# Asymptotic Behavior of Densities for Stochastic Functional Differential Equations

Authors: Akihiro Kitagawa; Atsushi Takeuchi
Journal: International Journal of Stochastic Analysis
 (2013)

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## Abstract

Consider stochastic functional differential equations depending on whole past histories in a finite time interval, which determine non-Markovian processes. Under the uniformly elliptic condition on the coefficients of the diffusion terms, the solution admits a smooth density with respect to the Lebesgue measure. In the present paper, we will study the large deviations for the family of the solution process and the asymptotic behaviors of the density. The Malliavin calculus plays a crucial role in our argument.


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## Body

## 1. Introduction

Stochastic functional differential equations, or stochastic delay differential equations, determine non-Markovian processes, because the current states of the process in the equation depend on the past histories of the process. Such kind of equations was initiated by Itô and Nisio [1] in their pioneering work about 50 years ago. As stated in [2], there are some difficulties to study such equations, because we cannot use any methods in analysis, partial differential equations, and potential theory at all. On the other hand, it seems to be more natural to consider the models determined by the solutions to the stochastic functional differential equations in finance, physics, biology, and so forth, because such processes include their past histories and can be recognized to reflect real phenomena in various fields much more exactly.The Malliavin calculus is well known as a powerful tool to study some properties on the density function by a probabilistic approach. There are a lot of works on the densities for diffusion processes by many authors, from the viewpoint of the Malliavin calculus (cf. [3]). Moreover it is also applicable to the case of solutions to stochastic functional differential equations, regarding as one of the examples of the Wiener functionals. Kusuoka and Stroock in [4] studied the application of the Malliavin calculus to the solutions to stochastic functional differential equations and obtained the result on the existence of the smooth density for the solution with respect to the Lebesgue measure. On the other hand, it is well known that the Malliavin calculus is very fruitful to study the asymptotic behavior of the density function related to the large deviations theory (cf. Léandre [5–8] and Nualart [9]). In fact, the Varadhan-type estimate of the density function for the diffusion processes can be also obtained from this viewpoint. Ferrante et al. in [10] discussed such problem in the case of stochastic delay differential equations, where the drift term depends on the whole past histories on the finite time interval, while the diffusion terms depend on the state only for the edges of the finite time interval. Mohammed and Zhang in [11] studied the large deviations for the solution process under a similar situation to [10]. But, the special forms on the diffusion terms play a crucial role throughout their arguments in [10, 11].In the present paper, we will study the large deviations on the solution process to the stochastic functional differential equations. Our stochastic functional differential equations are much more general, because they are time inhomogeneous, and they are not only the drift terms, but also the diffusion terms in the equation depend on the whole past histories of the process over a finite interval. Furthermore, as a typical application of the large deviation theory and the Malliavin calculus, we will study the asymptotic behavior, so-called the Varadhan-type estimate, of the density function for the solution process, which is quite similar to the case of diffusion processes. The effect of the time delay plays a crucial role in the behavior of the density function, and the obtained result can be also regarded as the natural extension of the estimate for diffusion processes, which are the most interesting points in the present paper.The paper is organized as follows. In Section2, we will prepare some notations and introduce our stochastic functional differential equations. Section 3 will be devoted to the brief summary on the Malliavin calculus and its application to our equations. We will consider some estimates which guarantee the smoothness of the solution process and the non degeneracy in the Malliavin sense. The existence of the smooth density will be also discussed in Section 3. The negative-order moments of the Malliavin covariance matrix will be studied there which is important in order to give the estimate of the density function. Sections 4 and 5 are our main goals in the present paper. In Section 4, we will focus on the large deviation principles on the solution processes. As an application of the result obtained in Section 4, we will study the asymptotic behavior on the density for the solution process. Moreover, we can also derive the short time asymptotics on the density function, which can be interpreted as the generalization of the Varadhan-type estimate on diffusion processes (cf. [5–9]).


