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)
Nils Berglund
nils.berglund@univ-orleans.fr
http://www.univ-orleans.fr/mapmo/membres/berglund/
Metastability for finite-dimensional SDEs
dxt = −∇V (xt ) dt +
√
2ε dWt
Mont
Puget
V ∶ Rd → R confining potential
Col de Sugiton
z
τyx = inf{t > 0∶ xt ∈ Bε (y )}
first-hitting time of small ball Bε (y ),
when starting in x
Luminy
Calanque de Sugiton
x
y
Arrhenius’ law (1889): E[τyx ] ≃ e[V (z)−V (x)]/ε
Eyring–Kramers law (1935, 1940):
Eigenvalues of Hessian of V at minimum x: 0 < ν1 6 ν2 6 ⋅ ⋅ ⋅ 6 νd
Eigenvalues of Hessian of V at saddle z: λ1 < 0 < λ2 6 ⋅ ⋅ ⋅ 6 λd
E[τyx ] = 2π
√
λ2 ...λd
∣λ1 ∣ν1 ...νd
e[V (z)−V (x)]/ε [1 + Oε (1)]
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 Laplacian, . . .
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
1/17(19)
Metastability for finite-dimensional SDEs
dxt = −∇V (xt ) dt +
√
2ε dWt
Mont
Puget
V ∶ Rd → R confining potential
Col de Sugiton
z
τyx = inf{t > 0∶ xt ∈ Bε (y )}
first-hitting time of small ball Bε (y ),
when starting in x
Luminy
Calanque de Sugiton
x
y
Arrhenius’ law (1889): E[τyx ] ≃ e[V (z)−V (x)]/ε
Eyring–Kramers law (1935, 1940):
Eigenvalues of Hessian of V at minimum x: 0 < ν1 6 ν2 6 ⋅ ⋅ ⋅ 6 νd
Eigenvalues of Hessian of V at saddle z: λ1 < 0 < λ2 6 ⋅ ⋅ ⋅ 6 λd
E[τyx ] = 2π
√
λ2 ...λd
∣λ1 ∣ν1 ...νd
e[V (z)−V (x)]/ε [1 + Oε (1)]
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 Laplacian, . . .
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
1/17(19)
Metastability for finite-dimensional SDEs
dxt = −∇V (xt ) dt +
√
2ε dWt
Mont
Puget
V ∶ Rd → R confining potential
Col de Sugiton
z
τyx = inf{t > 0∶ xt ∈ Bε (y )}
first-hitting time of small ball Bε (y ),
when starting in x
Luminy
Calanque de Sugiton
x
y
Arrhenius’ law (1889): E[τyx ] ≃ e[V (z)−V (x)]/ε
Eyring–Kramers law (1935, 1940):
Eigenvalues of Hessian of V at minimum x: 0 < ν1 6 ν2 6 ⋅ ⋅ ⋅ 6 νd
Eigenvalues of Hessian of V at saddle z: λ1 < 0 < λ2 6 ⋅ ⋅ ⋅ 6 λd
E[τyx ] = 2π
√
λ2 ...λd
∣λ1 ∣ν1 ...νd
e[V (z)−V (x)]/ε [1 + Oε (1)]
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 Laplacian, . . .
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
1/17(19)
Potential-theoretic proof
“Magic” formula: for A, B ⊂ Rd disjoint sets,
E νA,B [τB ] =
1
−V (y )/ε
dy
∫ hA,B (y ) e
capA (B) B c
▷
Equilibrium measure: νA,B proba measure concentrated on ∂A
▷
Committor function: hA,B (y ) = P y {τA < τB }
▷
Capacity: capA (B) = ε ∫
(A∪B)c
Property: capA (B) = ε
∥∇hA,B (y )∥2 e−V (y )/ε dy
inf
h∈H 1 ,h∣A =1,h∣B =0
∫
(A∪B)c
∥∇h(y )∥2 e−V (y )/ε dy
Proving Eyring–Kramers formula: A and B small sets around x and y
√ √
(2πε)d−1 −V (z)/ε
▷ Variational arguments: capA (B) ≃ ε ∣λ1 ∣
e
2πε
λ2 ...λd
√
(2πε)d −V (x)/ε
−V
(y
)/ε
▷ Laplace asymptotics: ∫ c hA,B (y ) e
dy ≃ ν1 ...νd e
B
ν
A,B
▷ Use Harnack inequalities to show that E
[τB ] ≃ E x [τB ]
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
2/17(19)
Potential-theoretic proof
“Magic” formula: for A, B ⊂ Rd disjoint sets,
E νA,B [τB ] =
1
−V (y )/ε
dy
∫ hA,B (y ) e
capA (B) B c
▷
Equilibrium measure: νA,B proba measure concentrated on ∂A
▷
Committor function: hA,B (y ) = P y {τA < τB }
▷
Capacity: capA (B) = ε ∫
(A∪B)c
Property: capA (B) = ε
∥∇hA,B (y )∥2 e−V (y )/ε dy
inf
h∈H 1 ,h∣A =1,h∣B =0
∫
(A∪B)c
∥∇h(y )∥2 e−V (y )/ε dy
