TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 364, Number 12, December 2012, Pages 6513–6529 S 0002-9947(2012)05680-7 Article electronically published on June 27, 2012
AN EXPLICIT SCHILDER TYPE THEOREM FOR SUPER-BROWNIAN MOTIONS: INFINITE INITIAL MEASURES
KAI-NAN XIANG
Abstract. In this paper, we prove that an explicit Schilder type theorem for super-Brownian motions holds for infinite initial measures, which concludes a long-time attacking on a conjecture of Fleischmann, G¨artner and Kaj (1996).
1. Introduction The aim of this paper is to conclude a long-time attacking on a conjecture of [17] by proving an explicit Schilder type (which is also a Cram´er type) theorem for super-Brownian motions holds for infinite initial measures. Super-Brownian motion (SBM) was introduced independently by S. Watanabe in 1968 and D. A. Dawson in 1975. The name “Super-Brownian motion (Super- process)” was coined by E. B. Dynkin in the late 1980s and has been studied by many authors (see [6], [23], [15], [14], [25], [24] and the references therein). SBM is the most fundamental branching measure-valued diffusion process (superprocess). Like ordinary Brownian motion, SBM is a universal object which arises in models from combinatorics (lattice tree and algebraic series), statistical mechanics (critical percolation), interacting particle systems (voter model and contact process), popu- lation theory and mathematical biology, and nonlinear partial differential equations ([23], [29], [3]). To begin our introduction of SBM, fix a natural number d and a real number r>d/2. Put −r φ (x)= 1+|x|2 , ∀x ∈ Rd. r B Rd Rd Let b resp. Cb be the set of all bounded measurable (resp. continuous) d d functions on R , and Cc R the set of all continuous functions with compact support on Rd. Write μ, f = μ(f)= f(x) μ(dx) Rd for the integral of a function f against a measure μ on Rd if it exists. Denote by d d Mr R the set of all Radon measures μ on R with μ, φr < ∞ endowed with the following τ topology: r d μn =⇒ μ if and only if lim μn,f = μ, f for any f ∈ Cc R ∪{φr}. n→∞
Received by the editors March 18, 2011. 2010 Mathematics Subject Classification. Primary 60J68, 60F10, 60G57, 49-XX. Key words and phrases. Large deviation, super-Brownian motion, calculus of variation. This project was partially supported by CNNSF (No.10971106).
c 2012 American Mathematical Society Reverts to public domain 28 years from publication 6513
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Rd Let Δ be the Laplace operator on . Then SBM X =(Xt)t≥0 is the unique M Rd ∈M Rd ≤ ∈ r -valued diffusion process such that for any μ r and any 0 f d Bb R , E {− } | {− σ,ρ } ∈ R ∞ [exp Xt,f X0 = μ]=exp μ, ut f ,t + := [0, ). ∈ ∞ σ,ρ Here σ, ρ (0, ) are fixed but arbitrary, and ut f is the unique nonnegative solution to the equation t σ − ρ σ 2 ∈ R ut = St f St−s (us) ds, t +, 2 0 { σ} and St t≥0 is the semigroup of the d-dimensional Brownian motion with generator ∈ R → ρ 2 ∈ R σΔ. Note X corresponds to the binary branching mechanism z + 2 z +. d For any μ ∈Mr R , let d Ωμ = Cμ [0, 1], Mr R d be the space of all continuous Mr R -valued paths on the interval [0, 1] starting σ,ρ at μ, equipped with the compact-open topology. Write Pμ for the distribution of (Xt)0≤t≤1 with X0 = μ on Ωμ. M Rd Rd Let F be the space of all finite measures on furnished with the weak d convergence topology. For any μ ∈MF R , denote by d Ωμ,F = Cμ [0, 1], MF R M Rd the set of all continuous maps ω =(ωt)0≤t≤1 from [0, 1] into F with ω0 = μ. M Rd Equip Ωμ,F with the corresponding compact-open topology. If X0 is in F , M Rd then the SBM (Xt)t≥0 is an F -valued diffusion process. Namely, for any ∈M Rd σ,ρ μ F ,Pμ is a probability on Ωμ,F . In this paper, we study the large deviation principle (LDP) for P σ,ρ on Ω μ >0 μ as ↓ 0 for any μ =0 . For LDPs, refer to [8], [9], [19] and [31]. On one hand, this is a Freidlin-Wentzell type LDP ([17] called it a Schilder type LDP by comparing the form of its (possible) rate function with that of the Schilder theorem for Brownian motions). On the other hand, since σ,ρ ∈· σ,ρ ∈· ∀ ∈ ∞ Pμ [(Xt)0≤t≤1 ]=Pμ/[(Xt)0≤t≤1 ], (0, ), σ,ρ 1 ∈· and for any natural number n, Pnμ n Xt 0≤t≤1 is the law of the empirical σ,ρ mean of n independent SBMs distributed as Pμ because of the branching property. The mentioned LDP is an infinite-dimensional version of the well-known classical Cram´er theorem with continuous parameters ([27], pp. 321-322). This is a fascinat- ing feature of the LDP for P σ,ρ . Notice this fascinating feature is precisely μ >0 the case for the Schilder’s theorem itself and generally stochastic processes possess- ing an appropriate “self-similarity” ([9]), and was told to the author by Professor O. Zeitouni. Without a doubt, SBM is one of the most important and fundamental processes with the mentioned fascinating feature. Note Schilder type theorem and Cram´er type theorem are two major topics for the large deviation theory. Recall the first statement of the Cram´er theorem (on R1) was due to [4], and [12] first extended Cram´er theorem to separable Banach spaces. While classical Schilder theorem was first derived by [28] and then based on Fernique’s inequality, it was generalized to abstract Wiener space by [9]. A nontopological form of Schilder theorem for centered Gaussian processes based on
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the isoperimetric inequality was established by [2]. For more information on these two theorems, see [8]. Rd Rd Let Φ := Φ be the set of all continuous functions f on satisfying that f(x) d has a finite limit as x goes to infinity, and · the uniform norm on Bb R . φr (x) Endow Φ with the norm
