Weak convergence and large deviation theory

Transcription

1 First Prev Next Go To Go Back Full Screen Close Quit 1 Weak convergence and large deviation theory Large deviation principle Convergence in distribution The Bryc-Varadhan theorem Tightness and Prohorov s theorem Exponential tightness Tightness for processes (Exponential) tightness and results for finite dimensional distributions Conditions for (exponential) tightness Joint work with Jin Feng Second Lecture

2 First Prev Next Go To Go Back Full Screen Close Quit 2 Large deviation principle (S, d) X n, n = 1, 2,... a (complete, separable) metric space. S-valued random variables {X n } satisfies a large deviation principle (LDP) if there exists a lower semicontinuous function I : S [0, ] such that for each open set A, lim inf n and for each closed set B, lim sup n 1 n log P {X n A} inf x A I(x) 1 n log P {X n B} inf x B I(x).

3 First Prev Next Go To Go Back Full Screen Close Quit 3 The rate function I is called the rate function for the large deviation principle. A rate function is good if for each a [0, ), {x : I(x) a} is compact. If I is a rate function for {X n }, then I (x) = lim ɛ 0 inf I(y) y B ɛ (x) is also a rate function for {X n }. I is lower semicontinuous. If the large deviation principle holds with lower semicontinuous rate I function, then I(x) = lim lim inf ɛ 0 n 1 n log P {X 1 n B ɛ (x)} = lim lim sup ɛ 0 n n log P {X n B ɛ (x)}

4 First Prev Next Go To Go Back Full Screen Close Quit 4 Convergence in distribution {X n } converges in distribution to X if and only if for each f C b (S) lim E[f(X n)] = E[f(X)] n

5 First Prev Next Go To Go Back Full Screen Close Quit 5 Equivalent statements: Large deviation principle {X n } satisfies an LDP with a good rate function if and only if {X n } is exponentially tight and for each f C b (S). Then 1 Λ(f) lim n n log E[enf(X n) ] I(x) = sup {f(x) Λ(f)} C b (S) and Λ(f) = sup{f(x) I(x)} x S Bryc, Varadhan

6 First Prev Next Go To Go Back Full Screen Close Quit 6 Equivalent statements: Convergence in distribution {X n } converges in distribution to X if and only if or equivalently lim inf n P {X n A} P {X A}, each open A, lim sup P {X n B} P {X B}, each closed B n LDP

7 First Prev Next Go To Go Back Full Screen Close Quit 7 Tightness A sequence {X n } is tight if for each ɛ > 0, there exists a compact set K ɛ S such that sup P {X n / K ɛ } ɛ. n Prohorov s theorem Theorem 1 Suppose that {X n } is tight. Then there exists a subsequence {n(k)} along which the sequence converges in distribution.

8 First Prev Next Go To Go Back Full Screen Close Quit 8 Exponential tightness {X n } is exponentially tight if for each a > 0, there exists a compact set K a S such that lim sup n 1 n log P {X n / K a } a. Analog of Prohorov s theorem Theorem 2 (Puhalskii, O Brien and Vervaat, de Acosta) Suppose that {X n } is exponentially tight. Then there exists a subsequence {n(k)} along which the large deviation principle holds with a good rate function.

9 First Prev Next Go To Go Back Full Screen Close Quit 9 Stochastic processes in D E [0, ) (E, r) complete, separable metric space S = D E [0, ) Modulus of continuity: w (x, δ, T ) = inf max {t i } i sup s,t [t i 1,t i ) r(x(s), x(t)) where the infimum is over {t i } satisfying 0 = t 0 < t 1 < < t m 1 < T t m and min 1 i n (t i t i 1 ) > δ X n stochastic process with sample paths in D E [0, ) X n adapted to {F n t }: For each t 0, X n (t) is F n t -measurable.

10 First Prev Next Go To Go Back Full Screen Close Quit 10 Tightness in D E [0, ) Theorem 3 (Skorohod) Suppose that for t T 0, a dense subset of [0, ), {X n (t)} is tight. Then {X n } is tight if and only if for each ɛ > 0 and T > 0 lim δ 0 lim sup P {w (X n, δ, T ) > ɛ} = 0. n Theorem 4 (Puhalskii) Suppose that for t T 0, a dense subset of [0, ), {X n (t)} is exponentially tight. Then {X n } is exponentially tight if and only if for each ɛ > 0 and T > 0 1 lim lim sup δ 0 n n log P {w (X n, δ, T ) > ɛ} =.