## 2. Preliminaries

Letr and T be positive constants, and denote an m-dimensional Brownian motion by W={W(t)=(W1(t),…,Wm(t));t∈[0,T]}. Let Ai(i=0,1,…,m) be ℝd-valued functions on [0,T]×C([-r,0];ℝd) such that, for each t∈[0,T], the mapping Ai(t,·):C([-r,0];ℝd)∋f↦Ai(t,f)∈ℝd is smooth in the Frechét sense and all Frechét derivatives of any orders greater than 1 are bounded. Under the conditions stated above, the functions Ai(i=0,1,…,m) satisfy the linear growth condition and the Lipschitz condition in the functional sense of the form:
(1)supt∈[0,T]∑i=0m‍|Ai(t,f)|≤C1,T(1+∥f∥∞),supt∈[0,T]∑i=0m‍|Ai(t,f)-Ai(t,g)|≤C2,T∥f-g∥∞,
for f,g∈C([-r,0];ℝd), where ∥f∥∞=supt∈[-r,0]|f(t)|. Denote by A=(A1,…,Am).Let0<ε≤1 be sufficiently small. For a deterministic path η∈C([-r,0];ℝd), we will consider the ℝd-valued process Xε={Xε(t);t∈[-r,T]} given by the stochastic functional differential equation of the form:
(2)Xε(t)=η(t)(t∈[-r,0]),dXε(t)=A0(t,Xtε)dt+εA(t,Xtε)dW(t)(t∈(0,T]),
where Xsε={Xε(s+u);u∈[-r,0]} is the segment. Since the current state of the solution depends on its past histories, the process Xε is non-Markovian clearly. Since the coefficients of (2) satisfy the Lipschitz and the linear growth condition in the functional sense, there exists a unique solution to (2), via the successive approximation Xε,(n)={Xε,(n)(t);t∈[-r,T]}(n∈ℤ+) of the solution process Xε to (2) as follows:
(3)Xε,(0)(t)=η(t)(t∈[-r,0]),Xε,(0)(t)=η(0)(t∈(0,T]),Xε,(n)(t)=η(t)(t∈[-r,0]),dXε,(n)(t)=A0(t,Xtε,(n-1))dt+εA(t,Xtε,(n-1))dW(t)(t∈(0,T]),
for n∈ℕ (cf. Itô and Nisio [1], Mohammed [2, 12]).Proposition 1.
For anyp>1, it holds that
(4)sup0<ε≤1𝔼[supt∈[-r,T]|Xε(t)|p]≤C3,p,T,η.Proof.
Letp>2 and t∈[0,T]. The Hölder inequality and the Burkholder inequality tell us to see that
(5)𝔼[supτ∈[-r,t]|Xε(τ)|p]≤C4,p∥η∥∞p+C4,p𝔼[supτ∈[0,t]|Xε(τ)|p]≤C4,p∥η∥∞p+C5,p𝔼[supτ∈[0,t]|∫0τ‍A0(s,Xsε)ds|p]+C5,pεp𝔼[supτ∈[0,t]|∫0τ‍A(s,Xsε)dW(s)|p]≤C4,p∥η∥∞p+C5,pTp-1∫0t‍𝔼[|A0(s,Xsε)|p]ds+C6,pεpTp/2-1∫0t‍∑i=1m‍𝔼[|Ai(s,Xsε)|p]ds≤C7,p,T,η+C8,p,T∫0t‍𝔼[supτ∈[-r,s]|Xε(τ)|p]ds,
from the linear growth condition on the coefficients Ai(i=0,1,…,m). Hence, the Gronwall inequality enables us to obtain the assertion for p>2.
As for1<p≤2, the Jensen inequality yields us to see that
(6)sup0<ε≤1𝔼[supt∈[-r,T]|Xε(t)|p]≤(sup0<ε≤1𝔼[supt∈[-r,T]|Xε(t)|2p])1/2,
which implies the assertion by using the consequence stated above. The proof is complete.


## 3. Applications of the Malliavin Calculus

At the beginning, we will introduce the outline of the Malliavin calculus on the Wiener spaceC0([0,T];ℝm), briefly, where C0([0,T];ℝm) is the set of ℝm-valued continuous functions on [0,T] starting from the origin. See Di Nunno et al. [13] and Nualart [9, 14] for details. Let H be the Cameron-Martin subspace of C0([0,T];ℝm) with the inner product
(7)〈g,h〉H=∫0T‍g˙(t)·h˙(t)dt(g,h∈H).