f Φ = f/φr ,f∈ Φ. · { σ} Then (Φ, Φ) is a separable Banach space. Note St t≥0 is a strongly continuous semigroup of bounded linear operators on (Φ, · Φ) (cf. [27] or Lemmas (3.2)-(3.3) in [5]). Let (σΔ,Dom(σΔ)) be its strong generator with domain Dom (σΔ) ⊂ Φ. Then Dom (σΔ) = Dom(Δ). D ∞ Rd Furnish the Schwartz space := Cc , the set of all smooth functions on Rd with compact supports, with its inductive topology via the subspaces
DK = {φ ∈D |φ(x)=0, ∀x ∈ K } , ∗ where sets K are compact in Rd. Let D be the dual space of D, namely the space Rd M Rd ⊂D∗ of all Schwartz distributions on . Then r . d ∗ ∗ For any ν ∈Mr R , define Δ ν ∈D , by (1.1) Δ∗ν, f = ν, Δf , ∀f ∈D. Then for any ϕ ∈ Dom(Δ), Δ∗ν, ϕ defined by ν, Δϕ is determined uniquely by (1.1). We call a map ∗ t ∈ [0, 1] → θt ∈D d absolutely continuous if for any compact set K in R there are a neighborhood VK of 0 in DK and an absolutely continuous real-valued function hK on [0, 1] satisfying
(1.2) |θt,f−θs,f|≤|hK (t) − hK (s)| , ∀s, t∈[0, 1], ∀f ∈VK . ˙ ∗ For such a map, the derivatives θt ∈D exist in the Schwartz distributional sense for almost all t ∈ [0, 1] with respect to the Lebesgue measure ([7]). σ Let Hμ be the space of all absolutely continuous (in time t in the sense of Schwartz distributions) paths ω =(ωt)0≤t≤1 ∈ Ωμ with derivativesω ˙ t such that for almost every t with respect to the Lebesgue measure, the Schwartz distribution ∗ ω˙ t − σΔ ωt is absolutely continuous with respect to ωt, and 1 − ∗ 2 d(ω ˙ t σΔ ωt) ωt, dt<∞. 0 dωt ∗ d Here ν ∈D is absolutely continuous with respect to w ∈Mr R if ν, f = w, hf, ∀f ∈D, Rd dν for some locally w-integrable measurable function h on , and write h = dw . For ∈ σ any ω Hμ , let − ∗ ω d(ω ˙ t σΔ ωt) ∈ (1.3) gt = gt := ,t [0, 1]. dωt Put 1 1 ω , |g |2 dt, if ω ∈ Hσ, Iσ,ρ(ω)= 2ρ 0 t t μ μ ∞ ∈ \ σ , if ω Ωμ Hμ .
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σ σ,ρ Note the introduction of Hμ and Iμ was attributed to [17] (or perhaps [16]). In d addition, for any μ ∈MF R \{0}, let σ σ ∩ σ,ρ ∞ σ,ρ ∈ Hμ,F = Hμ Ωμ,F ,I (ω)= I ∈ \ σ + Iμ (ω)I ∈ σ ,ω Ωμ,F . μ,F [ω Ωμ,F Hμ,F ] [ω Hμ,F ] Now a long-standing conjecture is stated as follows: Conjecture 1.1. For any μ ∈M Rd \M Rd on Ω , P σ,ρ satisfies an r F μ μ >0 σ,ρ ↓ ∈M Rd \{ } LDP with good rate function Iμ as 0. Whereas for any μ F 0 on Ω , P σ,ρ satisfies an LDP with good rate function Iσ,ρ as ↓ 0. μ,F μ >0 μ,F Notice the above important and deep conjecture was (implicitly) stated in [17] and the difference between the mentioned LDP for finite initial measures and for infinite initial measures was first noticed by [33]. Assume μ =0 . In Schied [27], LDP for P σ,ρ as ↓ 0 was given, and the μ >0 good rate function was represented by a variation formula, which was expected to be σ,ρ Iμ (see also Schied [26]). Notice that although the two variation formulae in [27] and [17] representing the good rate functions for the mentioned LDP respectively seem different, they are identical due to the fact that the rate function for any LDP on a Polish space is unique ([8], pp. 117-118, Remarks). Recall that along the lines of Fleischmann and Kaj [18], a weak LDP for P σ,ρ as ↓ 0onΩ was established in a topology weaker than the compact- μ >0 μ open topology by an original preprint of Fleischmann, G¨artner and Kaj [16]. Then Fleischmann, G¨artner and Kaj [17] also derived LDP for P σ,ρ as ↓ 0. When μ >0 replacing the underlying process, Brownian motion, by a killed or reflected Brow- nian motion in a bounded domain of Rd, they got the corresponding explicit rate σ,ρ function Iμ . However, for the Brownian motion case, relying on an additional unproven ‘Hypothesis of local blow-up’ ([17], Hypothesis 1.2.4), [17] proved the σ,ρ variation formula representing the rate function in [27] and [17] to be Iμ . To prove the conjecture, it suffices to calculate the variation formula for the good rate function. However, this variation formula is not covered by the standard theory of variational methods (e.g., [30]). As just mentioned, Hypothesis 1.2.4 (‘Hypoth- esis of local blow-up’) in [17] can be applied to calculate the mentioned variation. Though local blow-up properties for solutions to nonlinear partial differential equa- tions (PDEs) have been studied extensively by many references (see [20], [1], [32] and the references therein), there is no statement on local blow-up properties in the PDE theory which implies the mentioned local blow-up hypothesis. Recall that Djehiche and Kaj [10] took a Hamiltonian approach and a Girsanov transformation technique to prove an LDP result for some measure-valued jump processes, and they compared the rate function therein with that in [17] of the LDP for P σ,ρ as ↓ 0 and hoped to extend the method in [10] to the LDP μ >0 for P σ,ρ . See [10], pp. 1419 and 1437-1438. Then Djehiche and Kaj [11] took μ >0 the just mentioned method to prove directly upper and lower bounds of LDP for P σ,ρ as ↓ 0. In [11], though the rate function for the upper bound LDP was μ >0 σ,ρ Iμ , it required a restriction to more regular paths to get a lower large deviation bound. Thanks to previous results on the LDP for SBMs ([26], [27], [16], [17], [18], [10], [11]) and the Le Gall’s Brownian snake ([23]), an important breakthrough on Conjecture 1.1 was made by Xiang [33]. Recall [33] proved Conjecture 1.1 for any
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σ σ nonzero finite initial measures μ and a rather surprising consequence: Hμ,F = Hμ . However, for infinite initial measures, [33] was only able to prove Conjecture 1.1 for 1 ≤ the case d =1and 2
for any open subset O ⊆ Ωμ, σ,ρ ∈ ≥− σ,ρ lim inf log Pμ [ω O] inf Iμ (ω), ↓0 ω∈O σ,ρ ∈ σ,ρ ≤ Iμ is lower semicontinuous, and ω Ωμ Iμ (ω) a is compact in Ωμ for any a ∈ [0, ∞).