Proving Eyring–Kramers formula: A and B small sets around x and y
√ √
(2πε)d−1 −V (z)/ε
▷ Variational arguments: capA (B) ≃ ε ∣λ1 ∣
e
2πε
λ2 ...λd
√
(2πε)d −V (x)/ε
−V
(y
)/ε
▷ Laplace asymptotics: ∫ c hA,B (y ) e
dy ≃ ν1 ...νd e
B
ν
A,B
▷ Use Harnack inequalities to show that E
[τB ] ≃ E x [τB ]
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
2/17(19)
Deterministic Allen–Cahn PDE
[Chafee & Infante 74, Allen & Cahn 75]
∂t u(x, t) = ∂xx u(x, t) + f (u(x, t))
▷
x ∈ [0, L]
▷
u(x, t) ∈ R
▷
Either periodic or zero-flux Neumann boundary conditions
▷
In this talk: f (u) = u − u 3 (results more general)
Energy function:
L 1
1
1
→ min
V [u] = ∫ [ u ′ (x)2 − u(x)2 + u(x)4 ] dx
2
2
4
0
Scaling limit of particle system with potential
N
N
N2
1 2
1 4
2
V (y ) = 2L
2 ∑i=1 (yi+1 − yi ) + ∑i=1 [− 2 yi + 4 yi ]
Stationary solutions: u0′′ (x) = −u0 (x) + u0 (x)3
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
critical points of V
2 February 2016
3/17(19)
Deterministic Allen–Cahn PDE
[Chafee & Infante 74, Allen & Cahn 75]
∂t u(x, t) = ∂xx u(x, t) + f (u(x, t))
▷
x ∈ [0, L]
▷
u(x, t) ∈ R
▷
Either periodic or zero-flux Neumann boundary conditions
▷
In this talk: f (u) = u − u 3 (results more general)
Energy function:
L 1
1
1
→ min
V [u] = ∫ [ u ′ (x)2 − u(x)2 + u(x)4 ] dx
2
2
4
0
Scaling limit of particle system with potential
N
N
N2
1 2
1 4
2
V (y ) = 2L
2 ∑i=1 (yi+1 − yi ) + ∑i=1 [− 2 yi + 4 yi ]
Stationary solutions: u0′′ (x) = −u0 (x) + u0 (x)3
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
critical points of V
2 February 2016
3/17(19)
Stationary solutions
u0′′ (x) = −f (u0 (x)) = −u0 (x) + u0 (x)3
▷
H = 12 (u ′ )2 + 12 u 2 − 41 u 4
u′
u± (x) ≡ ±1
▷
u0 (x) ≡ 0
▷
Nonconstant solutions satisfying b.c.
H=
u
−1
1
(expressible in terms of Jacobi elliptic fcts)
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
4/17(19)
1
4
Stationary solutions
H = 12 (u ′ )2 + 12 u 2 − 41 u 4
u0′′ (x) = −f (u0 (x)) = −u0 (x) + u0 (x)3
▷
u′
u± (x) ≡ ±1
H=
u
▷
u0 (x) ≡ 0
▷
Nonconstant solutions satisfying b.c.
−1
1
(expressible in terms of Jacobi elliptic fcts)
▷
Neumann b.c: k nonconstant solutions when L > kπ
0
▷
π
2π
3π
4π
L
Periodic b.c: k families when L > 2kπ
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
4/17(19)
1
4
Stability of stationary solutions
u0′′ (x) = −u0 (x) + u0 (x)3
Variational eq around u0 : ∂t vt (x) = vt′′ (x) + [1 − 3u0 (x)2 ]vt (x)
Sturm–Liouville spectrum of RHS determines stability of u0
▷
u± ≡ ±1: always stable (global minima of V )
▷
) −1
u0 ≡ 0: always unstable, eigenvalues λk = ( βkπ
L
2
(Neumann b.c.: β = 1, periodic b.c.: β = 2)
0
Neumann b.c.:
Number of positive
eigenvalues
1
1
2
π
0
(= unstable directions)
2
3
2π
1
Transition state
3
4
3π
2
4π
L
3
0
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
5/17(19)
Coarsening dynamics
[Carr & Pego 89, Chen 04]
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
(Link to simulation)
2 February 2016
6/17(19)
Stochastic partial differential equations
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u 3 )
Ẅtx : space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution via Duhamel formula:
▷
u̇t = ∆ut
⇒
ut = e∆t u0
2
▷
) = e−(kπ/L) t cos( kπx
)
where e∆t cos( kπx
L
L
t
√
√
u̇t = ∆ut + 2ε Ẅtx ⇒ ut = e∆t u0 + 2ε ∫ e∆(t−s) Ẇx (ds)
0
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