11 First Prev Next Go To Go Back Full Screen Close Quit 11 Identification of limit distribution Theorem 5 If {X n } is tight in D E [0, ) and (X n (t 1 ),..., X n (t k )) (X(t 1 ),..., X(t k )) for t 1,..., t k T 0, T 0 dense in [0, ), then X n X. Identification of rate function Theorem 6 If {X n } is exponentially tight in D E [0, ) and for each 0 t 1 < < t m, {(X n (t 1 ),..., X n (t m ))} satisfies the large deviation principle in E m with rate function I t1,...,t m, then {X n } satisfies the large deviation principle in D E [0, ) with good rate function I(x) = sup {t i } c x I t1,...,t m (x(t 1 ),..., x(t m )), where x is the set of discontinuities of x.

12 First Prev Next Go To Go Back Full Screen Close Quit 12 Conditions for tightness S n 0 (T ) collection of discrete {F n t }-stopping times q(x, y) = 1 r(x, y) Suppose that for t T 0, a dense subset of [0, ), {X n (t)} is tight. Then the following are equivalent. a) {X n } is tight in D E [0, ).

13 Conditions for tightness b) For T > 0, there exist β > 0 and random variables γ n (δ, T ), δ > 0, satisfying E[q β (X n (t + u), X n (t)) q β (X n (t), X n (t v)) F n t ] E[γ n (δ, T ) F n t ] (1) for 0 t T, 0 u δ, and 0 v t δ such that and lim δ 0 lim δ 0 lim sup E[γ n (δ, T )] = 0 n lim sup E[q β (X n (δ), X n (0))] = 0. (2) n Kurtz First Prev Next Go To Go Back Full Screen Close Quit 13

14 Conditions for tightness c) Condition (2) holds, and for each T > 0, there exists β > 0 such that C n (δ, T ) sup sup τ S0 n(t ) u δ E[ sup q β (X n (τ + u), X n (τ)) v δ τ satisfies lim δ 0 lim sup n C n (δ, T ) = 0. q β (X n (τ), X n (τ v))] Aldous First Prev Next Go To Go Back Full Screen Close Quit 14

15 First Prev Next Go To Go Back Full Screen Close Quit 15 Conditions for exponential tightness S n 0 (T ) collection of discrete {F n t }-stopping times q(x, y) = 1 r(x, y) Suppose that for t T 0, a dense subset of [0, ), {X n (t)} is exponentially tight. Then the following are equivalent. a) {X n } is exponentially tight in D E [0, ).

16 First Prev Next Go To Go Back Full Screen Close Quit 16 Conditions for exponential tightness b) For T > 0, there exist β > 0 and random variables γ n (δ, λ, T ), δ, λ > 0, satisfying E[e nλqβ (X n (t+u),x n (t)) q β (X n (t),x n (t v)) F n t ] E[e γ n(δ,λ,t ) F n t ] for 0 t T, 0 u δ, and 0 v t δ such that and 1 lim lim sup δ 0 n n log E[eγ n(δ,λ,t ) ] = 0, 1 lim lim sup δ 0 n n log (X n (δ),x n (0)) ] = 0. (3) E[enλqβ

17 First Prev Next Go To Go Back Full Screen Close Quit 17 Conditions for exponential tightness c) Condition (3) holds, and for each T > 0, there exists β > 0 such that for each λ > 0 C n (δ, λ, T ) sup sup τ S0 n(t ) u δ E[ sup e nλqβ (X n (τ+u),x n (τ)) q β (X n (τ),x n (τ v)) ] v δ τ satisfies lim δ 0 lim sup n 1 n log C n(δ, λ, T ) = 0.

18 First Prev Next Go To Go Back Full Screen Close Quit 18 Example W standard Brownian motion X n = 1 n W E[e nλ X n(t+u) X n (t) F W t ] = E[e λ n W (t+u) W (t) F W t ] 2e 1 2 nλ2 u so 1 lim lim sup δ 0 n n log E[eγ n(δ,λ,t ) 1 ] = lim δ 0 2 λ2 δ = 0.