As mentioned in [33] Remark 1.3(1), to prove Theorem 1.2, it suffices to prove ∈ σ ∈M Rd \M Rd that for any ω Hμ with μ r F , (1.4) Iσ,ρ(ω) = lim inf Iσ,ρ (ω) , μ ω0,F δ↓0 ω∈Oδ,F (ω) where d σ O (ω)= ω ∈ ΩM Rd : q (ω, ω) <δ, ω ∈M R , ω ∈ H , δ,F r ( ) 0 F ω0,F where d ΩM Rd = C [0, 1], M R r ( ) r d is the space of all continuous maps from [0,1] into Mr R equipped with the compact-open topology, and where q(·, ·) is the corresponding compatible metric
d on ΩMr (R ). We will prove (1.4) in Section 2. Similarly, to prove Conjecture 1.1 in [33] and the present paper, one can prove an extended version of Conjecture 1.1 holds for super-α-stable processes and more general super-diffusions with the binary branching mechanism. Note unlike the local operator Δ, for super-α-stable processes with α ∈ (0, 2), the underlying generator Δα is not local, which will make the related proof slightly different. 2. Proof of Theorem 1.2 Sketch of the proof. By Lemma 2.2, it suffices to prove (1.4) holds for any ω ∈ Hσ ω0 such that σ d d (2.1) (ω − ηdx) ≤ ≤ ∈ H ,ω − ηdx ∈M R \M R , t 0 t 1 ω0−ηdx 0 r F where dx is the Lebesgue measure on Rd and η ∈ (0, 1). Note Lemma 2.1 is used to prove Lemma 2.2. By Lemma 2.3, ∈ σ σ,ρ σ,ρ ∈ σ ω Hω , lim Iω (ω )=Iω (ω),ω Hω , 0 ↓0 0 0 0 where for any ∈ (0, 1), σ ∗ ∈ ω =(ωt )0≤t≤1 ,ωt =(S ) ωt,t [0, 1],
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6518 KAI-NAN XIANG ∈M Rd σ ∗ ∈M Rd σ ∗ and for any ν r , (S ) ν r is defined by (S ) ν, f σ ∈ = ν, S [f] ,f Φ. ∈ σ ∈ Fix an ω Hω0 satisfying (2.1) and an (0, 1). Then the proof is boiled down ,n to find ω ∈ Ω ,n for any natural number n such that ω0 ,F ,n ∈ σ σ,ρ ,n σ,ρ (2.2) ω Hω,n,F , lim I ,n (ω )=Iω (ω ) . 0 n→∞ ω0 0 Note that ω = q(y)dy, t ∈ [0, 1], where q(y) is given by (2.6). By Lemma 2.3(i), t t t 1 σ,ρ 1 2 I (ω )= ω , |g | dt. ω0 t t 2ρ 0 To prove (2.2), it suffices to let ,n ∈ ωt = fn(t, y) ωt (dy)=fn(t, y)qt (y)dy, t [0, 1], ≤ ∈ where 0 fn Φ[0,1] satisfies that for almost all t with respect to the Lebesgue measure, ,n − ∗ ,n d(˙ωt σΔ ωt ) (2.3) ,n = gt , dωt ↑ ↑∞ and fn 1 pointwise as n ,andwhereΦ[0,1] := C ([0, 1], Φ) is the space of all Φ-valued continuous paths ψ =(ψt)0≤t≤1 on the interval [0, 1] equipped with { } the norm ψ Φ[0,1] =sup ψt Φ . However, notice that (2.9), a nonrigorous t∈[0,1] deduction, shows that if fn is smooth, then for almost all t with respect to the Lebesgue measure, ,n − ∗ ,n − d(˙ωt σΔ ωt ) ∂t (fn(t, y)qt (y)) σΔ(fn(t, y)qt (y)) ,n (y)= dωt fn(t, y)qt (y) − ∇ ·∇ ∂tfn(t, y) σΔfn(t, y) − 2σ fn(t, y) qt (y) = gt (y)+ , fn(t, y) fn(t, y)qt (y) which forces fn to satisfy ∇ ·∇ 2σ fn(t, y) qt (y) ∂tfn(t, y)=σΔfn(t, y)+ . qt (y) Thus stochastic differential equation (SDE) (2.10) comes into picture, and naturally fn should be defined by the first formula in (2.12). This