Augsburger Mathematisches Kolloquium
Sharp asymptotics for metastable transition times
in one- and two-dimensional Allen–Cahn SPDEs
Nils Berglund
MAPMO, Université d’Orléans
Augsburg, 10 May 2016
Joint works with Barbara Gentz (Bielefeld),
and with Giacomo Di Gesù (Paris) and Hendrik Weber (Warwick)
Nils Berglund
nils.berglund@univ-orleans.fr
http://www.univ-orleans.fr/mapmo/membres/berglund/
Metastability
A metastable system: supercooled water
(Source: https://youtu.be/fSPzMva9 CE)
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
1/21
Metastability for finite-dimensional SDEs
√
dxt = −∇V (xt ) dt + 2ε dWt
Mont
Puget
V ∶ Rd → R confining potential
= inf{t > 0∶ xt ∈ Bε (y )}
first-hitting time of small ball Bε (y ),
when starting in x
Col de Sugiton
z
τyx
Metastability of stochastic Allen-Cahn PDEs
Luminy
x
10 May 2016
Calanque de Sugiton
y
2/21
Metastability for finite-dimensional SDEs
√
dxt = −∇V (xt ) dt + 2ε dWt
Mont
Puget
V ∶ Rd → R confining potential
Col de Sugiton
z
= inf{t > 0∶ xt ∈ Bε (y )}
Luminy
first-hitting time of small ball Bε (y ),
x
when starting in x
Arrhenius’ law (1889): E[τyx ] ≃ e[V (z)−V (x)]/ε
τyx
Calanque de Sugiton
y
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
√
Metastability of stochastic Allen-Cahn PDEs
λ2 ...λd
∣λ1 ∣ν1 ...νd
e[V (z)−V (x)]/ε [1 + Oε (1)]
10 May 2016
2/21
Metastability for finite-dimensional SDEs
√
dxt = −∇V (xt ) dt + 2ε dWt
Mont
Puget
V ∶ Rd → R confining potential
Col de Sugiton
z
= inf{t > 0∶ xt ∈ Bε (y )}
Luminy
first-hitting time of small ball Bε (y ),
x
when starting in x
Arrhenius’ law (1889): E[τyx ] ≃ e[V (z)−V (x)]/ε
τyx
Calanque de Sugiton
y
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, . . .
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
2/21
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
Metastability of stochastic Allen-Cahn PDEs
∥∇hA,B (y )∥2 e−V (y )/ε dy
inf
h∈H 1 ,h∣A =1,h∣B =0
∫
(A∪B)c
10 May 2016
∥∇h(y )∥2 e−V (y )/ε dy
3/21
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
√
d
(2πε)
▷ Laplace asymptotics: ∫ c hA,B (y ) e−V (y )/ε dy ≃
e−V (x)/ε
B
ν1 ...νd
▷ Use Harnack inequalities to show that E νA,B [τB ] ≃ E x [τB ]
Alternative: coupling argument by [Martinelli, Olivieri & Scoppola]
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
3/21
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], L: bifurcation parameter
▷
u(x, t) ∈ R
▷
Either periodic or zero-flux Neumann boundary conditions
▷
In this talk: f (u) = u − u 3 (results more general)
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
4/21
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], L: bifurcation parameter
▷
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
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
critical points of V
4/21
Stationary solutions
H = 12 (u ′ )2 + 12 u 2 − 14 u 4
u0′′ (x) = −f (u0 (x)) = −u0 (x) + u0 (x)3
▷
u′
u± (x) ≡ ±1
▷
u0 (x) ≡ 0
▷
Nonconstant solutions satisfying b.c.
H=
u
−1
1
(expressible in terms of Jacobi elliptic fcts)
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
5/21
1
4
Stationary solutions
H = 12 (u ′ )2 + 12 u 2 − 14 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: 2k nonconstant solutions when L > kπ
0
▷
π
2π
3π
4π
L
Periodic b.c: k families when L > 2kπ
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
5/21
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 )
▷
)
u0 ≡ 0: always unstable, eigenvalues −λk = 1 − ( 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)
3
3
2π
1
Transition state
2
4
3π
2
4π
L
3
0
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
6/21
Coarsening dynamics
[Carr & Pego 89, Chen 04]
Metastability of stochastic Allen-Cahn PDEs
(Link to simulation)
10 May 2016
7/21
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)
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
8/21
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
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
8/21
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
t
−(kπ/L)2 (t−s)
e
0
Metastability of stochastic Allen-Cahn PDEs
(k)
)
dWs cos( kπx
L
10 May 2016
=∶wt (x)
α
∈ H ∩ C ∀s, α <
s
1
2
8/21
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 +
⇒
⇒
t
−(kπ/L)2 (t−s)
e
0
(k)
)
dWs cos( kπx
L
=∶wt (x)
α
∈ H ∩ C ∀s, α <
s
1
2
√
2ε Ẅtx + f (ut )
√
ut = e∆t u0 + 2ε wt + ∫
t
0
e∆(t−s) f (us ) ds =: Ft [u]
Existence and a.s. uniqueness [Faris & Jona-Lasinio 1982]
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
8/21
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)
zk1 zk2 zk3 dt + 2ε dWt
⇒
dzk = −λk zk dt −
∑
L k1 +k2 +k3 =k
(k)
Itō SDE, dWt : independent Wiener processes
λk = −1 + (πk/L)2
Energy functional:
1
1 ∞
2
zk zk zk zk
V [u] =
∑ λk ∣zk ∣ +
∑
2 k=−∞
4L k1 +k2 +k3 +k4 =0 1 2 3 4
√
⇒
dzt = −∇V (zt ) dt + 2ε dWt
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
9/21
Coarsening dynamics with noise
(Link to simulation)
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
10/21
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.)
