2 dx u(x)2 u ± (x) ±1: eigenvalues (2 + (πk/l) 2 ), k N u 0 (x) 0: eigenvalues 1 (πk/l) 2, k N Number of positive eigenvalues: 0 u + 1 u 1,+ u 2, u 3,+ u 4,+ 0 π 2π 3π L u u 4, u 3, 0 u 2, u 1, u 1 7-a

16 Allen-Cahn equation with noise u t (x) = u t (x) + f(u t (x)) + 2ε Ẅ tx ( xx, f(u) = u u 3 ) Ẅ tx : space time white noise (formal derivative of Brownian sheet) 8

17 Allen-Cahn equation with noise u t (x) = u t (x) + f(u t (x)) + 2ε Ẅ tx ( xx, f(u) = u u 3 ) Ẅ tx : space time white noise (formal derivative of Brownian sheet) Construction of mild solution: 1. u t = u t u t = e t u 0 where e t cos(kπx/l) = e (kπ/l)2t cos(kπx/l) 8-a

18 Allen-Cahn equation with noise u t (x) = u t (x) + f(u t (x)) + 2ε Ẅ tx ( xx, f(u) = u u 3 ) Ẅ tx : space time white noise (formal derivative of Brownian sheet) Construction of mild solution: 1. u t = u t u t = e t u 0 where e t cos(kπx/l) = e (kπ/l)2t cos(kπx/l) 2. u t = u t + 2ε Ẅ tx w t (x) = k u t = e t u 0 + 2ε t t 0 e (kπ/l)2 (t s) dw s (k) 0 e (t s) Ẇ x (ds) }{{} =: w t (x) cos(kπx/l) 8-b

19 Allen-Cahn equation with noise u t (x) = u t (x) + f(u t (x)) + 2ε Ẅ tx ( xx, f(u) = u u 3 ) Ẅ tx : space time white noise (formal derivative of Brownian sheet) Construction of mild solution: 1. u t = u t u t = e t u 0 where e t cos(kπx/l) = e (kπ/l)2t cos(kπx/l) 2. u t = u t + 2ε Ẅ tx w t (x) = k u t = e t u 0 + 2ε t t 0 e (kπ/l)2 (t s) dw s (k) 3. u t = u t + 2ε Ẅ tx + f(u t ) 0 e (t s) Ẇ x (ds) }{{} =: w t (x) t cos(kπx/l) 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-Lasinio 1982) 8-c

20 Allen-Cahn equation with noise u t (x) = u t (x) + f(u t (x)) + 2ε Ẅ tx ( xx, f(u) = u u 3 ) Fourier variables: u t (x) = dz k = λ k z k dt where λ k = 1 + (πk/l) 2 k= k 1 +k 2 +k 3 =k z k (t) e i πkx/l z k1 z k2 z k3 dt + 2ε dw (k) t 9

21 Allen-Cahn equation with noise u t (x) = u t (x) + f(u t (x)) + 2ε Ẅ tx ( xx, f(u) = u u 3 ) Fourier variables: u t (x) = dz k = λ k z k dt where λ k = 1 + (πk/l) 2 k= k 1 +k 2 +k 3 =k z k (t) e i πkx/l z k1 z k2 z k3 dt + 2ε dw (k) t Energy functional: 1 L V [u] = 1 λ k z k k= k 1 +k 2 +k 3 +k 4 =0 z k1 z k2 z k3 z k4 9-a

22 The question How long does the system take to get from u (x) 1 to (a neighbourhood of) u + (x) 1? Metastability: Time of order econst /ε (rate of order e const /ε ) 10

23 The question How long does the system take to get from u (x) 1 to (a neighbourhood of) u + (x) 1? Metastability: Time of order econst /ε (rate of order e const /ε ) We seek constants W (activation energy), Γ 0 and α such that the random transition time τ satisfies E[τ] = Γ 1 0 e W/ε[ 1 + O(ε α ) ] 10-a

24 The question How long does the system take to get from u (x) 1 to (a neighbourhood of) u + (x) 1? Metastability: Time of order econst /ε (rate of order e const /ε ) We seek constants W (activation energy), Γ 0 and α such that the random transition time τ satisfies E[τ] = Γ 1 0 e W/ε[ 1 + O(ε α ) ] Large deviations (Faris & Jona-Lasinio 1982) L π : W = V [u 0 ] V [u ] = L 4 L > π : W = V [u 1,± ] V [u ] = 1 [ 3 8 E(m) (1 m)(3m+5) 1+m 1+m K(m) ] 10-b