19 First Prev Next Go To Go Back Full Screen Close Quit 19 Equivalence to tightness for functions Theorem 7 {X n } is tight in D E [0, ) if and only if a) (Compact containment condition) For each T > 0 and ɛ > 0, there exists a compact K ɛ,t E such that lim sup P ( t T X n (t) K ɛ,t ) ɛ n b) There exists a family of functions F C(E) that is closed under addition and separates points in E such that for each f F, {f(x n )} is tight in D R [0, ). Kurtz, Jakubowski

20 First Prev Next Go To Go Back Full Screen Close Quit 20 Equivalence to exponential tightness for functions Theorem 8 {X n } is exponentially tight in D E [0, ) if and only if a) For each T > 0 and a > 0, there exists a compact K a,t E such that 1 lim sup n n log P ( t T X n(t) K a,t ) a (4) b) There exists a family of functions F C(E) that is closed under addition and separates points in E such that for each f F, {f(x n )} is exponentially tight in D R [0, ). Schied

21 First Prev Next Go To Go Back Full Screen Close Quit 21 Large Deviations for Markov Processes Martingale problems and semigroups Semigroup convergence and the LDP Control representation of the rate function Viscosity solutions and semigroup convergence Summary of method

22 First Prev Next Go To Go Back Full Screen Close Quit 22 Markov processes X n = {X n (t), t 0} is a Markov process if E[g(X n (t + s)) F n t ] = E[g(X n (t + s)) X n (t)] The generator of a Markov process determines its short time behavior E[g(X n (t + t)) g(x n (t)) F t ] A n g(x n (t)) t

23 First Prev Next Go To Go Back Full Screen Close Quit 23 Martingale problems X n is a solution of the martingale problem for A n if and only if g(x n (t)) g(x n (0)) t is an {F n t }-martingale for each g D(A n ). 0 A n g(x n (s))ds (5) If g is bounded away from zero, (5) is a martingale if and only if g(x n (t)) exp{ t 0 A n g(x n (s)) g(x n (s)) ds} is a martingale. (You can always add a constant to g.)

24 First Prev Next Go To Go Back Full Screen Close Quit 24 Nonlinear generator Define D(H n ) = {f B(E) : e nf D(A n )} and set H n f = 1 n e nf A n e nf. Then exp{nf(x n (t)) nf(x(0)) is a {Ft n }-martingale. t 0 nh n f(x(s))ds}

25 First Prev Next Go To Go Back Full Screen Close Quit 25 Tightness for solutions of MGPs E[f(X n (t + u)) f(x n (t)) F n t ] = E[ t+u t A n f(x n (s))ds F n t ] u A n f For γ n (δ, T ) = δ( A n f f A n f ) (see (1)) E[(f(X n (t + u)) f(x n (t))) 2 F n t ] = E[ t+u t A n f 2 (X n (s))ds F n t ] 2f(X n (t))e[ t+u t A n f(x n (s))ds F n t ] u( A n f f A n f ) γ n (δ, T )

26 First Prev Next Go To Go Back Full Screen Close Quit 26 Exponential tightness E[e n(λf(x n(t+u)) λf(x n (t)) t+u t H n [λf](x n (s))ds F n t ] = 1 so and E[e nλ(f(x n(t+u)) f(x n (t))) F n t ] e nu H nλf γ n (δ, λ, T ) = δn( H n [λf] + H n [ λf] ) ET Conditions

27 First Prev Next Go To Go Back Full Screen Close Quit 27 The Markov process semigroup Assume that the martingale problem for A n is well-posed. Define T n (t)f(x) = E[f(X n (t)) X n (0) = x] By the Markov property T n (s)t n (t)f(x) = T n (t + s)f(x) T n (t)f(x) f(x) lim t 0 t = A n f(x)