is a key novel point of our proof. Note by Lemma 2.3(ii) and Lemma 2.4, SDE (2.10) has a unique strong solution (Yt (x))0≤t≤1 . ∈ Rd Though Yt (x)issmoothinx for any fixed t, we cannot prove fn(t, x)isin C1,2-class in (t, x) ∈ [0, 1]×Rd. To obtain (2.3), we have to approximate SDE (2.10) ,m by SDE (2.11). The function fn specified in the second formula in (2.12) likes ↑∞ fn and converges to fn as m , but the former has better properties than the ,m ↑∞ latter. See Lemma 2.5. Since (2.16)-(2.18) hold for fn (let m ), we can easily get (2.19), which implies that (2.3) holds. This is another key novel aspect of our proof. Sofarwehaveproven(2.2). · V Vσ,ρ Note Φ[0,1], Φ[0,1] is a separable Banach space. Let := be the set of all (φ, ψ) ∈ Φ × Φ[0,1] such that the following nonlinear integral equation has a · unique solution Λ[φ, ψ]=(Λ[φ, ψ](s, ))0≤s≤1 in Φ[0,1] : 1 1 σ σ ρ σ 2 ∈ (2.4) vs = S1−sφ + St−sψt dt + St−s vt dt, s [0, 1]. s 2 s
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Then V is open in Φ × Φ[0,1], and d (φ, ψ) ∈ Φ × Φ[0,1] φ(x) ≤ 0,ψs(x) ≤ 0, ∀(s, x) ∈ [0, 1] × R ⊂V. d From [17], for any ω0 ∈Mr R \{0}, 1 − · σ,ρ ∀ ∈ (2.5) sup ωt,ψt dt ω0, Λ[0,ψ](0, ) = Iω0 (ω), ω Ωω0 . (0,ψ)∈V 0 In fact, the variation in (2.5) is equal to the good rate function of the LDP for P σ,ρ as ↓ 0 under the local blow-up hypothesis ([17]). By (2.5), we imme- ω0 >0 diately obtain [n] Lemma 2.1. For any sequence ω ⊂ ΩM Rd converging to ω ∈ ΩM Rd n≥1 r ( ) r( )
with ω0 =0, Iσ,ρ(ω) ≤ lim inf Iσ,ρ ω[n] . ω0 →∞ [n] n ω0 (η) d For any ω ∈ Ωμ and any η ∈ (0, 1), let ν = ν + ηdx for any ν ∈Mr R , and let (η) (η) ∈ ω := ωt =(ωt + ηdx)0≤t≤1 Ωμ(η) . 0≤t≤1 ∈ σ ∈ (η) ∈ σ Lemma 2.2. For any ω Hμ with μ =0and any η (0, 1),ω Hμ(η) . Further, σ,ρ (η) σ,ρ lim I (η) ω = Iμ (ω). η↓0 μ Proof. Let (η) dωt ∈ ht = ,t [0, 1]. dω(η) t ∈ → (η) ∈M Rd ⊂D∗ Clearly, t [0, 1] ωt r is absolutely continuous, and for almost all t ∈ [0, 1] with respect to the Lebesgue measure, (η) ∗ ω˙ =˙ωt = gt dωt + σΔ ωt. t ∈D ∗ Since for any f , Rd Δf(x)dx =0, we see that Δ (ηdx)=0. Thus for almost all t ∈ [0, 1] with respect to the Lebesgue measure, (η) ∗ (η) (η) (η) ∗ (η) ω˙ t = gt dωt + σΔ ωt = gtht dωt + σΔ ωt (η) − ∗ (η) d ω˙ t σΔ ωt andω ˙ (η) − σΔ∗ω(η) ω(η), = g h(η). t t t (η) t t dωt (η) ≤ (η) ≤ ∈ Notice that ht canbechosensatisfying0 ht 1,t [0, 1], and 1 2 1 1 (η) (η) ≤ (η) | |2 (η) | |2 ∞ ωt , gtht dt ωt , gt ht dt = ωt, gt dt< . 0 0 0 ∈ (η) ∈ σ σ,ρ (η) ≤ σ,ρ So for any η (0, 1),ω Hμ(η) , and lim sup Iμ(η) ω Iμ (ω). By Lemma η↓0 2.1, we have σ,ρ (η) σ,ρ lim I (η) ω = Iμ (ω). η↓0 μ
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As mentioned in the Introduction, to prove Theorem 1.2 it suffices to prove (1.4). (η) ∈ σ By Lemma 2.2, it only needs to show that (1.4) holds for ω , where ω Hω0 and η ∈ (0, 1).