25 Formal computation for Allen-Cahn (R.S. Maier, D. Stein, 01) Case L < π: Kramers: Γ 0 λ 1(u 0 ) 2π det( 2 V [u ]) det( 2 V [u 0 ]) Eigenvalues at V [u ] 1: µ k = 2 + (πk/l) 2 Eigenvalues at V [u 0 ] 0: λ k = 1 + (πk/l) 2 11

26 Formal computation for Allen-Cahn (R.S. Maier, D. Stein, 01) Case L < π: Kramers: Γ 0 λ 1(u 0 ) 2π det( 2 V [u ]) det( 2 V [u 0 ]) Eigenvalues at V [u ] 1: µ k = 2 + (πk/l) 2 Eigenvalues at V [u 0 ] 0: λ k = 1 + (πk/l) 2 Thus formally Γ 0 λ 0 2π Case L > π: k=0 µ k λ k = 1 sinh( 2L) 2 3/4 π sin L Spectral determinant computed by Gelfand s method 11-a

27 Formal computation for Allen-Cahn (R.S. Maier, D. Stein, 01) Case L < π: Kramers: Γ 0 λ 1(u 0 ) 2π det( 2 V [u ]) det( 2 V [u 0 ]) Eigenvalues at V [u ] 1: µ k = 2 + (πk/l) 2 Eigenvalues at V [u 0 ] 0: λ k = 1 + (πk/l) 2 Thus formally Γ 0 λ 0 2π Case L > π: k=0 µ k λ k = 1 sinh( 2L) 2 3/4 π sin L Spectral determinant computed by Gelfand s method Problems: 1. What happens when L π? (bifurcation) 2. Is the formal computation correct in infinite dimension? 11-b

28 Theorem: [Barret, B., Gentz 2010] E[τ] = Γ 1 0 e W/ε[ 1 + O(ε 1/4 log ε 5/4 ) ] 12

29 Theorem: [Barret, B., Gentz 2010] E[τ] = Γ 1 0 e W/ε[ 1 + O(ε 1/4 log ε 5/4 ) ] L < π: Γ 0 = 1 2 3/4 π sinh( 2L) sin L λ 1 λ 1 + 3ε/4L Ψ + where λ 1 = 1 + (π/l) 2 and α(1+α) Ψ + (α) = 8π e α2 /16 ( K α 2) 1/4 16 ( ) λ1 3ε/4L 12-a

30 Theorem: [Barret, B., Gentz 2010] E[τ] = Γ 1 0 e W/ε[ 1 + O(ε 1/4 log ε 5/4 ) ] L < π: Γ 0 = 1 2 3/4 π sinh( 2L) sin L λ 1 λ 1 + 3ε/4L Ψ + where λ 1 = 1 + (π/l) 2 and α(1+α) Ψ + (α) = 8π e α2 /16 ( K α 2) 1/4 16 In particular, lim Γ 0 = L π Γ(1/4) 2(3π 7 ) 1/4 sinh( 2π)ε 1/4 ( ) λ1 3ε/4L 12-b

31 Theorem: [Barret, B., Gentz 2010] E[τ] = Γ 1 0 e W/ε[ 1 + O(ε 1/4 log ε 5/4 ) ] L < π: Γ 0 = 1 2 3/4 π sinh( 2L) sin L λ 1 λ 1 + 3ε/4L Ψ + where λ 1 = 1 + (π/l) 2 and α(1+α) Ψ + (α) = 8π e α2 /16 ( K α 2) 1/4 16 ( ) λ1 3ε/4L In particular, lim Γ 0 = L π Γ(1/4) 2(3π 7 ) 1/4 sinh( 2π)ε 1/4 Similar expression for L > π (product of eigenvalues computed using path-integral techniques, cf. Maier and Stein) Γ 0 ε = ε = ε = ε = L π 12-c

32 Spectral Galerkin approximation u d t (x) = d k= d z k (t) e i πkx/l z k (t) solution of finite-dimensional SDE dz k (t) = V (z(t)) dt + 2ε dw k (t) 13

33 Spectral Galerkin approximation u d t (x) = d k= d z k (t) e i πkx/l z k (t) solution of finite-dimensional SDE dz k (t) = V (z(t)) dt + 2ε dw k (t) Theorem: [D. Blömker, A. Jentzen 2009] Assume Then sup d N sup u d t (ω) L < 0 t T sup u t (ω) u d t (ω) L < Z(ω)d γ 0 t T for all 0 < γ < 1/2, with Z a.s. finite r.v. ω Ω ω Ω 13-a