Denote by 𝒮 the set of ℝ-valued random variables such that a random variable F is represented as the following form:
(8)F(W)=f(W[h1],…,W[hn])
for W∈C0([0,T];ℝm), where h1,…,hn∈H, W[h]=∫0T‍h(s)·dW(s) for h∈H, and f∈Cp∞(ℝn;ℝ). Here, we will denote by Cp∞(ℝn;ℝ) the set of smooth functions on ℝn such that all derivatives of any orders have polynomial growth. For k∈ℕ, the k-times Malliavin-Shigekawa derivative DkF={Du1,…,ukkF;u1,…,uk∈[0,T]} for F∈𝒮 is defined by
(9)Du1,…,ukkF(W)={∑j=1n‍∂jf(W[h1],…,W[hn])×∫0u1‍hj(s)ds(k=1),Du1⋯DukF(W)(k≥2).
We will consider D0F=F, which helps us to define the operator Dk for k∈ℤ+. For p>1 and k∈ℤ+, let 𝔻k,p be the completion of 𝒮 with respect to the norm
(10)∥F∥k,p={(𝔼[|F|p])1/p(k=0),𝔼[|F|p]1/p+∑j=1k‍𝔼[∥DjF∥H⊗jp]1/p(k∈ℕ).
Let 𝔻k,p(ℝd) be the set of ℝd-valued random variables with the components of which belong to 𝔻k,p, and set 𝔻∞(ℝd)=⋂p>1‍⋂k∈ℤ+‍𝔻k,p(ℝd). For F∈𝔻1,2(ℝd), the ℝd⊗ℝd-valued random variable VF given by
(11)VF=〈DF,DF〉H=∫0T‍dduDuF·dduDuFdu
is well defined, which is called the Malliavin covariance matrix for F.Before studying the application of the Malliavin calculus to the solution processXε to (2), we will prepare two basic and well-known facts.Lemma 2 (cf. Kusuoka and Stroock [4], Lemma 2.1).
LetΓ be a real separable Hilbert space, and α:[0,T]×Ω→ℝm⊗Γ be a progressively measurable process such that
(12)𝔼[∫0T‍∥α(s)∥ℝm⊗Γpds]<+∞
for all p>1. Then, for any p>2 and τ∈[0,T], it holds that
(13)𝔼[supt∈[0,τ]∥∫0t‍α(s)dW(s)∥Γp]≤C9,p,τ∫0τ‍∑i=1m‍𝔼[∥αi(s)∥Γp]ds.Lemma 3 (cf. Nualart [9], Proposition 1.3.8).
Let{β(t);t∈[0,T]} be a (ℱt)-adapted, ℝm⊗ℝd-valued process such that β(t)∈𝔻1,2(ℝm⊗ℝd) for almost all t∈[0,T], and that
(14)𝔼[∫0T‍∫0T‍|Duβ(t)|2dudt]<+∞.
Then, for each t∈[0,T], it holds that ∫0t‍β(s)dW(s)∈𝔻1,2(ℝd), and that
(15)Du(∫0t‍β(s)dW(s))=∫0u∧t‍β(s)ds+∫0t‍Duβ(s)dW(s).Now, we will return our position to study the application of the Malliavin calculus to the solution processXε to (2).Proposition 4.
Letn∈ℤ+ and 0<ε≤1. Then, for each t∈[-r,T], the ℝd-valued random variable Xε,(n)(t) is in 𝔻∞(ℝd). Moreover, for each k∈ℤ+, it holds that
(16)𝔼[supt∈[-r,T]∥DkXε,(n)(t)∥H⊗k⊗ℝdp]≤C10,k,p,T,η,𝔼[supt∈[-r,T]∥DkXε,(n)(t)-DkXε,(n-1)(t)∥H⊗k⊗ℝdp]≤C11,k,p,T,η2n/2.Proof.