wt (x) = ∑ ∫
k
▷
u̇t = ∆ut +
⇒
0
t
e−(kπ/L)
2 (t−s)
=∶wt (x)
(k)
dWs
) ∈ H s ∩ C α ∀s, α <
cos( kπx
L
1
2
√
2ε Ẅtx + f (ut )
t
√
⇒
ut = e∆t u0 + 2ε wt + ∫ e∆(t−s) f (us ) ds =: Ft [u]
0
Existence and a.s. uniqueness [Faris & Jona-Lasinio 1982]
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
7/17(19)
Stochastic partial differential equations
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u 3 )
Ẅtx : space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution via Duhamel formula:
▷
u̇t = ∆ut
⇒
ut = e∆t u0
2
▷
) = e−(kπ/L) t cos( kπx
)
where e∆t cos( kπx
L
L
t
√
√
u̇t = ∆ut + 2ε Ẅtx ⇒ ut = e∆t u0 + 2ε ∫ e∆(t−s) Ẇx (ds)
0
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
wt (x) = ∑ ∫
k
▷
u̇t = ∆ut +
⇒
0
t
e−(kπ/L)
2 (t−s)
=∶wt (x)
(k)
dWs
) ∈ H s ∩ C α ∀s, α <
cos( kπx
L
1
2
√
2ε Ẅtx + f (ut )
t
√
⇒
ut = e∆t u0 + 2ε wt + ∫ e∆(t−s) f (us ) ds =: Ft [u]
0
Existence and a.s. uniqueness [Faris & Jona-Lasinio 1982]
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
7/17(19)
Stochastic partial differential equations
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u 3 )
Ẅtx : space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution via Duhamel formula:
▷
u̇t = ∆ut
⇒
ut = e∆t u0
2
▷
) = e−(kπ/L) t cos( kπx
)
where e∆t cos( kπx
L
L
t
√
√
u̇t = ∆ut + 2ε Ẅtx ⇒ ut = e∆t u0 + 2ε ∫ e∆(t−s) Ẇx (ds)
0
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
wt (x) = ∑ ∫
k
▷
u̇t = ∆ut +
⇒
0
t
e−(kπ/L)
2 (t−s)
=∶wt (x)
(k)
dWs
) ∈ H s ∩ C α ∀s, α <
cos( kπx
L
1
2
√
2ε Ẅtx + f (ut )
t
√
⇒
ut = e∆t u0 + 2ε wt + ∫ e∆(t−s) f (us ) ds =: Ft [u]
0
Existence and a.s. uniqueness [Faris & Jona-Lasinio 1982]
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
7/17(19)
Stochastic partial differential equations
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u 3 )
Ẅtx : space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution via Duhamel formula:
▷
u̇t = ∆ut
⇒
ut = e∆t u0
2
▷
) = e−(kπ/L) t cos( kπx
)
where e∆t cos( kπx
L
L
t
√
√
u̇t = ∆ut + 2ε Ẅtx ⇒ ut = e∆t u0 + 2ε ∫ e∆(t−s) Ẇx (ds)
0
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
wt (x) = ∑ ∫
k
▷
u̇t = ∆ut +
⇒
0
t
e−(kπ/L)
2 (t−s)
=∶wt (x)
(k)
dWs
) ∈ H s ∩ C α ∀s, α <
cos( kπx
L
1
2
√
2ε Ẅtx + f (ut )
t
√
⇒
ut = e∆t u0 + 2ε wt + ∫ e∆(t−s) f (us ) ds =: Ft [u]
0
Existence and a.s. uniqueness [Faris & Jona-Lasinio 1982]
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
7/17(19)
Stochastic partial differential equations
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u 3 )
1 ∞
Fourier variables: ut (x) = √ ∑ zk (t) ei πkx/L
L k=−∞
√
1
(k)
⇒
dzk = −λk zk dt −
zk1 zk2 zk3 dt + 2ε dWt
∑
L k1 +k2 +k3 =k
(k)
Itō SDE, dWt : independent Wiener processes
λk = −1 + (πk/L)2
Energy functional:
1 ∞
1
2
V [u] =
zk zk zk zk
∑ λk ∣zk ∣ +
∑
2 k=−∞
4L k1 +k2 +k3 +k4 =0 1 2 3 4
√
⇒
dzt = −∇V (zt ) dt + 2ε dWt
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
8/17(19)
Coarsening dynamics with noise
(Link to simulation)
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
9/17(19)
Eyring–Kramers law for 1D SPDEs: heuristics
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
Initial condition: uin near u− ≡ −1
(∆ ≡ ∂xx , f (u) = u − u 3 )
with eigenvalues νk
Target: u+ ≡ 1, τ+ = inf{t > 0∶ ∥ut − u+ ∥L∞ < ρ}
Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.)