34 Spectral Galerkin approximation u d t (x) = d k= d z k (t) e i πkx/l z k (t) solution of finite-dimensional SDE dz k (t) = V (z(t)) dt + 2ε dw k (t) Theorem: [D. Blömker, A. Jentzen 2009] Assume Then sup d N sup u d t (ω) L < 0 t T sup u t (ω) u d t (ω) L < Z(ω)d γ 0 t T for all 0 < γ < 1/2, with Z a.s. finite r.v. ω Ω ω Ω Rem: [Liu, 2003] similar result for u 2 = k (1 + k 2 ) r z k 2, r < b

35 Potential theory SDE: dz t = V (z t ) dt + 2ε dw t Equilibrium potential : h A,B (x) = P x {τ A < τ B } Then where cap A (B) = ε E[τ x B ] inf h H A,B B c h B ε (x),b (y) e V (y)/ε dy cap Bε (x) (B) (A B) c h(x) 2 e V (x)/ε dx 14

36 Potential theory SDE: dz t = V (z t ) dt + 2ε dw t Equilibrium potential : h A,B (x) = P x {τ A < τ B } Then where cap A (B) = ε E[τ x B ] inf h H A,B B c h B ε (x),b (y) e V (y)/ε dy cap Bε (x) (B) (A B) c h(x) 2 e V (x)/ε dx Rough a priori bounds on h show that if x potential minimum, Ac h B ε (x),b (y) e V (y)/ε dy (2πε)d/2 e V (x)/ε det( 2 V (x)) 14-a

37 Estimation of capacity Truncated energy functional: retain only modes with k d 1 L V [u] = 1 2 z2 0 + u 1(z 1 ) u 1 (z 1 ) = 1 2 λ 1z z4 1 d k=2 λ k z k

38 Estimation of capacity Truncated energy functional: retain only modes with k d 1 L V [u] = 1 2 z2 0 + u 1(z 1 ) u 1 (z 1 ) = 1 2 λ 1z z4 1 d k=2 λ k z k Theorem: For all L < π, cap Bε (u ) (B ε(u + )) = ε e u 1(z 1 )/ε dz 1 2πε d j=2 2πε [ ] 1 + R(ε) λ j where R(ε) = O(ε 1/4 log ε 5/4 ) is uniform in d. 15-a

39 Estimation of capacity Truncated energy functional: retain only modes with k d 1 L V [u] = 1 2 z2 0 + u 1(z 1 ) u 1 (z 1 ) = 1 2 λ 1z z4 1 d k=2 λ k z k Theorem: For all L < π, cap Bε (u ) (B ε(u + )) = ε e u 1(z 1 )/ε dz 1 2πε d j=2 2πε [ ] 1 + R(ε) λ j where R(ε) = O(ε 1/4 log ε 5/4 ) is uniform in d. Corollary: E Bε (u ) (τ B ε (u + ) ) = Γ 1 0 (d, ε) e W (d)/ε [1 + R(ε)] 15-b

40 Proposition: Let B = B ε (u + ). Assume sup d N E[(τ d B )2 ], E[τ 2 B ] < Then E[τ B ] = Γ 1 0 (, ε) e W ( )/ε [1 + O(R(ε))] 16

41 Proposition: Let B = B ε (u + ). Assume sup d N E[(τ d B )2 ], E[τ 2 B ] < Then E[τ B ] = Γ 1 0 (, ε) e W ( )/ε [1 + O(R(ε))] Proof: Set T Kr = Γ 1 0 (, ε) e W ( )/ε Fix sets B B B with boundaries at distance δ { } Let Ω K,d = sup u t u d t L δ, τb d t [0,KT Kr ] KT Kr P(Ω c K,d ) P{Z > δdγ } + E[τ d B ] KT Kr lim d P(Ω c K,d ) 2 K 16-a

42 Proposition: Let B = B ε (u + ). Assume sup d N E[(τ d B )2 ], E[τ 2 B ] < Then E[τ B ] = Γ 1 0 (, ε) e W ( )/ε [1 + O(R(ε))] Proof: Set T Kr = Γ 1 0 (, ε) e W ( )/ε Fix sets B B B with boundaries at distance δ { } Let Ω K,d = sup u t u d t L δ, τb d t [0,KT Kr ] KT Kr P(Ω c K,d ) P{Z > δdγ } + E[τ d B ] KT Kr lim d P(Ω c K,d ) 2 K On Ω K,d one has τ d B τ B τ d B E[τ d B ] E[τ d B 1 Ω c K,d ] E[τ B ] E[τ d B ] + E[τ B 1 Ω c K,d ] Use E[τ B 1 Ω c K,d ] E[τB 2] P(Ω c K,d ), take d and K large enough 16-b