28 Iterating the semigroup For 0 t 1 t 2, E[f 1 (X n (t 1 ))f 2 (X n (t 2 )) X n (0) = x] = T n (t 1 )(f 1 T n (t 2 t 1 )f 2 )(x) and in general E[f 1 (X n (t 1 )) f k (X n (t k )) X n (0) = x] = E[f 1 (X n (t 1 )) f k 1 (X n (t k 1 )) T n (t k t k 1 )f k (X n (t k 1 )) X n (0) = x] Convergence of the semigroups implies convergence of the finite dimensional distributions. First Prev Next Go To Go Back Full Screen Close Quit 28

29 First Prev Next Go To Go Back Full Screen Close Quit 29 A nonlinear semigroup (Fleming) Assume that the martingale problem for A n is well-posed. Define V n (t)f(x) = 1 n log E x[e nf(xn(t)) ] By the Markov property V n (s)v n (t)f(x) = V n (t + s)f(x) V n (t)f(x) f(x) lim t 0 t = 1 n e nf A n e nf (x) = H n f(x) Exponential generator

30 Iterating the semigroup For 0 t 1 t 2, define V n (t 1, t 2, f 1, f 2 )(x) = V n (t 1 )(f 1 + V n (t 2 t 1 )f 2 )(x) and inductively V n (t 1,..., t k, f 1,..., f k )(x) = V n (t 1 )(f 1 + V n (t 2,..., t k, f 2,..., f k ))(x). Then E[e n(f 1(X n (t 1 ))+ +f k (X n (t k ))) ] = E[e nv n(t 1,...,t k,f 1,...,f k )(X n (0)) ] By the Bryc-Varadhan result, convergence of semigroup should imply the finite dimensional LDP First Prev Next Go To Go Back Full Screen Close Quit 30

31 First Prev Next Go To Go Back Full Screen Close Quit 31 Weaker conditions for the LDP A collection of functions D C b (S) isolates points in S, if for each x S, each ɛ > 0, and each compact K S, there exists f D satisfying f(x) < ɛ, sup y K f(y) 0, and sup f(y) < 1 y K B ɛ. ɛ(x) c A collection of functions D C b (S) is bounded above if sup f D sup y f(y) <.

32 First Prev Next Go To Go Back Full Screen Close Quit 32 A rate determining class Proposition 9 Suppose {X n } is exponentially tight, and let 1 Γ = {f C b (S) : Λ(f) = lim n n log E[enf(X n) ] exists}. If D Γ is bounded above and isolates points, then Γ = C b (S) and I(x) = sup{f(x) Λ(f)}. f D

33 First Prev Next Go To Go Back Full Screen Close Quit 33 Semigroup convergence and the LDP Suppose D C b (E) contains a set that is bounded above and isolates points. Suppose X n (0) = x and {X n (t)} is exponentially tight. If V n (t)f(x) V (t)f(x) for each f D, then {X n (t)} satisfies a LDP with rate function I t (y x) = sup{f(y) V (t)f(x)}, f D and hence V (t)f(x) = sup{f(y) I t (y x)} y Think of I t (y x) as the large deviation analog of a transition density.

34 Iterating the semigroup Suppose D is closed under addition, V (t) : D D, t 0, and 0 t 1 t 2. Define and inductively V (t 1, t 2, f 1, f 2 )(x) = V (t 1 )(f 1 + V (t 2 t 1 )f 2 )(x) V (t 1,..., t k, f 1,..., f k )(x) = V (t 1 )(f 1 + V (t 2,..., t k, f 2,..., f k ))(x) First Prev Next Go To Go Back Full Screen Close Quit 34

35 First Prev Next Go To Go Back Full Screen Close Quit 35 Semigroup convergence and the LDP Theorem 10 For each n, let A n C b (E) B(E), and suppose that existence and uniqueness holds for the D E [0, )-martingale problem for (A n, µ) for each initial distribution µ P(E). Let D C b (E) be closed under addition and contain a set that is bounded above and isolates points, and suppose that there exists an operator semigroup {V (t)} on D such that for each compact K E sup V (t)f(x) V n (t)f(x) 0, f D. x K