Hence, throughout the rest of the paper, assume ∈ σ satisfies (2.1). ω Hω0 Then there is a measurable subset N of [0, 1] with zero Lebesgue measure such that ∗ ∗ for any t ∈ [0, 1] \N, ω˙ t ∈D exists andω ˙ t − σΔ ωt is absolutely continuous with respect to ωt and
− ∗ 2 d(˙ωt σΔ ωt) 2 ωt, = ωt, |gt| < ∞, dωt
where gt is defined by (1.3). For any s ∈ (0, ∞), let | |2 σ 1 − x ∀ ∈ Rd Ps (x)= d exp , x . (4πσs) 2 4σs
Put σ − ∈ ∈ Rd (2.6) qt (y):= P (y x) ωt(dx),t [0, 1],y . Rd Then q(y) ∈ [η, ∞) is continuous in (t, y) ∈ [0, 1] × Rd, and ω(dy)=q(y)dy. Let t t t σ ·− d gt(x)P ( x) ωt(dx) (2.7) g(·)= R ,t∈ [0, 1], t · qt ( ) ≡ ∈N where gt 0ift . To state Lemma 2.3, introduce the following notation. For any multi-index d α =(α1, ··· ,αd) of nonnegative integers, any smooth function f on R and any d x =(x1, ··· ,xd) ∈ R , let | | ··· ··· α α1 ··· αd α = α1 + + αd,α!=α1! αd!,x = x1 xd , ∂|α|f(x) Dαf(x)=Dαf(x)= . x α1 ··· αd ∂x1 ∂xd σ Lemma 2.3. (i) For any ∈ (0, 1),ω ∈ H . Namely, there is a measurable ω0 subset N of [0, 1] with zero Lebesgue measure such that N⊆N, and for every ∈ \N ∈D∗ − ∗ t [0, 1] , ω˙ t exists and ω˙ t σΔ ωt is absolutely continuous with respect ∗ d(ω ˙ −σΔ ω) 1 2 t t | | ∞ to ωt , and gt = .Further ωt , gt dt< . dωt 0 (ii) For any ∈ (0, 1),q(y) is absolutely continuous in t and of the derivatives t ω˙ ,Pσ(y −·)=ω ,g P σ(y−·)+ω ,σΔP σ(y−·) ,t∈ [0, 1] \N, ∂ q(y)= t t t t t t 0,t∈N in the sense that ∂tqt (y) is absolutely integrable in t with respect to the Lebesgue t ∈ measure, and qt (y)=q0(y)+ 0 ∂sqs(y)ds, t [0, 1]. In addition, qt (y) is smooth in y =(y , ··· ,y ) ∈ Rd, and for any multi-index α of nonnegative integers, 1 d α α σ − Dy qt (y)= Dy P (y x) ωt(dx). Rd Particularly, for any 1 ≤ i, j ≤ d, σ − 2 2 σ − (2.8) ∂yi qt (y)= ∂yi P (y x) ωt(dx),∂yiyj qt (y)= ∂yiyj P (y x) ωt(dx), Rd Rd
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α ∈ ×Rd ∈ \N and clearly each Dy qt (y) is continuous in (t, y) [0, 1] . So for any t [0, 1] ,
− ∂tqt (y) σΔqt (y) ∈ Rd (2.9) = gt (y),y . qt (y)
σ,ρ σ,ρ (iii) lim Iω (ω )=Iω (ω). ↓0 0 0
Proof. By Lemmas 9.1-9.2 in [33] and Lemma 2.1, we have that Lemma 2.3(i) and Lemma 2.3(iii) hold. For the first part of Lemma 2.3(ii) and (2.8), see Lemma 9.3 and (9.2) in [33]. Note the idea of proving [33] Lemma 9.3 can be used to deduce α α σ − that qt (y)isinfactsmoothiny and Dy qt (y)= Rd Dy P (y x) ωt(dx).
Lemma 2.4. For any ∈ (0, 1), there is a positive constant C0 = C0() such that for any 1 ≤ i ≤ d,
|∂ q(y)| yi t ≤ | | ∀ ∈ Rd ∀ ∈ C0(1 + y ), y , t [0, 1]. qt (y)
Generally, for any multi-index α =(α1, ··· ,αd) of nonnegative integers, there exists a positive constant Cα = Cα() satisfying that for any 1 ≤ i ≤ d, ∂ q(y) α yi t ≤ | ||α|+1 ∀ ∈ Rd ∀ ∈ Dy Cα 1+ y , y , t [0, 1]. qt (y)
Proof. By (2.8), for any y ∈ Rd and t ∈ [0, 1], − 1 − σ − ∂yi qt (y)= (yi xi)P (y x) ωt(dx). 2σ Rd
Thus for any y ∈ Rd and t ∈ [0, 1], | − | σ − |∂ q (y)| 1 d y x P (y x) ω (dx) yi t ≤ R t qt (y) 2σ qt (y) |y|P σ(y − x) ω (dx) |x|P σ(y − x) ω (dx) ≤ 1 Rd t 1 Rd t + 2σ qt (y) 2σ qt (y) | | | | σ − y 1 Rd x P (y x) ωt(dx) = + . 2σ 2σ qt (y)
Note for any y ∈ Rd and a ∈ (0, ∞), σ |x|P (y − x) ωt(dx) {|x|≤a|y|} ≤ | | ≥ ∀ ∈ a y ,qt (y) η, t [0, 1]. qt (y)
To prove the first part of the lemma, it suffices to verify that for any c1 ∈ [2, ∞), there is a positive constant c2 = c2(, c1), | | σ − ≤ ∈ Rd \{ } ∈ x P (y x) ωt(dx) c2,y 0 ,t [0, 1]. {|x|>c1|y|}
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In fact, for any y ∈ Rd \{0} and t ∈ [0, 1], | | σ − x P (y x) ωt(dx) {| | | |} x >c1 y − 2| |2 ≤ 1 | | −(1 1/c1) x d x exp ωt(dx) (4πσ) 2 {|x|>c |y|} 4σ 1 − 2| |2 ≤ 1 | | −(1 1/c1) x d x exp ωt(dx) (4πσ) 2 Rd 4σ − 2| |2 ≤ 1 | | −(1 1/c1) x ∞ d sup x exp ωs(dx) < . (4πσ) 2 0≤s≤1 Rd 4σ
Inductively, similar to the first part of the lemma, it is easy to check that the second part of the lemma holds by some tedious calculuses.