⎧
⎪
if L 6 βπ with ev λk =( βkπ
)2 −1
⎪u0 (x) ≡ 0
L
uts (x) = ⎨
⎪u1 (x) β-kink stationary sol. if L > βπ with ev λ′k
⎪
⎩
[Faris & Jona-Lasinio 82]: large-deviation principle
⇒ Arrhenius law: E uin [τ+ ] ≃ e(V [uts ]−V [u− ])/ε
[Maier & Stein 01]: formal computation; for Neumann b.c.
√
λk (V [uts ]−V [u− ])/ε
⇒ E uin [τ+ ] = 2π ∣λ01∣ν0 ∏∞
k=1 νk e
▷
Rigorous proof?
▷
What happens when L → π as then λ1 → 0?
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
10/17(19)
Eyring–Kramers law for 1D SPDEs: heuristics
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
Initial condition: uin near u− ≡ −1
(∆ ≡ ∂xx , f (u) = u − u 3 )
with eigenvalues νk
Target: u+ ≡ 1, τ+ = inf{t > 0∶ ∥ut − u+ ∥L∞ < ρ}
Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.)
⎧
⎪
if L 6 βπ with ev λk =( βkπ
)2 −1
⎪u0 (x) ≡ 0
L
uts (x) = ⎨
⎪u1 (x) β-kink stationary sol. if L > βπ with ev λ′k
⎪
⎩
[Faris & Jona-Lasinio 82]: large-deviation principle
⇒ Arrhenius law: E uin [τ+ ] ≃ e(V [uts ]−V [u− ])/ε
[Maier & Stein 01]: formal computation; for Neumann b.c.
√
λk (V [uts ]−V [u− ])/ε
⇒ E uin [τ+ ] ≃ 2π ∣λ01∣ν0 ∏∞
k=1 νk e
▷
Rigorous proof?
▷
What happens when L → π as then λ1 → 0?
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
10/17(19)
Eyring–Kramers law for 1D SPDEs: heuristics
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
Initial condition: uin near u− ≡ −1
(∆ ≡ ∂xx , f (u) = u − u 3 )
with eigenvalues νk
Target: u+ ≡ 1, τ+ = inf{t > 0∶ ∥ut − u+ ∥L∞ < ρ}
Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.)
⎧
⎪
if L 6 βπ with ev λk =( βkπ
)2 −1
⎪u0 (x) ≡ 0
L
uts (x) = ⎨
⎪u1 (x) β-kink stationary sol. if L > βπ with ev λ′k
⎪
⎩
[Faris & Jona-Lasinio 82]: large-deviation principle
⇒ Arrhenius law: E uin [τ+ ] ≃ e(V [uts ]−V [u− ])/ε
[Maier & Stein 01]: formal computation; for Neumann b.c.
√
λk (V [uts ]−V [u− ])/ε
⇒ E uin [τ+ ] ≃ 2π ∣λ01∣ν0 ∏∞
k=1 νk e
▷
Rigorous proof?
▷
What happens when L → βπ as then λ1 → 0?
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
10/17(19)
Eyring–Kramers law for 1D SPDEs: main result
Theorem: Neumann b.c. [B & Gentz, Elec J Proba 2013]
▷
▷
▷
If L < π − c with c > 0, then
¿
Á 1 ∞ λk (V [uts ]−V [u− ])/ε
uin
À
[1 + O(ε1/2 ∣log ε∣3/2 )]
e
E [τ+ ] = 2π Á
∏
∣λ0 ∣ν0 k=1 νk
If L > π + c, then same formula with extra factor
and λ′k instead of λk
1
2
(since 2 saddles)
If π − c 6 L 6 π, then
¿
√
Á λ1 + 3ε/2L ∞ λk e(V [uts ]−V [u− ])/ε
uin
Á
À
[1 + R(ε)]
√
E [τ+ ] = 2π
∏
∣λ0 ∣ν0 ν1 k=2 νk Ψ+ (λ1 / 3ε/2L)
where Ψ+ explicit, involves Bessel function K1/4 , limα→∞ Ψ+ (α) = 1
▷
If π 6 L 6 π + c, then same formula, with another function Ψ− ,
involving Bessel functions I±1/4 , limα→∞ Ψ− (α) = 2
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
11/17(19)
Eyring–Kramers law for 1D SPDEs: main result
Theorem: Neumann b.c. [B & Gentz, Elec J Proba 2013]
▷
▷
▷
If L < π − c with c > 0, then
¿
Á 1 ∞ λk (V [uts ]−V [u− ])/ε
uin
À
[1 + O(ε1/2 ∣log ε∣3/2 )]
e
E [τ+ ] = 2π Á
∏
∣λ0 ∣ν0 k=1 νk
If L > π + c, then same formula with extra factor
and λ′k instead of λk
1
2
(since 2 saddles)
If π − c 6 L 6 π, then
