Large Deviations from the Hydrodynamic Limit for a System with Nearest Neighbor Interactions

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1 Large Deviations from the Hydrodynamic Limit for a System with Nearest Neighbor Interactions Amarjit Budhiraja Department of Statistics & Operations Research University of North Carolina at Chapel Hill Joint work with yan Banerjee and Michael Perlmutter University of Southern California University of Southern California 1 /

2 Background Let {Y i } i 1 be a sequence of R d -valued iid zero mean random variables with common probability law ρ. Let S n = n i=1 Y i. Then S n /n 0 a.s. by LLN. Large Deviation Principle: For c > 0 where for y R d, P( S n > nc) exp{ ninf{i(y) : y c}}, I(y) = sup { α,y log exp α,y ρ(dy)}. α R d R d University of Southern California 2 /

3 Large Deviation Principle. Definition. Consider a sequence {X ε } ε>0 of E valued r.vs. I : E [0, ] is a rate function on E if for each M <, {x E : I(x) M} is compact. {X ǫ } is said to satisfy the large deviation principle on E (as ε 0) with rate function I if: For each closed F E For each open G E Formally, for small ε: limsup ǫ 0 ǫlogp(x ǫ F) inf x F I(x). liminf ǫ 0 ǫlogp(x ǫ G) inf x G I(x). P(X ǫ A) exp { inf } x AI(x), A B(E). ε University of Southern California 3 /

4 Stochastic Control Connection (Fleming 1978) Consider a small noise n-dimensional SDE: dx ε (t) = b(x ε (t))dt + εσ(x ε (t))dw(t), X ε (0) = x. b, σ suitable coefficients... W a f.d. BM. Let G R n be bounded open. Let x G and τ ε = inf{t : X ε (t) G}. Interested in lim ε 0 εlogp x (X ε (τ ε ) N), where N G. University of Southern California 4 /

5 Stochastic Control Connection (Ctd.) Formally, with Φ a nonnegative C 2 function, Φ(x) M1 N c(x), M a large scaler, { } lim εlogp x(x ε (τ ε ) N) lim εloge x e Φ(Xε (τ ε ))/ε. ε 0 ε 0 Then g ε (x) = E x { e Φ(X ε (τ ε ))/ε } solves where L ε g = ε 2 Tr(σD2 gσ )+b g. Interested in asymptotics of εlogg ε. L ε g ε (x) = 0, x G g ε (x) = e Φ(x)/ε, x G University of Southern California 5 /

6 Stochastic Control Connection (ctd.) log transform: Let J ε = εlogg ε. Then J ε solves where ε 2 Tr(σD2 J ε σ )+H(x, J ε ) = 0 H(x,p) = min v Rn[L(x,v)+p v], x G, p Rn and L(x,v) = 1 2 (b(x) v) [σ(x)σ (x)] 1 (b(x) v). J ε can be characterized as the value function of the stochastic control problem: { τ ε } J ε (x) = inf E x L( X ε (t),u(t))dt +Φ( X ε ( τ ε )) u A 0 d X ε (t) = u(t)dt + εσ(x ε (t))dw(t), Xε (0) = x University of Southern California 6 /

7 Stochastic Control Connection (ctd.) One can argue J ε J, where J(x) is the value function of the deterministic control problem: [ θ ] J(x) = inf L(φ(t), φ(t))dt +Φ(φ(θ)), φ,θ 0 where inf is over all abs. cts. φ such that φ(0) = x, and θ = inf{t : φ(t) G}. Later works: Sheu (1985), Dupuis and Ellis(1997), Feng and Kurtz (2005). University of Southern California 7 /

8 LDP and Laplace Principle. LDP is equivalent to Laplace principle if the state space is Polish (Varadhan(1966), Bryc(1990)): A collection of E valued random variables {X ε } is said to satisfy Laplace principle with rate function I, if for all h C b (E) lim ǫlog E ǫ 0 From Donsker-Varadhan: { exp [ 1 ǫ h(xǫ ) ]} = inf x E {h(x)+i(x)}. ǫlog E{exp[ 1 ǫ h(xǫ )]}=inf Q P(E)[ h(x)dq(x)+r(q P ε )]. University of Southern California 8 /