d d For any 1 ≤ m<∞, let ζm : R → R be a smooth map such that ζm(x)=x if |x|≤m and = 0 if |x|≥2m. For any s ∈ [0, 1] and x ∈ Rd, put
∇ ,m ∇ ,m ,m qs (x) qs(x) qs (x)=qs(ζm(x)),bs (x)= ,m ,bs(x)= . qs (x) qs(x)
Then by Lemma 2.3(ii), for any (s, x) ∈ [0, 1] × Rd,
≤ ,m ≤ | | ∞ η qs (x) max qt (ζm(y)) < , 0≤t≤1,|y|≤2m
,m ∈ ×Rd ∈ ,m qs (x) is continuous in (s, x) [0, 1] , for any fixed s [0, 1],qs (x)issmooth ∈ Rd α ,m ∈ × Rd in x ,andD qs (x) is bounded and continuous in (s, x) [0, 1] for any multi-index α =(α1, ··· ,αd) of nonnegative integers. By Lemma 2.3(ii) and Lemma 2.4, uniformly in time s, bs(x) is locally Lips- ,m chitzian and satisfies the linear growth condition in spatial variable x, and bs (x) is Lipschitzian in x.Also,
| ,m | ∞ max bs (x) < . (s,x)∈[0,1]×Rd
Let B =(Bt)t≥0 (B0 = 0) be the standard d-dimensional Brownian motion. Then by the standard SDE theory, √ t ∈ ∈ Rd (2.10) Yt (x)=x + 2σBt + 2σbs (Ys (x)) ds, t [0, 1],x , 0 ∈ Rd has a unique strong solution (Yt (x))0≤t≤1,x , √ t ,m ,m ,m ∈ ∈ Rd (2.11) Yt (x)=x + 2σBt + 2σbs (Ys (x)) ds, t [0, 1],x , 0 ,m ∈ Rd ∈ Rd has a unique strong solution (Yt (x))0≤t≤1,x ,andforanyx , almost surely, ,m ≤ λm(x)=λm(x), and Yt (x)=Yt (x) for all t λm(x).
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Here { ∈ | |≥ } λm(x) = inf t [0, 1] : Yt (x) m { ∈ | |≥ } ∅ (λm(x)=1if t [0, 1] : Yt (x) m = ) , λ (x) = inf {t ∈ [0, 1] : |Y ,m(x)|≥m} m t { ∈ | ,m |≥ } ∅ λm(x)=1if t [0, 1] : Yt (x) m = . Clearly, for any x ∈ Rd,
lim P [λm(x)=1]=1. m↑∞ Following the same line for proving Proposition 2.2 in [21], Chapter V, Section 2, though the drift coefficient is time-inhomogeneous in the considered setting, one ∈ ,m ∈ Rd can prove that for any t [0, 1], almost surely, Yt (x)issmoothinx . ∈ ∈ Rd Therefore, for any t [0, 1], almost surely, Yt (x) is also smooth in x . Let { n}n≥1 ⊂Dbe a sequence such that
0 ≤ n ≤ 1, n(x)=1if|x|≤n and = 0 if |x|≥n +1, ∀n ≥ 1. ∈ Rd 2 Rd Clearly, lim k(x) = 1 for any x . Write Cb for the family of all bounded k↑∞ continuous functions on Rd with bounded continuous derivatives of orders one and two. For any n ≥ 1andm ≥ 1, and any (t, x) ∈ [0, 1] × Rd, let E ,m E ,m (2.12) fn(t, x)= [ n(Yt (x))],fn (t, x)= [ n(Yt (x))], ∇ ·∇ 2σ f(x) qt (x) ∀ ∈ 2 Rd Atf(x)=σΔf(x)+ , f Cb , qt (x) ∇ ·∇ ,m ,m 2σ f(x) qt (x) ∀ ∈ 2 Rd At f(x)=σΔf(x)+ ,m , f Cb . qt (x) Lemma 2.5. Given any ∈ (0, 1) and any n ≥ 1 and m ≥ 1,then ,k ∈ × Rd · ∈ lim fn (t, x)=fn(t, x) for any (t, x) [0, 1] , (fn(t, ))0≤t≤1 Φ[0,1], k↑∞ ,m · ∈ ,m 1,2 ∈ × Rd (fn (t, ))0≤t≤1 Φ[0,1],fn (t, x) is in C -class in (t, x) [0, 1] , ,m ,m ,m · ∈ ∈ Rd ∂tfn (t, x)=At [fn (t, )](x),t [0, 1],x . ∈ × Rd ,k d Proof. Clearly, for any (t, x) [0, 1] , lim fn (t, x)=fn(t, x). For any u> , k↑∞ 2 put d f(x) Φu = f ∈ Cb R lim exists and is finite , |x|→∞ φu(x)
f Φ = f/φu ,f∈ Φu, u − 2 u d d Δφu where φu(x)= 1+|x| ,x∈ R . Notice for any u> , < ∞ and 2 φu |∇ | | | φu(x) ≤ 2 x ∈ Rd 2 ,x . φu(x) 1+|x| d ∈ By Lemma 2.4, for any u> 2 and t [0, 1], |∇φ (x)| |∇q(x)| √ |x|(1 + |x|) √ u t ≤ 2 dC ≤ 4 dC, ∀x ∈ Rd. 0 | |2 0 φu(x) qt (x) 1+ x