43 References W. G. Faris and G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A 15, (1982) R. S. Maier and D. L. Stein, Droplet nucleation and domain wall motion in a bounded interval, Phys. Rev. Lett. 87, (2001) Anton Bovier, Michael Eckhoff, Véronique Gayrard and Markus Klein, Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times, J. Eur. Math. Soc. 6, (2004) N. B. and Barbara Gentz, Anomalous behavior of the Kramers rate at bifurcations in classical field theories, J. Phys. A: Math. Theor. 42, (2009) N. B. and Barbara Gentz, The Eyring Kramers law for potentials with nonquadratic saddles, Markov Processes Relat. Fields 16, (2010) Florent Barret, N. B. and Barbara Gentz, in preparation 17

44 Potential theory Consider first Brownian motion Wt x = x + W t Let τa x = inf{t > 0: W t x A}, A R d 18

45 Potential theory Consider first Brownian motion Wt x = x + W t Let τa x = inf{t > 0: W t x A}, A R d Fact 1: w A (x) = E[τA x] satisfies w A (x) = 1 w A (x) = 0 G A c(x, y) Green s function w A (x) = x A c x A A c G A c(x, y) dy 18-a

46 Potential theory Consider first Brownian motion Wt x = x + W t Let τa x = inf{t > 0: W t x A}, A R d Fact 1: w A (x) = E[τA x] satisfies w A (x) = 1 w A (x) = 0 G A c(x, y) Green s function w A (x) = x A c x A A c G A c(x, y) dy y A 18-b

47 Potential theory Fact 2: h A,B (x) = P[τ x A < τ x B ] satisfies h A,B (x) = h A,B (x) = 0 h A,B (x) = 1 h A,B (x) = 0 A G B c(x, y) ρ A,B(dy) x (A B) c x A x B 19

48 Potential theory Fact 2: h A,B (x) = P[τ x A < τ x B ] satisfies h A,B (x) = 0 h A,B (x) = 1 h A,B (x) = 0 h A,B (x) = G B c(x, y) ρ A,B(dy) A ρ A,B : surface charge density on A x (A B) c x A x B B + A V 19-a

49 Potential theory Capacity: cap A (B) = A ρ A,B(dy) 20

50 Potential theory Capacity: cap A (B) = A ρ A,B(dy) Key observation: let C = B ε (x), then (using G(y, z) = G(z, y)) A c h C,A(y) dy = A c C G A c(y, z)ρ C,A(dz) dy h C,A(y) dy = w A(z)ρ C,A (dz) w A (x) cap C (A) A c C 20-a

51 Potential theory Capacity: cap A (B) = A ρ A,B(dy) Key observation: let C = B ε (x), then (using G(y, z) = G(z, y)) A c h C,A(y) dy = A c C G A c(y, z)ρ C,A(dz) dy h C,A(y) dy = w A(z)ρ C,A (dz) w A (x) cap C (A) A c C E[τA x ] = w A(x) A c h B ε (x),a (y) dy cap Bε (x) (A) 20-b

52 Potential theory Capacity: cap A (B) = A ρ A,B(dy) Key observation: let C = B ε (x), then (using G(y, z) = G(z, y)) A c h C,A(y) dy = A c C G A c(y, z)ρ C,A(dz) dy h C,A(y) dy = w A(z)ρ C,A (dz) w A (x) cap C (A) A c C E[τA x ] = w A(x) A c h B ε (x),a (y) dy cap Bε (x) (A) Variational representation: Dirichlet form cap A (B) = (A B) c h A,B(x) 2 dx = inf h H A,B (H A,B : set of sufficiently smooth functions satisfying b.c.) (A B) c h(x) 2 dx 20-c

53 Sketch of proof Upper bound: cap = inf h Φ(h) Φ(h +) Let δ = cε log ε, choose h + (z) = Φ(h) = ε 1 for z 0 < δ f(z 0 ) h(z) 2 e V (z)/ε dz for δ < z 0 < δ 0 for z 0 > δ where εf (z 0 ) + z0 V (z 0, 0)f (z 0 ) = 0 with b.c. f(±δ) = 0, 1 21

54 Sketch of proof Upper bound: cap = inf h Φ(h) Φ(h +) Let δ = cε log ε, choose h + (z) = Φ(h) = ε 1 for z 0 < δ f(z 0 ) h(z) 2 e V (z)/ε dz for δ < z 0 < δ 0 for z 0 > δ where εf (z 0 ) + z0 V (z 0, 0)f (z 0 ) = 0 with b.c. f(±δ) = 0, 1 Lower bound: Bound Dirichlet Φ form below by restricting domain, taking only 1st component of gradient and use for b.c. a priori bound on h 21-a