9 LDP and Laplace Principle. Goal is to show the convergence of variational expressions: inf Q P(E)[ h(x)dq(x)+r(q P ε )] ε 0 inf x E {h(x)+i(x)}. University of Southern California 9 /

10 Some Settings where the Approach Works. Small Noise SPDE (B., Dupuis and Maroulas (2008)). Stochastic Flows of Diffeomorphisms (B., Dupuis and Maroulas (2010)). Finite and Infinite Dimensional Jump-Stochastic Dynamical Systems with Small Noise (B., Chen and Dupuis (2013)). Moderate deviation principles for SDE w/ Jumps in Finite and Infinite Dimensions (B., Dupuis and Ganguly (2016)). Component Size Large Deviations for Configuration Model (Bhamidi, B., Dupuis and Wu (2017)). Multiscale jump-diffusions Large Deviations from Stochastic Averaging Principle (B., Dupuis and Ganguly (2017)). Weakly Interacting Diffusions Large and Moderate Deviations (B., Dupuis and Fischer (2012), B. and Wu (2016)). University of Southern California 10 /

11 A System with Nearest Neighbor Interactions. Ginzburg-Landau in Finite Volume: For t [0,T] and i = 1,...N dxi N (t) = N2 [ φ ( ) X N 2 i 1(t) 2φ ( Xi N (t)+φ ( ))] Xi+1(t) N dt +N[dB i (t) db i+1 (t)] {1/N,...,(N 1)/N,1} is the periodic lattice. I.e. identify X N N+1 with XN 1. {B i (t)} i=1 are independent standard one-dimensional Brownian motions given on some probability space (V,F,P). University of Southern California 11 /

12 A System with Nearest Neighbor Interactions. φ : R R is C 2 and exp( φ(x))dx = 1, M(λ) =. R for all λ R, and for all σ < exp(σ φ (x) φ(x))dx <. R R exp(λx φ(x)) < University of Southern California 12 /

13 Model Description (ctd.) Invariant measure for X N : Φ N (dx). = Φ(dx 1 )Φ(dx 2 )...Φ(dx N ) where Φ(dx). = e φ(x) dx. Consider X N with X N (0) Φ N and process µ N with values in M S (the space of signed measures on the unit circle S): µ N (t,dθ). = 1 N N i=1 X N i (t)δ i/n (dθ). A LLN for µ N shown in Guo-Papanicolaou-Varadhan (1988) and a LDP proved in Donsker-Varadhan (1989). A new proof... University of Southern California 13 /

14 Remarks The original proof [DW(1989)] requires control on exponential moments and exponential probability estimates. This approach has been extended to many different systems. Exponential estimates are the hardest parts of the proof. The new proof uses stochastic control representations and weak convergence methods. Proof techniques similar to that for LLN analysis. No exponential estimates are invoked. Key Technical Step: Suitable Regularity of Densities of Controlled Processes. Bounds on certain Dirichlet Forms. University of Southern California 14 /

15 Main Result Let M l S be elements in M S with total variation bounded by l. Let for l N, Ω l. = C([0,T] : M l S ) the Polish space of continuous paths of signed measures with total variation bounded by l. Then Ω. = C([0,T] : M S ) = l N C([0,T] : M l S ) = l NΩ l. This space is equipped with the direct limit topology. Theorem {µ N } satisfies a LDP in C([0,T] : M S ) with rate function I. University of Southern California 15 /

16 Rate Function Let ρ(λ) =. logm(λ), λ R, h(x) =. sup{λx ρ(λ)}. λ R Let Ω be the collection of all µ in Ω such that for all t, µ(t,dθ) = m(t,θ)dθ and m satisfies [h(m(t,θ))+[h (m(t,θ))] 2 θ ]dtdθ <, [0,T] S Let P (R S) be all π P(R S) such that π(dx dθ) = π 1 (dx θ)dθ, with m 0 (θ) = R xπ 1 (dx θ), S h(m 0 (θ))dθ <. University of Southern California 16 /