At the beginning, we will consider the casep>2 inductively on k∈ℤ+. As for k=0, it is a routine work to check the assertion via the Hölder inequality and the Burkholder inequality, from the Lipschitz condition and the linear growth condition on the coefficients Ai(i=0,1,…,m), similarly to Proposition 1. Next, we will discuss the case k=1. Let n∈ℕ, because the assertion for n=0 is trivial. Since DXε,(n)=0 for t∈[-r,0], we have only to prove the assertion for t∈(0,T]. The chain rule on the operator D and Lemma 3 tell us to see that
(17)DuXε,(n)(t)=ε∫0u‍A(s,Xsε,(n-1))𝕀(s≤t)ds+∫0t‍∇A0(s,Xsε,(n-1))DuXsε,(n-1)ds+ε∫0t‍∇A(s,Xsε,(n-1))DuXsε,(n-1)dW(s)
for u∈[0,T] (cf. Ferrante et al. [10], Lemma 6.1), where the symbol ∇ is the Frechét derivative in C([-r,0];ℝd). Thus, the Hölder inequality and Lemma 2 enable us to get the assertions. Finally, we will discuss the general case k∈ℤ+. Suppose that the assertions are right until the case n-1. Remark that
(18)Du1,…,ukk(∫0t‍A(s,Xsε,(n-1))dW(s))=Du1,…,uk-1k-1(∫0t‍Duk(A(s,Xsε,(n-1)))dW(s))+Du1,…,uk-1k-1(∫0uk∧t‍A(s,Xsε,(n-1))ds)=⋯=∫0t‍Du1,…,ukk(A(s,Xsε,(n-1)))dW(s)+∑σ∈𝔖k‍∫0uσ(k)∧t‍Duσ(1),…,uσ(k-1)k-1(A(s,Xsε,(n-1)))ds,
from Lemma 3, where 𝔖k is the set of permutations of {1,…,k}. Since
(19)Du1,…,ukk(Ai(s,Xsε,(n-1)))=Du1,…,uk-1k-1(∇Ai(s,Xsε,(n-1))DukXsε,(n-1))=∑σ∈𝔖k‍∑j=0k-1‍(k-1j)Duσ(1),…,uσ(k-j)k-1-j(∇Ai(s,Xsε,(n-1)))×Duσ(k-j+1),…,uσ(k-1),ukj+1Xsε,(n-1)
for i=0,1,…,m, and
(20)Du1,…,ukkXε,(n)(t)=Du1,…,ukk(∫0t‍A0(s,Xsε,(n-1))ds)+Du1,…,ukk(ε∫0t‍A(s,Xsε,(n-1))dW(s))=∑σ∈𝔖k‍ε∫0uσ(k)∧t‍Duσ(1),…,uσ(k-1)k-1(A(s,Xsε,(n-1)))ds+∫0t‍Du1,…,ukk(A0(s,Xsε,(n-1)))ds+ε∫0t‍Du1,…,ukk(A(s,Xsε,(n-1)))dW(s),
we can get the assertion by using the Hölder inequality, Lemma 2, and the assumption on the case until k-1 of the induction.
The case1<p≤2 is the direct consequence by the Jensen inequality. The proof is complete.Proposition 5.
Fort∈[-r,T], the ℝd-valued random variable Xε(t) is in 𝔻∞(ℝm). Moreover, for each u∈[0,T], the ℝm⊗ℝd-valued process {DuXε(t);t∈[-r,T]} satisfies the equation of the form:
(21)DuXε(t)=0(t∈[-r,0]ort<u),DuXε(t)=ε∫0u∧t‍A(s,Xsε)ds+∫0t‍∇A0(s,Xsε)DuXsεds+ε∫0t‍∇A(s,Xsε)DuXsεdW(s)(t∈[u,T]).Proof.