¿
√
Á λ1 + 3ε/2L ∞ λk e(V [uts ]−V [u− ])/ε
uin
Á
À
[1 + R(ε)]
√
E [τ+ ] = 2π
∏
∣λ0 ∣ν0 ν1 k=2 νk Ψ+ (λ1 / 3ε/2L)
where Ψ+ explicit, involves Bessel function K1/4 , limα→∞ Ψ+ (α) = 1
▷
If π 6 L 6 π + c, then same formula, with another function Ψ− ,
involving Bessel functions I±1/4 , limα→∞ Ψ− (α) = 2
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
11/17(19)
Eyring–Kramers law for 1D SPDEs: comments
▷
Periodic b.c.: similar result [B & Gentz, Elec J Proba 2013]
√
For L > 2π: extra factor ε because saddle is a whole curve
▷
Proof: relies on spectral Galerkin approximation
▷
Similar results by F. Barret [Annales IHP, 2015]
using different method (away from bifurcation points)
▷
Details
For Neumann b.c. and L < π: spectral determinant in prefactor is
explicitly computable (Euler product formulas)
2
2
kπ
L
sin(L)
1 ∞ λk 1 ∞ ( L ) − 1 1 ∞ 1 − ( kπ )
√
= ∏
= ∏
=√
∏
2
2
L
kπ
∣λ0 ∣ν0 k=1 νk 2 k=1 ( ) + 2 2 k=1 2 + ( )
2 sinh( 2L)
L
kπ
Similar expression for periodic b.c. and L < 2π
▷
For larger L, 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
12/17(19)
Eyring–Kramers law for 1D SPDEs: comments
▷
Periodic b.c.: similar result [B & Gentz, Elec J Proba 2013]
√
For L > 2π: extra factor ε because saddle is a whole curve
▷
Proof: relies on spectral Galerkin approximation
▷
Similar results by F. Barret [Annales IHP, 2015]
using different method (away from bifurcation points)
▷
Details
For Neumann b.c. and L < π: spectral determinant in prefactor is
explicitly computable (Euler product formulas)
2
2
kπ
L
sin(L)
1 ∞ λk 1 ∞ ( L ) − 1 1 ∞ 1 − ( kπ )
√
= ∏
= ∏
=√
∏
2
2
L
kπ
∣λ0 ∣ν0 k=1 νk 2 k=1 ( ) + 2 2 k=1 1 + 2( )
2 sinh( 2L)
L
kπ
Similar expression for periodic b.c. and L < 2π
▷
For larger L, 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
12/17(19)
The two-dimensional case
(Link to simulation)
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
13/17(19)
The two-dimensional case
▷
Large-deviation principle: [Hairer & Weber, Ann. Fac. Sc. Toulouse, 2015]
▷
Naive computation of prefactor fails:
2
log ∏
k∈(N2 )∗
L
)
1 − ( ∣k∣π
2
L
)
1 + 2( ∣k∣π
≃
∑ log(1 −
k∈(N2 )∗
≃− ∑
k∈(N2 )∗
▷
3L2
)
∣k∣2 π 2
∞ r dr
3L2
3L2
≃
−
= −∞
∫
2
2
2
∣k∣ π
π
r2
1
In fact, the equation needs to be renormalised
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
14/17(19)
The two-dimensional case
▷
Large-deviation principle: [Hairer & Weber, Ann. Fac. Sc. Toulouse, 2015]
▷
Naive computation of prefactor fails:
2
log ∏
k∈(N2 )∗
L
)
1 − ( ∣k∣π
2
L
)
1 + 2( ∣k∣π
≃
∑ log(1 −
k∈(N2 )∗
≃− ∑
k∈(N2 )∗
▷
3L2
)
∣k∣2 π 2
∞ r dr
3L2
3L2
≃
−
= −∞
∫
2
2
2
∣k∣ π
π
r2
1
In fact, the equation needs to be renormalised
Theorem: [Da Prato & Debussche 2003]
Let 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
14/17(19)
Renormalisation
t
Problem: For d = 2, stoch. convolution ∫0 e∆(t−s) Ẇx (ds) is a distribution
▷ δ-mollification should be equivalent to Galerkin approx. ∣k∣ 6 N = δ −1 :
t
1
(k)
wN (x, t) = ∑ ak (t) ei Ωk⋅x
ak = ∫ e−µk (t−s) dWs
L
0
∣k∣6N
µk = (Ω∣k∣)2
▷
Ω = βπ/L
t
limt→∞ ∫0 e(∆−1l)(t−s) Ẇx (ds) = φN is a Gaussian free field, s.t.