17 Rate Function For u L 2 ([0,T] S : R) and π P (R S) let M (u,π) be all µ Ω, s.t. µ(t,dθ) = m(t,θ)dθ, and m solves weakly t m(t,θ) = 1 2 [ h (m(t,θ)) ] θθ θu(t,θ), m(0,θ) = m 0 (θ) Letting π 0 (dx dθ) = Φ(dx)dθ, define I : Ω [0, ] by I(µ) = [ 1 T ] inf u(s,θ) 2 dθds +R(π π 0 ) {(u,π):µ M (u,π)} 2 0 S for µ Ω, and set I(µ) = otherwise. University of Southern California 17 /

18 Main Steps in Proof Compact level sets: I is a rate function on Ω, namely for every M < {µ Ω : I(µ) M} is compact. Laplace upper bound: For all F C b (Ω) lim sup 1 N N loge[exp( NF(µN ))] inf {F(µ)+I(µ)}. µ Ω Laplace lower bound: For all F C b (Ω) lim inf N 1 N loge[exp( NF(µN ))] inf µ Ω {F(µ)+I(µ)}. University of Southern California 18 /

19 Variational Representation Boué-Dupuis (1998), B., Fan and Wu (2017). Let ( V, F, P) be a probability space with an N-dimensional Brownian motion, B N = (B 1,...B N ), and a R N -valued random variable X N (0) independent of B N and with probability law Π N. Let { F t } be any filtration satisfying the usual conditions such that B N is a { F t }-Brownian motion and X N (0) is F 0 measurable. Let K Π N. = ( V, F,{ F t }, P, X N (0),B N ) and let A N (K Π N). = {ψ : ψ = (ψ i ) N i=1,ψ i is simple and F t adapted}. University of Southern California 19 /

20 Variational Representation For a ψ N A N (K Π N), let B i N (t) =. t B i (t)+ ψi N (s)ds, t [0,T], i = 1,...N. 0 Let d X i N (t) = N2 [ ) ))] φ ( XN 2 i 1 (t) 2φ ( XN i (t)+φ ( XN i+1 (t) dt +N [ d B i (t) d B i+1 (t) ] Disintegrate Π N P(R N ), as Π N (dx) =. Π 1 (dx 1 )Π 2 (dx 2 x 1 )...Π N (dx N dx 1,...,dx N 1 ) =. N Φ N i (x,dx i ), and with X N (0) distributed as Π N, let Φ N i (dz). = Φ N i ( X N (0),dz). i=1 University of Southern California 20 /

21 Variational Representation Let F C b (Ω). Then for all N N 1 N logeexp( NF(µN )) = inf Π N,K Π N inf Ē Π ψ N A N N (K Π N) [ 1 N N i=1 ( R( Φ N i Φ)+ 1 T ) ] ψi N (s) 2 ds +F( µ N ). 2 0 University of Southern California 21 /

22 Laplace Upper Bound Fix F C b (Ω) and let ǫ (0,1). Choose for each N N, Π N P(R N ), a system K Π N and ψ N A N (K Π N) such that 1 N logeexp( NF(µN )) Ē 1 ( N Π N[ N i=1 R( Φ N i Φ)+ 1 2 Since F is bounded, there is a C (0, ) such that sup N N Ē Π N( 1 N N i=1 R( Φ N i Φ)) C, sup N N Ē Π N ) ] T 0 ψn i (s) 2 ds +F( µ N ) ǫ. ( 1 N ) T 2N i=1 0 ψn i (s) 2 ds C. By a localization argument, we can assume that for every N 1 2N N i=1 T 0 ψ N i (s) 2 ds C a.s. University of Southern California 22 /

23 Consequences of Bounded Costs Suppose sup N N Ē Π N( 1 ( N N i=1 R( Φ N i Φ)) C 0, sup 1 N ) T N N 2N i=1 0 ψn i (s) 2 ds C 0. Lemma 1 Let Q Π N(t) = L( X N (t)). Then, there exists C T (0, ) s.t. for t [0,T], H N (t). = R( Q Π N(t) Φ N ) C T N for all N N. Let V i = i i+1 and a positive f on R N that is continuously differentiable along V 1,...,V N, define I N (f) = 4D N ( f) = N RN (V i f(x)) 2 Φ N (dx). f(x) i=1 Lemma 2 For t [0,T] and N N, X N (t) has a density p N (t, ) w.r.t Φ N which is continuously differentiable, once in time and twice along V 1,...,V N, and satisfies for some C (0, ): I N ( 1 T T 0 ) p N (s, )ds C for all N 1. N University of Southern California 23 /