Letp>1 and k∈ℤ+ be arbitrary. For each t∈[-r,T], the sequence {Xε,(n)(t);n∈ℕ} is the Cauchy one in 𝔻k,p(ℝd), from Proposition 4. Hence, we can find the limit, denoted by X~ε(t), in 𝔻k,p(ℝd). Then, it is a routine work to see that the process {X~ε(t);t∈[-r,T]} satisfies (2), via the Hölder inequality and the Burkholder inequality, from the conditions on the coefficients Ai(i=0,1,…,m), which implies X~ε(t)=Xε(t) for t∈[-r,T] from the uniqueness of the solutions. Thus, we can get X(t)∈𝔻k,p(ℝd) for t∈[-r,T]. Similarly, we can check that {DuX(t);u∈[0,T]} satisfies (21), by taking the limit in each term of (17) via the Hölder inequality and Lemma 2.Foru∈[0,T], denote by {Zε(t,u);t∈[-r,T]} the ℝd⊗ℝd-valued process determined by the following equation:
(22)Zε(t,u)=0(t∈[-r,0]ort<u),Zε(u,u)=Id,dZε(t,u)=∇A0(t,Xtε)Ztε(·,u)dt+ε∇A(t,Xtε)Ztε(·,u)dW(t)(t∈(u,T]),
where Ztε(·,u)={Zε(t+τ,u);τ∈[-r,0]}.Corollary 6.
(23)
D
u
X
ε
(
t
)
=
ε
∫
0
u
∧
t
‍
Z
ε
(
t
,
s
)
A
(
s
,
X
s
ε
)
d
s
.Proof.
Direct consequence of Proposition5 and the uniqueness of the solution to (21).Finally, we will introduce the well-known criterion on the existence of the smooth density for the probability law ofXε(t) with respect to the Lebesgue measure on ℝd.Lemma 7 (cf. Kusuoka and Stroock [4]).
Suppose the uniformly elliptic condition on the coefficientsAi(i=1,…,m) of (2) as follows:
(24)infζ∈𝕊d-1inft∈[0,T]inff∈C([-r,0];ℝd)∑i=1m‍(ζ·Ai(t,f))2>0.
Then, for each t∈(0,T] and 0<ε≤1, there exists a smooth density pε(t,y) for the probability law of Xε(t) with respect to the Lebesgue measure over ℝd.Proof.
SinceX(t)∈𝔻∞(ℝd) from Proposition 5, it is sufficiently to study that (detVε(t))-1∈⋂p>1‍𝕃p(Ω) under the uniformly elliptic condition (24), where Vε(t) is the Malliavin covariance matrix for Xε(t). Denote by
(25)V~ε(t)=∫0t‍∑i=1m‍Zε(t,u)Ai(u,Xuε)Ai(u,Xuε)*Zε(t,u)*du.
Then, Vε(t)=ε2V~ε(t), so we have only to discuss the moment estimate on V~ε(t). As stated in Lemma 1 of Komatsu and Takeuchi [15], we will pay attention to the boundedness of
(26)supζ∈𝕊d-1𝔼[(ζ·V~ε(t)ζ)-p],
for any p>1, which is sufficient to our goal. Since
(27)𝔼[(ζ·V~ε(t)ζ)-p]=1Γ(p)∫0+∞‍λp-1×𝔼[exp(-λζ·V~ε(t)ζ)]dλ,
we have to study the decay order of supζ∈𝕊d-1𝔼[exp(-λζ·V~ε(t)ζ)] as λ→+∞.
Letλ>1 be sufficiently large. Remark that
(28)𝔼[∥Zε(t,u)-Id∥ℝd⊗ℝdp]≤C12,p,T(t-u)p/2,
for any p>1, from the Burkholder inequality and the Hölder inequality. Let ξ>1/2, 1<γ<2ξ and 0<σ<(γ-1)/2. Write tξ:=t-λ-ξ, and let ζ∈𝕊d-1. Then, we see that
(29)𝔼[exp(-ζ·V~ε(t)ζ)]≤𝔼1[exp(-λζ·V~ε(t)ζ)]+ℙ[∫tξt‍∥Zε(t,u)-Id∥ℝd⊗ℝd2du≥λ-γ]+ℙ[sups∈[-r,t]|Xε(s)|≥λσ]=:I1+I2+I3,
where
(30)𝔼1[·]≔𝔼[·:∫tξt‍∥Zε(t,u)-Id∥ℝd⊗ℝd2du<λ-γ,sups∈[-r,t]|Xε(s)|<λσ].