36 Suppose that {X n } is exponentially tight, and that {X n (0)} satisfies a large deviation principle with good rate function I 0. Define 1 Λ 0 (f) = lim n n log E[enf(X n(0)) ], f C b (E). a) For each 0 t 1 < < t k and f 1,..., f k D, Recall 1 lim n n log E[enf 1(X n (t 1 ))+...+nf k (X n (t k )) ] = Λ 0 (V (t 1,..., t k, f 1,..., f k )). E[e n(f 1(X n (t 1 ))+ +f k (X n (t k ))) ] = E[e nv n(t 1,...,t k,f 1,...,f k )(X n (0)) ] First Prev Next Go To Go Back Full Screen Close Quit 36

37 First Prev Next Go To Go Back Full Screen Close Quit 37 b) For 0 t 1 <... < t k {(X n (t 1 ),..., X n (t k ))} satisfies the large deviation principle with rate function I t1,...,t k (x 1,..., x k ) (6) = sup {f 1 (x 1 ) f k (x k ) f 1,...,f k D C b (E) Λ 0 (V (t 1,..., t k, f 1,..., f k ))} = inf x 0 E (I 0(x 0 ) + k I ti t i 1 (x i x i 1 )) i=1 c) {X n } satisfies the large deviation principle in D E [0, ) with rate function I(x) = sup {t i } c x I t1,...,t k (x(t 1 ),..., x(t k )) k = sup {t i } c x (I 0 (x(0)) + I ti t i 1 (x(t i ) x(t i 1 ))) i=1

38 First Prev Next Go To Go Back Full Screen Close Quit 38 Example: Freidlin and Wentzell small diffusion Let X n satisfying the Itô equation X n (t) = x + 1 n t 0 σ(x n (s ))dw (s) + and define a(x) = σ T (x) σ(x). Then A n g(x) = 1 a ij (x) i j g(x) + 2n i ij t 0 b(x n (s))ds, b i (x) i g(x), Take D(A n ) to be the collection of functions of the form c + f, c R and f C 2 c (R d ).

39 First Prev Next Go To Go Back Full Screen Close Quit 39 Convergence of the nonlinear generator H n f(x) = 1 2n and Hf = lim n H n f is a ij (x) ij f(x) + 1 a ij (x) i f(x) j f(x) 2 ij ij + i b i (x) i f(x). Hf(x) = 1 2 ( f(x))t a(x) f(x) + b(x) f(x).

40 First Prev Next Go To Go Back Full Screen Close Quit 40 A control problem Let (E, r) and (U, q) be complete, separable metric spaces, and let A : D(A) C b (E) C(E U) Let H be as above, and suppose that that there is a nonnegative, lower semicontinuous function L on E U such that Hf(x) = sup(af(x, u) L(x, u)). u U {V (t)} should be the Nisio semigroup corresponding to an optimal control problem with reward function L. (cf. Book by Dupuis and Ellis)

41 First Prev Next Go To Go Back Full Screen Close Quit 41 Dynamics of control problem Require f(x(t)) f(x(0)) U [0,t] Af(x(s), u)λ s (du ds) = 0, for each f D(A) and t 0, where x D E [0, ) and λ M m (U), the space of measures on U [0, ) satisfying λ(u [0, t]) = t.) For each x 0 E, we should have V (t)g(x 0 ) = sup (x,λ) J t x 0 {g(x(t)) [0,t] U L(x(s), u)λ(du ds)}

42 First Prev Next Go To Go Back Full Screen Close Quit 42 Representation theorem Theorem 11 Suppose (E, r) and (U, q) are complete, separable, metric spaces. Let A : D(A) C b (E) C(E U) and lower semicontinuous L(x, u) 0 satisfy 1. D(A) is convergence determining. 2. For each x 0 E, there exists (x, λ) J such that x(0) = x 0 and L(x(s), u)λ(du ds) = 0, t 0. U [0,t] 3. For each f D(A), there exists a nondecreasing function ψ f : [0, ) [0, ) such that Af(x, u) ψ f (L(x, u)), (x, u) E U, and lim r r 1 ψ f (r) = 0.