⎧
⎪u0 (x) ≡ 0
if L 6 βπ with ev λk =( βkπ
)2 −1
⎪
L
uts (x) = ⎨
′
⎪
⎪
⎩u1 (x) β-kink stationary sol. if L > βπ with ev λk
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
11/21
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.)
⎧
⎪u0 (x) ≡ 0
if L 6 βπ with ev λk =( βkπ
)2 −1
⎪
L
uts (x) = ⎨
′
⎪
⎪
⎩u1 (x) β-kink stationary sol. if L > βπ with ev λk
[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
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
11/21
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.)
⎧
⎪u0 (x) ≡ 0
if L 6 βπ with ev λk =( βkπ
)2 −1
⎪
L
uts (x) = ⎨
′
⎪
⎪
⎩u1 (x) β-kink stationary sol. if L > βπ with ev λk
[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?
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
11/21
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
À
[1 + O(ε1/2 ∣log ε∣3/2 )]
E uin [τ+ ] = 2π Á
e
∏
∣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
À
[1 + R(ε)]
√
E uin [τ+ ] = 2π Á
∏
∣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
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
12/21
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
À
[1 + O(ε1/2 ∣log ε∣3/2 )]
E uin [τ+ ] = 2π Á
e
∏
∣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
À
[1 + R(ε)]
√
E uin [τ+ ] = 2π Á
∏
∣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
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
12/21
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 (no bifurcation points, f more general)
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
13/21
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 (no bifurcation points, f more general)
▷
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
kπ
L
∣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]
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
13/21
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
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
∀d ∈ N
14/21
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− )
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
14/21
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− )
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
Metastability of stochastic Allen-Cahn PDEs
sup
d>d0 v0 ∈Br (u− )
E v0 [(τ+ )2 ] 6 T12 e2H1 /ε
10 May 2016
(d)
14/21
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 } +
Metastability of stochastic Allen-Cahn PDEs
E v0 [τB− ]
(d)
KTKr
∥L∞ 6 δ, τB− 6 KTKr }
(d)
Cauchy–
Schwarz
.
⇒
lim sup P(ΩcK ,d ) =
d→∞
10 May 2016
M(ε)
K
15/21
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 } +
(d)
E v0 [τB− ]
(d)
KTKr
∥L∞ 6 δ, τB− 6 KTKr }
(d)
Cauchy–
Schwarz
.
⇒
lim sup P(ΩcK ,d ) =
d→∞
M(ε)
K
(d)
On ΩK ,d one has τB+ 6 τB 6 τB−
(d)
(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 } ]
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
15/21
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 } +
(d)
E v0 [τB− ]
(d)
KTKr
∥L∞ 6 δ, τB− 6 KTKr }
(d)
Cauchy–
Schwarz
.
⇒
lim sup P(ΩcK ,d ) =
d→∞
M(ε)
K
(d)
On ΩK ,d one has τB+ 6 τB 6 τB−
(d)
(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:
E v0 [τB 1{ΩcK ,d } ] 6 E v0 [τB2 ]P(ΩcK ,d ), take d → ∞ and finally K large
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
15/21
The two-dimensional case
(Link to simulation)
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
16/21
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 )∗
Metastability of stochastic Allen-Cahn PDEs
3L2
)
∣k∣2 π 2
∞ r dr
3L2
3L2
≃
−
= −∞
∫
∣k∣2 π 2
π2 1
r2
10 May 2016
17/21
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
≃
−
= −∞
∫
∣k∣2 π 2
π2 1
r2
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 + 3εC (δ)]u − u 3 + 2εξ δ
with C (δ) ≃ log(δ −1 ) admits local solution converging as δ → 0
(Global version: [Mourrat & Weber 2015])
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
17/21
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
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