The Chebyshev inequality yields that
(31)I2≤λγp𝔼[(∫tξt‍∥Zε(t,u)-Id∥ℝd⊗ℝd2du)p]≤C13,p,Tλ-(2ξ-γ)p.
Similarly, the Chebyshev inequality leads to
(32)I3≤λ-σp𝔼[sups∈[-r,t]|Xε(s)|p]≤C14,p,T,ηλ-σp,
from Proposition 1. On the other hand, as for I1, we have
(33)I1≤𝔼1[exp(-λ∫tξt‍∑i=1m‍|ζ·Zε(t,u)Ai(u,Xuε)|2du)]≤𝔼1[exp(-λ2infζ∈𝕊d-1∫tξt‍∑i=1m‍|ζ·Ai(u,Xuε)|2du)×exp(λ∫tξt‍∥Zε(t,u)-Id∥ℝd⊗ℝd2∑i=1m‍|Ai(u,Xuε)|2du)]≤exp(λ1-γ+2σ)×exp(-λ2infζ∈𝕊d-1infu∈[0,T]inff∈C([-r,0];ℝd)∑i=1m‍|ζ·Ai(u,f)|2)≤C15exp(-C16λ).
Therefore, we can get
(34)𝔼[exp(-λζ·V~ε(t)ζ)]≤C17,p,T,ηλ-C18p,
so we have
(35)supζ∈𝕊d-1𝔼[(ζ·Vε(t)ζ)-p]=ε-2dpsupζ∈𝕊d-1𝔼[(ζ·V~ε(t)ζ)-p]≤C19,p,Tε-2dp,
for any p>1. The proof is complete.Remark 8.
Consider the case(36)Ai(t,f)=A~i(t,f(0))(i=1,…,m),
where A~i:[0,T]×ℝd→ℝd with the good conditions on the boundedness and the regularity. Now, our stochastic functional differential equation is as follows:
(37)Xε(t)=η(t)(t∈[-r,0]),dXε(t)=A0(t,Xtε)dt+εA~(t,Xε(t))dW(t)(t∈(0,T]),
where A~=(A~1,…,A~m). Then, we can get the same upper estimate of the inverse of the Malliavin covariance matrix Vε(t) for Xε(t) in the hypoelliptic situation, which means that the linear space generated by the vectors A~i(i=1,…,m), and their Lie brackets span the space ℝd (cf. Takeuchi [16]).


## 4. Large Deviation Principles forXε

At the beginning, we will introduce the well-known fact on the sample-path large deviations for Brownian motions. See also [8]. Recall that H is the Cameron-Martin space of C0([0,T];ℝm).Lemma 9 (cf. Dembo and Zeitouni [17], Theorem  5.2.3).
The family{ℙ∘(εW)-1;0<ε≤1} of the laws of εW over C0([0,T];ℝm) satisfies the large deviation principle with the good rate function I, where
(38)I(f)={∥f∥H22(f∈H),+∞(f∉H).Forf∈H, let xf={xf(t);t∈[-r,T]} be the solution to the following functional differential equation:
(39)xf(t)=η(t)(t∈[-r,0]),dxf(t)=A0(t,xtf)dt+A(t,xtf)f˙(t)dt(t∈(0,T]).
Denote by
(40)Cη([-r,T];ℝd)={w∈C([-r,T];ℝd);w(t)=η(t)(t∈[-r,0])}.Theorem 10.
The family{ℙ∘(Xε)-1;0<ε≤1} of the laws of Xε over Cη([-r,T];ℝd) satisfies the large deviation principle with the good rate function I~, where
(41)I~(g)=inf{I(f);f∈H,g=xf},
and I is the function given in Lemma 9.Theorem10 tells us to see, via the contraction principle (cf. Dembo and Zeitouni [17], Theorem 4.2.1).Corollary 11.