L2 CN ∶= L2 Eφ2N = E∥φN ∥2L2 = ∑
1
2(µk +1)
=
Tr(PN [−∆+1l]−1 )
2
≃ log(N)
∣k∣6N
▷
Wick powers
∶ φ2N ∶ = φ2N − CN
∶ φ3N ∶ = φ3N − 3CN φN
∶ φ4N ∶ = φ4N − 6CN φ2N + 3CN2
have zero mean and uniformly bounded variance
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
15/17(19)
Renormalisation
t
Problem: For d = 2, stoch. convolution ∫0 e∆(t−s) Ẇx (ds) is a distribution
▷ δ-mollification should be equivalent to Galerkin approx. ∣k∣ 6 N = δ −1 :
t
1
(k)
wN (x, t) = ∑ ak (t) ei Ωk⋅x
ak = ∫ e−µk (t−s) dWs
L
0
∣k∣6N
µk = (Ω∣k∣)2
▷
Ω = βπ/L
t
limt→∞ ∫0 e(∆−1l)(t−s) Ẇx (ds) = φN is a Gaussian free field, s.t.
L2 CN ∶= L2 Eφ2N = E∥φN ∥2L2 = ∑
1
2(µk +1)
=
Tr(PN [−∆+1l]−1 )
2
≃ log(N)
∣k∣6N
▷
Wick powers
∶ φ2N ∶ = φ2N − CN
∶ φ3N ∶ = φ3N − 3CN φN
∶ φ4N ∶ = φ4N − 6CN φ2N + 3CN2
have zero mean and uniformly bounded variance
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
15/17(19)
Renormalisation
t
Problem: For d = 2, stoch. convolution ∫0 e∆(t−s) Ẇx (ds) is a distribution
▷ δ-mollification should be equivalent to Galerkin approx. ∣k∣ 6 N = δ −1 :
t
1
(k)
wN (x, t) = ∑ ak (t) ei Ωk⋅x
ak = ∫ e−µk (t−s) dWs
L
0
∣k∣6N
µk = (Ω∣k∣)2
▷
Ω = βπ/L
t
limt→∞ ∫0 e(∆−1l)(t−s) Ẇx (ds) = φN is a Gaussian free field, s.t.
L2 CN ∶= L2 Eφ2N = E∥φN ∥2L2 = ∑
1
2(µk +1)
=
Tr(PN [−∆+1l]−1 )
2
≃ log(N)
∣k∣6N
▷
Wick powers
∶ φ2N ∶ = φ2N − CN
∶ φ3N ∶ = φ3N − 3CN φN
∶ φ4N ∶ = φ4N − 6CN φ2N + 3CN2
have zero mean and uniformly bounded variance
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
15/17(19)
Computation of the prefactor
Work in progress with Giacomo Di Gesu and Hendrik Weber
▷
▷
▷
Consider for simplicity L < βπ ⇒ transition state in 0
Galerkin-truncated renormalised potential
1
1
′
(x)2 − uN (x)2 ] dx + ∫ ∶ uN (x)4 ∶ dx
VN = ∫ [uN
2 T2
4 T2
√
√
∣λ0 ∣ε
2πε
Using Nelson estimate: capA (B) ≃
∏
2π
λk
0<∣k∣6N
▷
Symmetry argument:
1
1
hA,B (z) e−VN (z)/ε dz = ∫ e−VN (z)/ε dz = ZN (ε)
2
2
√
2πε −VN (L,0)/ε
ZN (ε) ≃ 2 ∏
where −VN (L, 0) = 41 L2 + 23 L2 CN ε
νk e
∫
Bc
▷
∣k∣6N
▷
Prefactor proportionnal to (since νk = λk + 3)
∏
0<∣k∣6N
λk
3/λk
λk +3 e
converges since
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
k
log[ λλk +3
e3/λk ] = O( ∣k∣14 )
2 February 2016
16/17(19)
Computation of the prefactor
Work in progress with Giacomo Di Gesu and Hendrik Weber
▷
▷
▷
Consider for simplicity L < βπ ⇒ transition state in 0
Galerkin-truncated renormalised potential
1
1
′
(x)2 − uN (x)2 ] dx + ∫ ∶ uN (x)4 ∶ dx
VN = ∫ [uN
2 T2
4 T2
√
√
∣λ0 ∣ε
2πε
Using Nelson estimate: capA (B) ≃
∏
2π
λk
0<∣k∣6N
▷
Symmetry argument:
1
1
hA,B (z) e−VN (z)/ε dz = ∫ e−VN (z)/ε dz = ZN (ε)
2
2
√
2πε −VN (L,0)/ε
ZN (ε) ≃ 2 ∏
where −VN (L, 0) = 41 L2 + 23 L2 CN ε
νk e
∫
Bc
▷
∣k∣6N
▷
Prefactor proportionnal to (since νk = λk + 3)
∏
0<∣k∣6N
λk
3/λk
λk +3 e
converges since
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
k
log[ λλk +3
e3/λk ] = O( ∣k∣14 )
2 February 2016
16/17(19)
Computation of the prefactor
Work in progress with Giacomo Di Gesu and Hendrik Weber
▷
▷
▷
Consider for simplicity L < βπ ⇒ transition state in 0
Galerkin-truncated renormalised potential
1
1
′
(x)2 − uN (x)2 ] dx + ∫ ∶ uN (x)4 ∶ dx
VN = ∫ [uN
2 T2
4 T2
√
√
∣λ0 ∣ε
2πε
Using Nelson estimate: capA (B) ≃
∏
2π
λk
0<∣k∣6N
▷
Symmetry argument:
1
1
hA,B (z) e−VN (z)/ε dz = ∫ e−VN (z)/ε dz = ZN (ε)
2
2
√
2πε −VN (L,0)/ε
ZN (ε) ≃ 2 ∏
where −VN (L, 0) = 41 L2 + 23 L2 CN ε
νk e
∫
Bc
▷
∣k∣6N
▷
Prefactor proportionnal to (since νk = λk + 3)
∏
0<∣k∣6N
λk
3/λk
λk +3 e
converges since
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
k
log[ λλk +3
e3/λk ] = O( ∣k∣14 )
2 February 2016
16/17(19)
Computation of the prefactor
Work in progress with Giacomo Di Gesu and Hendrik Weber
▷
▷
▷
Consider for simplicity L < βπ ⇒ transition state in 0
Galerkin-truncated renormalised potential
1
1
′
(x)2 − uN (x)2 ] dx + ∫ ∶ uN (x)4 ∶ dx
VN = ∫ [uN
2 T2
4 T2
√
√
∣λ0 ∣ε
2πε
Using Nelson estimate: capA (B) ≃
∏
2π
λk
0<∣k∣6N
▷
Symmetry argument:
1
1
hA,B (z) e−VN (z)/ε dz = ∫ e−VN (z)/ε dz = ZN (ε)
2
2
√
2πε −VN (L,0)/ε
ZN (ε) ≃ 2 ∏
where −VN (L, 0) = 41 L2 + 23 L2 CN ε
νk e
∫
Bc
▷
∣k∣6N
▷
Prefactor proportionnal to (since νk = λk + 3)
∏
0<∣k∣6N
λk 3/λk
λk +3 e
converges since
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
k
log[ λλk +3
e3/λk ] = O( ∣k∣14 )
2 February 2016
16/17(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)
, The Eyring–Kramers law for potentials with nonquadratic saddles,
2.
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
17/17(19)
Sketch of proof: Spectral Galerkin approximation
ut (x) =
(d)
√1
L
∑dk=−d zk (t) ei πkx/L
⇒
dzt = −∇V (zt ) dt +
√
2ε dWt
Theorem [Blömker & Jentzen 13]
For all γ ∈ (0, 12 ) there exists an a.s. finite r.v. Z ∶ Ω → R+ s.t. ∀ω ∈ Ω
sup ∥ut (ω) − ut (ω)∥L∞ < Z (ω)d −γ
(d)
06t6T
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
∀d ∈ N
2 February 2016
18/17(19)
Sketch of proof: Spectral Galerkin approximation
ut (x) =
(d)
√1
L
∑dk=−d zk (t) ei πkx/L
⇒
dzt = −∇V (zt ) dt +
√
2ε dWt
Theorem [Blömker & Jentzen 13]
For all γ ∈ (0, 12 ) there exists an a.s. finite r.v. Z ∶ Ω → R+ s.t. ∀ω ∈ Ω
sup ∥ut (ω) − ut (ω)∥L∞ < Z (ω)d −γ
(d)
06t6T
∀d ∈ N
Proposition (using potential theory)
∃ε0 > 0∶ ∀ε < ε0 ∃d0 (ε) < ∞∶ ∀d > d0 ∃νd proba measure on ∂Br (u− )
∫
E v0 [τ+ ]νd (dv0 ) = C (d, ε) eH(d)/ε [1 + R(ε)]
(d)
∂Br (u− )
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
2 February 2016
18/17(19)
Sketch of proof: Spectral Galerkin approximation
ut (x) =
(d)
√1
L
∑dk=−d zk (t) ei πkx/L
⇒
dzt = −∇V (zt ) dt +
√
2ε dWt
Theorem [Blömker & Jentzen 13]
For all γ ∈ (0, 12 ) there exists an a.s. finite r.v. Z ∶ Ω → R+ s.t. ∀ω ∈ Ω
sup ∥ut (ω) − ut (ω)∥L∞ < Z (ω)d −γ
∀d ∈ N
(d)
06t6T
Proposition (using potential theory)
∃ε0 > 0∶ ∀ε < ε0 ∃d0 (ε) < ∞∶ ∀d > d0 ∃νd proba measure on ∂Br (u− )
∫
E v0 [τ+ ]νd (dv0 ) = C (d, ε) eH(d)/ε [1 + R(ε)]
(d)
∂Br (u− )
Proposition (using large deviations and lots of other stuff)
H0 ∶= V [uts ] − V [u− ]. ∀η > 0 ∃ε0 , T1 , H1 ∶ ∀ε < ε0 ∃d0 < ∞ s.t. ∀d > d0
sup
v0 ∈Br (u− )
E v0 [τ+2 ] 6 T12 e2(H0 +η)/ε , sup
sup
d>d0 v0 ∈Br (u− )
Sharp estimates on metastable lifetimes for 1D and 2D Allen–Cahn SPDEs
E v0 [(τ+ )2 ] 6 T12 e2H1 /ε
(d)
2 February 2016
18/17(19)
Main step of the proof
Set TKr = C (∞, ε) eH0 /ε
Let B = Bρ (u+ ) and define nested sets B− ⊂ B ⊂ B+ at L∞ -distance δ
ΩK ,d = {
sup
∥vt − vt
(d)
t∈[0,KTKr ]
(d)
P(ΩcK ,d ) 6 P{Z > δd γ } +
On ΩK ,d one has
(d)
(d)
τB+
E v0 [τB− ]
(d)
KTKr
6 τB 6
∥L∞ 6 δ, τB− 6 KTKr }
(d)
Cauchy–
Schwarz
.
