Capacitary inequalities in discrete setting and application to metastable Markov chains. André Schlichting

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1 Capacitary inequalities in discrete setting and application to metastable Markov chains André Schlichting Institute for Applied Mathematics, University of Bonn joint work with M. Slowik (TU Berlin) 12 th German Probability and Statistics Days, Bochum

2 Metastability: A common phenomenon The paradigm. Related to the dynamics of first order phase transitions Change parameters quickly across the line of first order phase transition, the system reveals the existence of multiple time scales: Short time scales. Existence of disjoint subsets S i trapping effectively the system S 1 S 4 S 3 Quasi-equilibrium ( ˆ= metastable states) is reached within S i S 2 Larger time scales. Rapid transitions between S i and S j occur induced by random fluctuations André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 2 (14)

3 Spectrum and metastability Heuristic. Reversible Markov process {X t : t 0}, generator L, λ i spec( L) 0 = λ 1 λ 2,... λ 4 O(ε) λ 5 λ1 =... = λ 4 = 0 λ5 The goal. Understanding of quantitative aspects of dynamical phase transitions: expected time of a transition from a metastable to a stable state distribution of the exit time from a metastable state spectral properties of the generator and mixing times André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 3 (14)

4 Spectrum and metastability Heuristic. Reversible Markov process {X t : t 0}, generator L, λ i spec( L) 0 = λ 1 λ 2,... λ 4 O(ε) λ 5 λ1 =... = λ 4 = 0 λ5 The goal. Understanding of quantitative aspects of dynamical phase transitions: expected time of a transition from a metastable to a stable state distribution of the exit time from a metastable state spectral properties of the generator and mixing times André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 3 (14)

5 Spectrum and metastability Heuristic. Reversible Markov process {X t : t 0}, generator L, λ i spec( L) 0 = λ 1 λ 2,... λ 4 O(ε) λ 5 λ1 =... = λ 4 = 0 λ5 The goal. Understanding of quantitative aspects of dynamical phase transitions: expected time of a transition from a metastable to a stable state distribution of the exit time from a metastable state spectral properties of the generator and mixing times André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 3 (14)

6 Reversible Markov chains Setting. state space S (finite or countable infinite) µ measure on S ( ) p(x, y) : x, y S stochastic matrix, irreducible (positive recurrent) Dynamics. Discrete-time Markov chain X = {X t : t 0} on S with generator (Lf)(x) = p(x, y) ( f(y) f(x) ) y Assumption: The Markov process X is reversible with respect to µ. First return time. For any A S, let τ A = inf{t > 0 : X t A} André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 4 (14)

7 Reversible Markov chains Setting. state space S (finite or countable infinite) µ measure on S ( ) p(x, y) : x, y S stochastic matrix, irreducible (positive recurrent) Dynamics. Discrete-time Markov chain X = {X t : t 0} on S with generator (Lf)(x) = p(x, y) ( f(y) f(x) ) y Assumption: The Markov process X is reversible with respect to µ. First return time. For any A S, let τ A = inf{t > 0 : X t A} André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 4 (14)

8 Reversible Markov chains Setting. state space S (finite or countable infinite) µ measure on S ( ) p(x, y) : x, y S stochastic matrix, irreducible (positive recurrent) Dynamics. Discrete-time Markov chain X = {X t : t 0} on S with generator (Lf)(x) = p(x, y) ( f(y) f(x) ) y Assumption: The Markov process X is reversible with respect to µ. First return time. For any A S, let τ A = inf{t > 0 : X t A} André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 4 (14)

9 Poincaré and logarithmic Sobolev inequality E(f, f) = 1 2 Poincaré inequality. µ(x) p(x, y) ( f(x) f(y) ) 2 x,y S f : S R : var µ[f] C PI E(f, f). PI(C PI) Logarithmic Sobolev inequality. f : S R : Ent µ[f 2 ] = E µ [f 2 ln f 2 ] E µ[f 2 ] C LSI E(f, f). LSI(C LSI) The goal: Compute for metastable Markov chains in terms of capacities the optimal constant C PI in the Poincaré inequality (inverse spectral gap) the optimal constant C LSI in the logarithmic Sobolev inequality André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 5 (14)