43 First Prev Next Go To Go Back Full Screen Close Quit There exists a tightness function Φ on E U, such that Φ(x, u) L(x, u) for (x, u) E U. Let {V (t)} be an LDP limit semigroup and satisfy the control identity. Then I(x) = I 0 (x(0)) + inf { L(x(s), u)λ(du ds)}. λ:(x,λ) J U [0, )

44 First Prev Next Go To Go Back Full Screen Close Quit 44 Small diffusion Hf(x) = 1 2 ( f(x))t a(x) f(x) + b(x) f(x) For and Af(x, u) = u f(x) L(x, u) = 1 2 (u b(x)a(x) 1 (u b(x)), I(x) = 0 Hf(x) = sup u R d (Af(x, u) L(x, u)) 1 2 (ẋ(s) b(x(s))a(x(s)) 1 (ẋ(s) b(x(s)))ds

45 First Prev Next Go To Go Back Full Screen Close Quit 45 Alternative representation For and again Af(x, u) = (u T σ(x) + b(x)) f(x) L(x, u) = 1 2 u 2, Hf(x) = sup(af(x, u) L(x, u)) u R d 1 I(x) = inf{ 0 2 u(s) 2 ds : ẋ(t) = u T (t)σ(x(t)) + b(x(t))}

46 First Prev Next Go To Go Back Full Screen Close Quit 46 Legendre transform approach If Hf(x) = H(x, f(x)), where H(x, p) is convex and continuous in p, then L(x, u) = sup p R d {p u H(x, p)} and H(x, p) = sup{p u L(x, u)}, u R d so taking Af(x, u) = u f(x), Hf(x) = sup u R d {u f(x) L(x, u)}

47 First Prev Next Go To Go Back Full Screen Close Quit 47 Viscosity solutions Let E be compact, H C(E) B(E), and (f, g) H imply (f +c, g) H. Fix h C(E) and α > 0. f B(E) is a viscosity subsolution of f αhf = h (7) if and only if f is upper semicontinuous and for each (f 0, g 0 ) H there exists x 0 E satisfying (f f 0 )(x 0 ) = sup x (f(x) f 0 (x)) and or equivalently f(x 0 ) h(x 0 ) α (g 0 ) (x 0 ) f(x 0 ) α(g 0 ) (x 0 ) + h(x 0 )

48 First Prev Next Go To Go Back Full Screen Close Quit 48 f B(E) is a viscosity supersolution of (7) if and only if f is lower semicontinuous and for each (f 0, g 0 ) H there exists x 0 E satisfying (f 0 f)(x 0 ) = sup x (f 0 (x) f(x)) and or f(x 0 ) h(x 0 ) (g 0 ) (x 0 ) α f(x 0 ) α(g 0 ) (x 0 ) + h(x 0 ) A function f C(E) is a viscosity solution of f αhf = h if it is both a subsolution and a supersolution.

49 First Prev Next Go To Go Back Full Screen Close Quit 49 Comparison principle The equation f αhf = h satisfies a comparison principle, if f a viscosity subsolution and f a viscosity supersolution implies f f on E.

50 First Prev Next Go To Go Back Full Screen Close Quit 50 Viscosity approach to semigroup convergence Theorem 12 Let (E, r) be a compact metric space, and for n = 1, 2,..., assume that the martingale problem for A n B(E) B(E) is well-posed. Let H n f = 1 n e nf A n e nf, e nf D(A n ), and let H C(E) B(E) with D(H) dense in C(E). Suppose that for each (f, g) H, there exists (f n, g n ) H n such that f f n 0 and g g n 0.

51 First Prev Next Go To Go Back Full Screen Close Quit 51 Fix α 0 > 0. Suppose that for each 0 < α < α 0, there exists a dense subset D α C(E) such that for each h D α, the comparison principle holds for (I αh)f = h. Then there exists {V (t)} on C(E) such that sup V (t)f(x) V n (t)f(x) 0, f C(E). x If {X n (0)} satisfies a large deviation principle with a good rate function. Then {X n } is exponentially tight and satisfies a large deviation principle with rate function I given above (6).

52 First Prev Next Go To Go Back Full Screen Close Quit 52 Proof of a large deviation principle 1. Verify convergence of the sequence of operators H n and derive the limit operator H. In general, convergence will be in the extended limit or graph sense. 2. Verify exponential tightness. Given the convergence of H n, exponential tightness typically follows provided one can verify the exponential compact containment condition. 3. Verify the range condition or the comparison principle for the limiting operator H. The rate function is characterized by the limiting semigroup. 4. Construct a variational representation for H. This representation typically gives a more explicit representation of the rate function.