Ω = βπ/L
18/21
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(∆−1)(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
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
18/21
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(∆−1)(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 (when integrated)
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
18/21
Computation of the prefactor
▷
▷
Consider for simplicity L < βπ ⇒ transition state in 0
Galerkin-truncated renormalised potential
1
1
VN = ∫ [∥∇uN (x)∥2 − uN (x)2 ] dx + ∫ ∶ uN (x)4 ∶ dx
2 T2
4 T2
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
19/21
Computation of the prefactor
▷
▷
▷
Consider for simplicity L < βπ ⇒ transition state in 0
Galerkin-truncated renormalised potential
1
1
VN = ∫ [∥∇uN (x)∥2 − uN (x)2 ] dx + ∫ ∶ uN (x)4 ∶ dx
2 T2
4 T2
√
√
∣λ0 ∣ε
2πε
Using Nelson estimate: capA (B) ≃
∏
2π
λk
0<∣k∣6N
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
19/21
Computation of the prefactor
▷
▷
▷
Consider for simplicity L < βπ ⇒ transition state in 0
Galerkin-truncated renormalised potential
1
1
VN = ∫ [∥∇uN (x)∥2 − uN (x)2 ] dx + ∫ ∶ uN (x)4 ∶ dx
2 T2
4 T2
√
√
∣λ0 ∣ε
2πε
Using Nelson estimate: capA (B) ≃
∏
2π
λk
0<∣k∣6N
▷
Symmetry argument:
∫
Bc
▷
hA,B (z) e−VN (z)/ε dz =
ZN (ε) ≃ 2 ∏
√
2πε −VN (L,0)/ε
νk e
1
1
−VN (z)/ε
dz = ZN (ε)
∫ e
2
2
where −VN (L, 0) = 41 L2 + 23 L2 CN ε
∣k∣6N
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
19/21
Computation of the prefactor
▷
▷
▷
Consider for simplicity L < βπ ⇒ transition state in 0
Galerkin-truncated renormalised potential
1
1
VN = ∫ [∥∇uN (x)∥2 − uN (x)2 ] dx + ∫ ∶ uN (x)4 ∶ dx
2 T2
4 T2
√
√
∣λ0 ∣ε
2πε
Using Nelson estimate: capA (B) ≃
∏
2π
λk
0<∣k∣6N
▷
Symmetry argument:
∫
Bc
▷
hA,B (z) e−VN (z)/ε dz =
ZN (ε) ≃ 2 ∏
√
2πε −VN (L,0)/ε
νk e
1
1
−VN (z)/ε
dz = ZN (ε)
∫ e
2
2
where −VN (L, 0) = 41 L2 + 23 L2 CN ε
∣k∣6N
▷
Prefactor proportional to (since νk = λk + 3)
∏
0<∣k∣6N
λk 3/λk
λk +3 e
Metastability of stochastic Allen-Cahn PDEs
converges since
k
log[ λλk +3
e3/λk ] = O( ∣k∣14 )
10 May 2016
19/21
Main result in dimension 2
Theorem: [B, Di Gesù, Weber 2016]
For appropriate A ∋ u− , B ∋ u+ , ∃µN probability measures on ∂A:
¿
νk −λk
Á
√
2π
Á
À ∏ ∣λk ∣ e ∣λk ∣ e(V [uts ]−V [u− ])/ε [1 + c+ ε ]
lim sup E µN [τB ] 6
∣λ0 ∣ k∈Z2 νk
N→∞
¿
νk −λk
2π Á
µN
Á
À ∏ ∣λk ∣ e ∣λk ∣ e(V [uts ]−V [u− ])/ε [1 − c− ε]
lim inf E [τB ] >
N→∞
∣λ0 ∣ k∈Z2 νk
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
20/21
Main result in dimension 2
Theorem: [B, Di Gesù, Weber 2016]
For appropriate A ∋ u− , B ∋ u+ , ∃µN probability measures on ∂A:
¿
νk −λk
Á
√
2π
Á
À ∏ ∣λk ∣ e ∣λk ∣ e(V [uts ]−V [u− ])/ε [1 + c+ ε ]
lim sup E µN [τB ] 6
∣λ0 ∣ k∈Z2 νk
N→∞
¿
νk −λk
2π Á
µN
Á
À ∏ ∣λk ∣ e ∣λk ∣ e(V [uts ]−V [u− ])/ε [1 − c− ε]
lim inf E [τB ] >
N→∞
∣λ0 ∣ k∈Z2 νk
(Inverse of) prefactor involves Carleman–Fredholm determinant:
det2 (Id + T ) = det(Id + T )e− Tr T
Defined whenever T is Hilbert–Schmidt, but not necessarily trace class
Applied here to T = [(−∆ + 2) − (−∆ − 1)](∣−∆ − 1∣)−1 = 3(∣−∆ − 1∣)−1
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
20/21
Outlook
▷
Dim d = 3: more difficult because 2 renormalisation constants needed
▷
Potentials with more than two wells: understood in d = 1
▷
Potentials invariant under symmetry group: understood in Rn
▷
More difficult: non-gradient systems
References: For this talk: [1,2,3,5]; Overview: [4]; Non-gradient: [6];
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., Giacomo Di Gesù, Hendrik Weber, An Eyring–Kramers law for the stochastic
Allen–Cahn equation in dimension two, preprint (2016), arXiv/1604.05742
6. N. B. and Christian Kuehn, Regularity structures and renormalisation of FitzHugh–
Nagumo SPDEs in three space dimensions, Elec J. Proba 21, (18):1-48 (2016)
Metastability of stochastic Allen-Cahn PDEs
10 May 2016
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