10 Equilibrium potential and capacities Equilibrium potential. Given A, B S disjoint h { A,B (x) = 0 x B LhA,B = 0, on (A B) c x (A B) c [ ] : h A,B(x) = P x τa < τ B h A,B = 1 A, h A,B on A(x) B= G B c(x, y) ρ A,B(dy) A ρ A,B : surface charge density on A Capacity. cap(a, B) = x A µ(x) ( Lh A,B)(x) h A,B (x) = 0 h A,B (x) = 1 x (A B) c x A = h A,B, Lh A,B µ = E(h A,B, h A,B) = [ ] µ(x) P x τb < τ A x A B + A Fact. cap(a, B) = cap(b, A) and cap(a, B) cap(a, B), A A 1V André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 6 (14)

11 Computation of capacities Variational principles. Allows to bound capacities from above and from below Dirichlet principle. 1 cap(a, B) = inf h H A,B 2 µ(x) p(x, y) ( h(x) h(y) ) 2 x,y H A,B : space of functions with boundary constraints; minimizer harmonic function Thomson principle. 1 cap(a, B) = inf 1 f U A,B 2 x,y f(x, y) 2 µ(x) p(x, y) U A,B : space of unit AB-flows; maximizer harmonic flow. André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 7 (14)

12 Capacitary inequalities h, Lg µ = 1 µ(x) p(x, y) ( h(x) h(y) )( g(x) g(y) ) 2 x,y S Proposition Let B S be non-empty. For any f : S R with f 0 on B set A t := {x S : f(x) > t}. Then, 0 2t cap(a t, B) dt 4 E(f, f). Previous and related work Maz ya (1972), operators in divergence form on R d André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 8 (14)

13 Capacitary inequalities h, Lg µ = 1 µ(x) p(x, y) ( h(x) h(y) )( g(x) g(y) ) 2 x,y S Proposition Let B S be non-empty. For any f : S R with f 0 on B set A t := {x S : f(x) > t}. Then, 0 2t cap(a t, B) dt 4 E(f, f). Previous and related work Maz ya (1972), operators in divergence form on R d André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 8 (14)

14 Consequences Proposition Let B S be non-empty and ν P 1(S). Then, there exist C 1, C 2 (0, ) satisfying C 1 C 2 4C 1 such that the following statements are equivalent: (i) For all A S \ B it holds (ii) For all f : S R with f B 0 holds ν[a] C 1 cap(a, B). f 2 l 1 (ν) C 2 E(f, f). André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 9 (14)

15 Consequences Proposition f Φ,ν := sup { E ν[ f g] : g 0, E ν[ψ(g)] 1 } Let B S be non-empty and ν P 1(S). Then, for any Orlicz pair (Φ, Ψ), there exist C 1, C 2 (0, ) satisfying C 1 C 2 4C 1 such that the following statements are equivalent: (i) For all A S \ B it holds (ii) For all f : S R with f B 0 holds ν[a] Ψ 1( 1/ν[A] ) C 1 cap(a, B). f 2 Φ,ν C 2 E(f, f). Examples: ( Φp(r), Ψ ) p(r) := ( 1 p rp, 1 p r p ), ( ΦEnt(r), Ψ ) Ent(r) := ( 1 [1, ) (r)(r ln r r + 1), e r 1 ) André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 9 (14)

16 Definition of metastability Definition Let ρ > 0 and M S be finite. {X t : t 0} is ρ-metastable with respect to M (set of metastable points), if max m M P m [ τm\m < τ m ] min A S\M P µa [ τm < τ A ] ρ 1. m 1 S 1 S 4 m 4 m 2 m 3 S 2 S 3 Previous and related definition Bovier (2006), reversible Markov chains with finite state space; reversible diffusions Bovier, den Hollander (2016): Metastability - a portential theoretic approach see Springer bookshelf Metastable partition. S = m M Sm, the sets Sm, m M are mutually disjoint { [ ] [ ] } S m x S : P x τm < τ M\m max m M\m Px τm < τ M\m André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 10 (14)

17 Definition of metastability Definition Let ρ > 0 and M S be finite. {X t : t 0} is ρ-metastable with respect to M (set of metastable points), if max m M P m [ τm\m < τ m ] min A S\M P µa [ τm < τ A ] ρ 1. m 1 S 1 S 4 m 4 m 2 m 3 S 2 S 3 Previous and related definition Bovier (2006), reversible Markov chains with finite state space; reversible diffusions Bovier, den Hollander (2016): Metastability - a portential theoretic approach see Springer bookshelf Metastable partition. S = m M Sm, the sets Sm, m M are mutually disjoint { [ ] [ ] } S m x S : P x τm < τ M\m max m M\m Px τm < τ M\m André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 10 (14)