53 First Prev Next Go To Go Back Full Screen Close Quit 53 R d -valued processes Let a = σσ T, and define A n f(x) = n (f(x + 1 R d n z) f(x) 1 z f(x))η(x, dz) n +b(x) f(x) + 1 a ij (x) i j f(x) 2n ij

54 First Prev Next Go To Go Back Full Screen Close Quit 54 Nonlinear generator The operator H n f = 1 n e nf A n e nf is given by H n f(x) = (e n(f(x+ 1 z) f(x)) n 1 z f(x))η(x, dz) R d + 1 a ij (x) i j f(x) 2n ij + 1 a ij (x) i f(x) j f(x) + b(x) f(x) 2 ij

55 First Prev Next Go To Go Back Full Screen Close Quit 55 Limiting operator Hf(x) = (e f(x) z 1 z f(x))η(x, dz) R d + 1 a ij (x) i f(x) j f(x) + b(x) f(x) 2 ij Note that H has the form Hf(x) = H(x, f(x)) for H(x, p) = 1 2 σt (x)p 2 + b(x) p + (e p z 1 p z)η(x, dz) R d pc

56 First Prev Next Go To Go Back Full Screen Close Quit 56 Gradient limit operators Condition For each compact Γ R d, there exist µ m + and ω : (0, ) [0, ] such that {(x m, y m )} Γ Γ, µ m x m y m 2 0, and imply sup H (y m, µ m (x m y m )) < m lim inf m [λh (x m, µ m(x m y m ) ) H (y m, µ m (x m y m ))] ω(λ) λ and inf ω(λ) 0. lim ɛ 0 λ 1 ɛ 2. If x m and p m 0, then lim m H(x m, p m ) = 0.

57 First Prev Next Go To Go Back Full Screen Close Quit 57 Conditions for comparison principle Lemma 14 If Condition 13 is satisfied, then for h C(E) and α > 0, the comparison principle holds for (I αh)f = h.

58 First Prev Next Go To Go Back Full Screen Close Quit 58 Sufficient conditions Lemma 15 Suppose σ and b are bounded and Lipschitz and η = 0. Then Condition 13 holds with { 0 λ > 1 ω(λ) = λ 1. If H is continuous and for each x, p R d lim r H(x, rp) =, then Condition 13.1 holds with { 0 λ = 1 ω(λ) = λ 1. If σ and b are bounded and lim sup (e p z 1 p z)η(x, dz) = 0, p 0 x R d then Condition 13.2 holds.

59 First Prev Next Go To Go Back Full Screen Close Quit 59 Diffusions with periodic coefficients (Baldi) Let σ be periodic (for each 1 i d, there is a period p i > 0 such that σ(y) = σ(y + p i e i ) for all y R d ), and let X n satisfy the Itô equation dx n (t) = 1 n σ(α n X n (t))dw (t), where α n > 0 and lim n n 1 α n =. Let a = σσ T. Then and H n f(x) = 1 2n A n f(x) = 1 2 a ij (α n x) f(x), n x i x j ij a ij (α n x) ij f(x) + 1 a ij (α n x) i f(x) j f(x). 2 ij ij

60 First Prev Next Go To Go Back Full Screen Close Quit 60 Limit operator Let f n (x) = f(x) + ɛ n h(x, α n x), where ɛ n = nα 2 n. ɛ n α n = nα 1 n 0 If h has the same periods in y as the a ij and 1 ( ) 2 a ij (y) h(x, y) + i f(x) j f(x) = g(x) 2 y i y j ij for some g independent of y, then lim H nf n (x, y) = g(x). n

61 First Prev Next Go To Go Back Full Screen Close Quit 61 It follows that g(x) = 1 a ij i f(x) j f(x), 2 ij where a ij is the average of a ij with respect to the stationary distribution for the diffusion on [0, p 1 ] [0, p d ] whose generator is A 0 f(y) = 1 2 a ij (y) f(y) 2 y i y j with periodic boundary conditions. In particular, h(x, y) = 1 h ij (y) i f(x) j f(x), 2 where h ij satisfies ij i,j A 0 h ij (y) = a ij a ij (y).