⇒
lim sup P(ΩcK ,d ) =
d→∞
M(ε)
K
(d)
τB−
(d)
⇒ E v0 [τB+ 1{ΩK ,d } ] 6 E v0 [τB 1{ΩK ,d } ] 6 E v0 [τB− 1{ΩK ,d } ]
(d)
(d)
⇒ (d)
(d)
(d)
(d)
(d)
(d)
E v0 [τB+ ] − E v0 [τB+ 1{ΩcK ,d } ] 6 E v0 [τB ] 6 E v0 [τB− ] + E v0 [τB 1{ΩcK ,d } ]
Integrate against ν√
d and use Cauchy–Schwarz to bound error terms:
v0
E [τB 1{ΩcK ,d } ] 6 E v0 [τB2 ]P(ΩcK ,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)
Main step of the proof
Set TKr = C (∞, ε) eH0 /ε
Let B = Bρ (u+ ) and define nested sets B− ⊂ B ⊂ B+ at L∞ -distance δ
ΩK ,d = {
sup
∥vt − vt
(d)
t∈[0,KTKr ]
(d)
P(ΩcK ,d ) 6 P{Z > δd γ } +
On ΩK ,d one has
(d)
(d)
τB+
E v0 [τB− ]
(d)
KTKr
6 τB 6
∥L∞ 6 δ, τB− 6 KTKr }
(d)
Cauchy–
Schwarz
.
⇒
lim sup P(ΩcK ,d ) =
d→∞
M(ε)
K
(d)
τB−
(d)
⇒ E v0 [τB+ 1{ΩK ,d } ] 6 E v0 [τB 1{ΩK ,d } ] 6 E v0 [τB− 1{ΩK ,d } ]
(d)
(d)
⇒ (d)
(d)
(d)
(d)
(d)
(d)
E v0 [τB+ ] − E v0 [τB+ 1{ΩcK ,d } ] 6 E v0 [τB ] 6 E v0 [τB− ] + E v0 [τB 1{ΩcK ,d } ]
Integrate against ν√
d and use Cauchy–Schwarz to bound error terms:
v0
E [τB 1{ΩcK ,d } ] 6 E v0 [τB2 ]P(ΩcK ,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)
Main step of the proof
Set TKr = C (∞, ε) eH0 /ε
Let B = Bρ (u+ ) and define nested sets B− ⊂ B ⊂ B+ at L∞ -distance δ
ΩK ,d = {
sup
∥vt − vt
(d)
t∈[0,KTKr ]
(d)
P(ΩcK ,d ) 6 P{Z > δd γ } +
On ΩK ,d one has
(d)
(d)
τB+
E v0 [τB− ]
(d)
KTKr
6 τB 6
∥L∞ 6 δ, τB− 6 KTKr }
(d)
Cauchy–
Schwarz
.
⇒
lim sup P(ΩcK ,d ) =
d→∞
M(ε)
K
(d)
τB−
(d)
⇒ E v0 [τB+ 1{ΩK ,d } ] 6 E v0 [τB 1{ΩK ,d } ] 6 E v0 [τB− 1{ΩK ,d } ]
(d)
(d)
⇒ (d)
(d)
(d)
(d)
(d)
(d)
E v0 [τB+ ] − E v0 [τB+ 1{ΩcK ,d } ] 6 E v0 [τB ] 6 E v0 [τB− ] + E v0 [τB 1{ΩcK ,d } ]
Integrate against ν√
d and use Cauchy–Schwarz to bound error terms:
v0
E [τB 1{ΩcK ,d } ] 6 E v0 [τB2 ]P(ΩcK ,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)