18 Main result Theorem Suppose {X t : t 0} is a ρ-metastable Markov chain with M = {m 1, m 2}. Then, C PI = µ[s1] µ[s2] ( ) 1 + O( ρ). cap(m 1, m 2) Moreover, under further conditions on µ[ S i], it holds C LSI = µ[s 1] µ[s 2] 1 ( ) 1 + O( ρ), Λ(µ[S 1], µ[s 2]) cap(m 1, m 2) where Λ(s, t) = (s t)/(ln s ln t) denotes the logarithmic mean. Previous and related results Bovier, Eckhoff, Gayrard, Klein (2002), low lying spectrum, reversible Markov chains Bovier, Gayrard, Klein (2005), low lying spectrum, reversible diffusion Bianchi, Gaudilliére (2011), spectral gap, reversible Markov chains Menz, S. (2014), PI and LSI, reversible diffusions André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 11 (14)

19 Main result Theorem Suppose {X t : t 0} is a ρ-metastable Markov chain with M = {m 1, m 2}. Then, C PI = µ[s1] µ[s2] ( ) 1 + O( ρ). cap(m 1, m 2) Moreover, under further conditions on µ[ S i], it holds C LSI = µ[s 1] µ[s 2] 1 ( ) 1 + O( ρ), Λ(µ[S 1], µ[s 2]) cap(m 1, m 2) where Λ(s, t) = (s t)/(ln s ln t) denotes the logarithmic mean. Previous and related results Bovier, Eckhoff, Gayrard, Klein (2002), low lying spectrum, reversible Markov chains Bovier, Gayrard, Klein (2005), low lying spectrum, reversible diffusion Bianchi, Gaudilliére (2011), spectral gap, reversible Markov chains Menz, S. (2014), PI and LSI, reversible diffusions André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 11 (14)

20 Two-scale decomposition Splitting the variance. µ i[ ] := µ[ S i] and µ := µ[s 1] δ m1 + µ[s 2] δ m2 var µ[f] = µ[s 1] var µ1 [f] }{{} local variance + µ[s 2] var µ2 [f] }{{} local variance + µ[s 1] µ[s ( ) 2 2] E µ1 [f] E µ2 [f] }{{} mean difference Splitting the entropy. Ent µ[f 2 ] = µ[s 1] Ent µ1 [f 2 ] }{{} local entropy + µ[s 2] Ent µ2 [f 2 [ ] + Ent µ Eµ [f 2 ] ] }{{}}{{} local entropy macroscopic entropy Ent µ [ Eµ [f 2 ] ] µ[s 1] µ[s ( 2] var µ1 [f] + var µ2 [f] + ( E µ1 [f] E µ2 [f] ) ) 2 Λ(µ[S 1], µ[s 2]) The strategy. rough bounds for local quantities, sharp bounds for the mean difference André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 12 (14)

21 Local variances and mean difference estimate Fact. [ ] 1 [ P µa τmi < τ A M Pµ A τm < τ A] A S i \ {m i} Assumption of metastability. For all A S i \ {m i} µ i[a] ρ M ( [ ] ) max P m τm\{m} < τ m cap(a, m i) µ[s i] m M\{m i } Local variances. M = {m 1, m 2} µ[s i] var µi [f] 4 ρ M µ[s 1] µ[s 2] E(f, f) cap(m 1, m 2) Mean difference estimate. µ[s 1] µ[s 2] ( E µ1 [f] E µ2 [f] )2 µ[s1] µ[s2] cap(m 1, m E(f, f) ( 1 + O( ρ M ) ) 2) André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 13 (14)

22 Summary and open problems What has been done so far. Capacitary inequality that allows to establish a local PI and LSI inequality Method can be applied beyond the situation of metastable points (e.g. RFCW) Generalization to metastable sets under suitable regularity assumptions on the metastable sets Possible extensions and future challenges Continuous high dimensional systems (approximation of stochastic 1d-Allen-Cahn equation) certain types of entropic metastability non-reversible systems André Schlichting Capacitary inequalities and metastable Markov chains March 2, 2016 Page 14 (14)