24 Consequences of Bounds on RE and Dirichlet Forms { µ N } is a tight sequence of Ω-valued random variables. {ν N } is a tight sequence of P(R S) valued r.v., where Define u N as u N (t,θ). = ν N (dx dθ). = N i=1 Φ N i (dx)i (i/n,(i+1)/n] (θ)dθ. N ψi N (t)i ((i 1)/N,i/N] (θ), (t,θ) [0,T] S. i=1 Then {u N } is a sequence of r.v. with values in }. S C0 = {u L 2 ([0,T] S) : u(t,θ) 2 dθdt C 0. [0,T] S With the weak topology on the Hilbert space S C0 is compact and thus {u N } N N is a tight sequence of S C0 -valued random variables. Suppose ( µ N,u N,ν N ) converges in distribution along a subsequence to ( µ,u,ν). Then µ M (u,ν), a.s. University of Southern California 24 /

25 Completing the Proof of Upper Bound. lim inf 1 ( N N logeexp NF ( ( liminfē Π N F µ N) + 1 N N Ē Π N N = liminf Ē ( µ N)) +ǫ N i=1 ( ( F µ N) +R(ν N π 0 )+ 1 2 ( F ( µ)+r(ν π 0 )+ 1 2 T Ē[F( µ)+i( µ)] inf µ Ω [F(µ)+I(µ)], 0 ( R( Φ N i Φ)+ 1 T ) ) ψi N (s) 2 ds 2 0 ) S T 0 S ) u(s,θ) 2 dsdθ u N (s,θ) 2 dsdθ where the next to last inequality follows from µ M (u,ν) a.s. University of Southern California 25 /

26 Laplace Lower Bound. Fix F C b (Ω), and let ǫ > 0. Choose µ Ω such that F( µ )+I( µ ) inf {F(µ)+I(µ)}+ǫ <, µ Ω Choose u L 2 ([0,T] S) and π P (R S) such that µ M (u,π ) and I( µ )+ǫ 1 [ T ] u (s,θ) 2 dθds +R(π π 0 ). 2 0 Fix δ (0,1) and let u C be such that u u 2 S δ 2(1+ u 2 ). Lemma There is a unique µ Ω s.t. µ M(u,π ). Furthermore, as δ 0, µ µ (δ) µ. University of Southern California 26 /

27 Control Synthesis. Let π (dx,dθ) = π1 (dx θ)dθ and define Φ N i (dx) =. i/n N π1 (dx θ)dθ, 1 i N, (i 1)/N Let X N (0) be a R N -valued r.v. with distribution Define ψ N i L 2 ([0,T] : R) as ψ N i (t). = j=1 Π N (dx). = Φ N 1 (dx 1)... Φ N N (dx N). N ( jt u N, i ) I N (jt/n,(j+1)t/n] (t), t [0,T]. University of Southern California 27 /

28 Control Synthesis. Lemma Let µ N be constructed using Π N and {ψ N i }. Then 1 T lim N N 0 N ψi N (t) 2 dt = i=1 T 0 S u (t,θ) 2 dθdt, 1 N N R( Φ N i Φ) R(π π 0 ), for all N N i=1 and µ N converges to µ in distribution in Ω. University of Southern California 28 /

29 Completing the Proof of Lower Bound. Choose δ small s.t. F( µ ) F( µ ) ǫ. Then lim sup 1 ( NF N N logeexp ( limsupē Π N N F ( µ N) + 1 N F( µ )+R(π π 0 )+ 1 2 F( µ )+R(π π 0 )+ 1 2 F( µ )+I( µ )+2ǫ+2δ (µ N)) N i=1 T S 0 T inf µ Ω {F(µ)+I(µ)}+3ǫ+2δ. S 0 ( R( Φ N i Φ)+ 1 T ) ) ψi N (s) 2 ds 2 0 u (s,θ) 2 dsdθ u (s,θ) 2 dsdθ+ǫ+2δ University of Southern California 29 /