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Thus,itiseasytoseethatforanyu> d , 2 Atφu c1 := c1(u, ):= sup < ∞. 0≤t≤1 φu d ∈ ∈ Rd Since for any u> 2 ,t [0, 1] and x , t E E [φu (Yt (x))] = φu(x)+ [Asφu (Ys (x))] ds 0 t ≤ E φu(x)+c1 [φu (Ys (x))] ds, 0 by the Gronwall inequality, E ≤ tc1 ∈ Rd ∈ [φu (Yt (x))] φu(x)e ,x ,t [0, 1]. d ∈ Therefore, for any u> 2 and t [0, 1], ≤ E ≤ c1 ∈ Rd fn(t, x) n Φu [φu (Yt (x))] n Φu e φu(x),x , · ∞ · ∞ which implies sup fn(t, )/φu < . Particularly, by sup fn(t, )/φr+1 < , 0≤t≤1 0≤t≤1 we obtain that · ∈ (fn(t, ))0≤t≤1 Φ[0,1]. Similarly, we can prove that ,m · ∈ (fn (t, ))0≤t≤1 Φ[0,1]. To prove the lemma, we need to prove that for any natural number N and any multi-index α of nonnegative integers, E | α ,m |p ∞ ∀ ∈ ∀ (2.13) sup Dx n (Yt (x)) < , t [0, 1], p>2. |x|≤N Since the drift coefficient in SDE (2.11) is bounded, applying Itˆo’s formula to SDE (2.11), by a routine argument, we can prove that for some positive constant c2 = c2(, p, m), E | ,m |p ≤ | |p ∈ Rd (2.14) sup Yt (x) c2 (1 + x ) ,x . 0≤t≤1 ,m · Notice the derivatives of the drift coefficient 2σbs ( ) for SDE (2.11) with any order are bounded uniformly in s ∈ [0, 1]. Keeping (2.14) in mind, we can easily prove that for any natural number N and any multi-index α of nonnegative integers, E | α ,m |p ∞ ∀ ∈ ∀ (2.15) sup Dx Yt (x) < , t [0, 1], p>2. |x|≤N To see the idea for proving (2.15), which is standard (see [22], pp. 175-176, Corollary 4.6.7 for a stronger related result on general time-homogenous SDEs for instance; clearly in our case the related proof is simpler than that of the just-mentioned !d E | ,m |p ∞ corollary), we prove sup ∂xi Yt (x) < for example. For simplicity, |x|≤N i=1 assume d =1forthetimebeing.Then t ,m ,m ,m ,m ∈ ∂xYt (x)=1+2σ ∂xYs (x)(bs ) (Ys (x)) ds, t [0, 1]. 0
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By Itˆo’s formula, t | ,m |p | ,m |p ,m ,m ∈ ∂xYt (x) =1+2σp ∂xYs (x) (bs ) (Ys (x)) ds, t [0, 1]. 0 ,m ∈ ≥ Since ∂xYt (x)iscontinuousint [0, 1], for any k 2, define the following stopping time: { ∈ | ,m |≥ } { ∈ | ,m |≥ } ∅ τk =inf s [0, 1] : ∂xYs (x) k (τk =1if s [0, 1] : ∂xYs (x) k = ),
then almost surely, τk ↑ 1ask ↑∞. So ∧ t τk ,m p ,m p ,m ,m ∂ Y (x) =1+2σp ∂ Y ∧ (x) (b ) Y ∧ (x) ds, t ∈ [0, 1]. x t∧τk x s τk s s τk 0 Write ,m · ∞ C1 := C1(, m):= sup (bs ) ( ) < . s∈[0,1] Then t ,m p ,m p E ∂ Y (x) ≤ 1+2σpC E ∂ Y ∧ (x) ds, t ∈ [0, 1]. x t∧τk 1 x s τk 0 By Gronwall’s inequality, for any x ∈ Rd, E ,m p ≤ { } ∈ ∂xYt∧τk (x) exp 2σpC1t ,t [0, 1]. By Fatou’s lemma, for any x ∈ Rd, E | ,m |p ≤ E ,m p ≤ { } ∈ ∂xYt (x) lim inf ∂xYt∧τ (x) exp 2σpC1t ,t [0, 1]. k↑∞ k By (2.14)-(2.15), (2.13) holds immediately. It is well known that the following Taylor formula holds ([13], p. 70, Theorem 2.8.4): for any smooth function f on Rd, any nonnegative integer k, and any x, y ∈ Rd, " yα " yα 1 f(x + y)= (Dαf)(x)+(k +1) (1 − s)k (Dαf)(x + sy)ds. α! α! |α|≤k |α|=k+1 0 Combining this with (2.13), it is easy to see that for any multi-index α of non- negative integers, we can interchange the order of the expectation E[·]andthe α E ,m differentiation D when considering the differentiation of [ n (Yt (x))] , namely, α ,m Dx fn (t, x)existsand α ,m E α ,m Dx fn (t, x)= [Dx n (Yt (x))] . Note that t ,m E ,m ,m ∈ ∈ Rd fn (t, x)= n(x)+ [As n (Ys (x))] ds, t [0, 1],x . 0 Now the lemma holds immediately. Remark 2.6. By Lemma 2.4, the derivatives of the drift coefficient for SDE (2.10) with any order satisfy a polynomial growth condition. It is unknown whether (2.13) ∈ Rd 1,2 holds for (Yt (x))0≤t≤1 ,x , and further whether fn(t, x)isinC -class in (t, x) ∈ [0, 1] × Rd. This makes the proof of (2.19) somewhat indirect. In fact, we prove (2.19) by an approximation argument.
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Let ,n ,n ,n ω =(ωt )0≤t≤1 ,ωt (dy)=fn(t, y)qt (y)dy. By Lemma 2.5, we obtain ,n d ,n Corollary 2.7. ω ∈M R ,ω ∈ Ω ,n . 0 F ω0 ,F Lemma 2.8. Given any ∈ (0, 1). Then (2.2) holds.
σ Proof. By Lemma 2.3(i), ω ∈ H ,andforalmosteveryt ∈ [0, 1] with respect to ω0 d(ω ˙ −σΔ∗ω) t t ∈D the Lebesgue measure, = gt . By Lemma 2.5, for any h , dωt ,m · ∈ ∈ fn (t, )h Dom(Δ),t [0, 1]; ,m · ∈ ,m · ∈ (Δ (fn (t, )h))0≤t≤1 Φ[0,1], (∂t (fn (t, )h))0≤t≤1 Φ[0,1]. So by [33] Proposition 5.2, for any t ∈ [0, 1] and h ∈D, and any m ≥ 1, t ,m · ,m · ,m · ωt ,fn (t, )h = ω0,fn (0, )h + ωs,gsfn (s, )h ds 0 t t ,m · ,m · (2.16) + ωs,σΔ(fn (s, )h) ds + ωs,∂s(fn (s, )h) ds. 0 0 ∈ ∈D · For any s [0, 1] and h , by Lemma 2.3(ii), qs( ) is a smooth function, and further ,m · ,m · (2.17) ωs,σΔ(fn (s, )h) = σΔ(fn (s, )h)(y)qs(y)dy Rd ,m = fn (s, y)h(y)σΔqs(y)dy Rd ,m · − ,m = h(y)σΔ(fn (s, )qs)(y)dy h(y)qs(y)σΔfn (s, y)dy Rd Rd "d − ,m 2σh(y) ∂ifn (s, y)∂iqs(y)dy Rd i=1 ,m · − ,m = σΔh(y)(fn (s, )qs)(y)dy h(y)qs(y)σΔfn (s, y)dy Rd Rd "d ,m + 2σfn (s, y) ∂i(h∂iqs)(y)dy. Rd i=1 By Lemma 2.5, for any s ∈ [0, 1] and h ∈D, we have (2.18) "d ,m ,m · ,m · ,m · ∂iqs ∂s (fn (s, )h)=hσΔfn (s, )+2σh ∂ifn (s, ) ,m , qs i=1 ,m · ωs,∂s (fn (s, )h) "d ,m qs(y) ,m ,m = h(y)qs(y)σΔfn (s, y)dy + 2σh(y) ,m ∂ifn (s, y)∂iqs (y)dy Rd Rd qs (y) i=1 "d ,m − ,m qs ,m = h(y)qs(y)σΔfn (s, y)dy 2σfn (s, y) ∂i ,m h∂iqs (y)dy. Rd Rd qs i=1
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Since h ∈D, by the dominated convergence theorem, for any t ∈ [0, 1], as m ↑∞, t "d ,m fn (s, y) ∂i(h∂iqs)(y)dyds Rd 0 i=1 t "d − ,m qs ,m → fn (s, y) ∂i ,m h∂iqs (y)dyds 0, 0 Rd qs i=1 t t t ,m · → · ,n ωs,fn (s, )σΔh ds ωs,fn(s, )σΔh ds = ωs ,σΔh ds. 0 0 0 Thus for any t ∈ [0, 1] and any h ∈D, as m ↑∞, t t t ,m · ,m · → ,n ωs,σΔ(fn (s, )h) ds + ωs,∂s(fn (s, )h) ds ωs ,σΔh ds. 0 0 0 Combining this with that for any t ∈ [0, 1], t t t lim ω,gf ,m(s, ·)h ds = ω,gf (s, ·)h ds = ω,n,gh ds, ↑∞ s s n s s n s s m 0 0 0 ,m · · ,n lim ωt ,fn (t, )h = ωt ,fn(t, )h = ωt ,h , m↑∞ by (2.16)-(2.18), we obtain that for any h ∈D, t t ,n ,n ,n ,n ∈ (2.19) ωt ,h = ω0 ,h + ωs ,gsh ds + ωs ,σΔh ds, t [0, 1]. 0 0 Therefore, for any compact set K in Rd, let
VK = VK,n = h ∈DK :sup{|h|(x)+|Δh|(x)} < 1 , ∈ x K t t ,n | | ,n ∈ hK (t)= ωs , gs IK ds + ωs ,σIK ds, t [0, 1]. 0 0
Then VK is a neighborhood of 0 in DK and hK is an absolutely continuous real- valued function, and ∈ → ,n ∈M Rd ⊂D∗ t [0, 1] ωt r is absolutely continuous due to the fact that (1.2) holds for ω,n. Furthermore, by (2.19), for almost all t ∈ [0, 1] with respect to the Lebesgue measure, ,n ,n ∗ ,n ∈D∗ ω˙ t = gt dωt + σΔ ωt exists ,n − ∗ ,n ,n andω ˙ t σΔ ωt is absolutely continuous with respect to ωt , and ,n − ∗ ,n d(˙ωt σΔ ωt ) ,n = gt . dωt Since 1 ,n − ∗ ,n 2 1 ,n d(˙ωt σΔ ωt ) · | |2 ωt , ,n dt = ωt ,fn(t, ) gt dt 0 dωt 0 1 ≤ | |2 ∞ ωt , gt dt< , 0
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,n σ by Corollary 2.7, we obtain that ω ∈ H ,n , and by the dominated convergence ω0 ,F theorem,
1 d(˙ω,n − σΔ∗ω,n) 2 1 lim ω,n, t t dt = ω, |g|2 dt. →∞ t ,n t t n 0 dωt 0 Lemma 2.8 holds.
Proof of Theorem 1.2. As mentioned before, to prove Theorem 1.2 it suffices to ∈ σ ∈ prove that (1.4) holds for any ω Hμ satisfying (2.1). Clearly, for any (0, 1), ,n →∞ → d as n ,ω ω in ΩMr (R ).Also, ↓ → d as 0,ω ω in ΩMr (R ). Therefore, by Lemma 2.3(iii), Lemma 2.8 and Lemma 2.1, we see that (1.4) holds ∈ σ for any ω Hμ satisfying (2.1).
Acknowledgement The author thanks the anonymous referee for his valuable comments.
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School of Mathematical Sciences, LPMC, Nankai University, Tianjin City, 300071, People’s Republic of China E-mail address: [email protected]
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