For eacht∈[0,T], the family {ℙ∘(Xε(t))-1;0<ε≤1} of the laws of Xε(t) over ℝd satisfies the large deviation principle with the good rate function I-, where
(42)I-(y)=inf{I~(g);g∈Cη([-r,T];ℝd),y=g(t)},
and I~ is the function given in Theorem 10.Now, we will prove Theorem10, according to Azencott [18] and Léandre [5–8]. Our strategy stated here is almost parallel to [10, 11].Proposition 12.
For anya>0, the mapping
(43)Ha:={f∈H;∥f∥H≤a}∋f⟼xf∈Cη([-r,T];ℝd)
is continuous.Proof.
Letf,g∈Ha. Since
(44)xf(t)=η(0)+∫0t‍A0(s,xsf)ds+∫0t‍A(s,xsf)f˙(s)ds,
we see that
(45)supτ∈[-r,t]|xf(τ)|≤∥η∥∞+supτ∈[0,t]|xf(τ)|≤2∥η∥∞+∫0t‍|A0(s,xsf)|ds+∫0t‍∥A(s,xsf)∥ℝm⊗ℝd|f˙(s)|ds≤C20,T,η+C21,T∫0t‍(1+|f˙(s)|)×(1+supτ∈[-r,s]|xf(τ)|)ds,
from the linear growth condition on Ai(i=0,1,…,m), which tells us to see that
(46)supτ∈[-r,T]|xf(τ)|≤C22,T,η,a.
On the other hand, since
(47)xf(t)-xg(t)=∫0t‍{A0(s,xsf)-A0(s,xsg)}ds+∫0t‍{A(s,xsf)f˙(s)-A(s,xsg)g˙(s)}ds,
for t∈(0,T], and the ℝd-valued functions Ai(i=0,1,…,m) satisfy the Lipschitz condition and the linear growth condition, we have
(48)supτ∈[-r,t]|xf(τ)-xg(τ)|=supτ∈[0,t]|xf(τ)-xg(τ)|≤∫0t‍|A0(s,xsf)-A0(s,xsg)|ds+∫0t‍∥A(s,xsf)-A(s,xsg)∥ℝm⊗ℝd|f˙(s)|ds+∫0t‍∥A(s,xsg)∥ℝm⊗ℝd|f˙(s)-g˙(s)|ds≤C23,T∫0t‍supτ∈[-r,s]|xf(τ)-xg(τ)|(1+∑i=1m‍|f˙i(s)|)ds+C24,T,η,a∥f-g∥H.
The Gronwall inequality tells us to see that
(49)supτ∈[-r,t]|xf(τ)-xg(τ)|≤C24,T,η,a∥f-g∥H×exp[C23,T∫0t‍(1+∑i=1m‍|f˙i(s)|)ds]≤C25,T,η,a∥f-g∥H,
which completes the proof.Proposition 13.
Suppose that theℝd-valued functions Ai(i=1,…,m) are bounded. Then, for any f∈H and ρ>0, there exist αρ>0 and ερ>0 such that
(50)ℙ[supτ∈[-r,T]|Xε(τ)-xf(τ)|>ρ,supτ∈[0,T]|εW(τ)-f(τ)|≤αρ]≤C26,T,f,ρexp[-C27,T,fρ2ε2],
for any 0<ε≤ερ.Proof.
Define a new probability measuredℙ~ by
(51)dℙ~dℙ|ℱT=exp[∫0T‍∑i=1m‍f˙i(s)εdWi(s)-∥f∥H22ε2].
The Girsanov theorem tells us to see that the ℝm-valued process {W~(t):=W(t)-f(t)/ε;t∈[0,T]} is also the m-dimensional Brownian motion under the probability measure dℙ~. Let {Xε,f(t);t∈[-r,T]} be the ℝd-valued process determined by the following equation:
(52)Xε,f(t)=η(t)(t∈[-r,0]),dXε,f(t)=A0(t,Xtε,f)dt+A(t,Xtε,f){εdW~(t)+f˙(t)dt}(t∈(0,T]).
Write M(t):=∫0t‍A(s,Xsε,f)dW~(s). Remark that
(53)supτ∈[-r,t]|Xε,f(τ)-xf(τ)|=supτ∈[0,t]|Xε,f(τ)-xf(τ)|≤∫0t‍|A0(s,Xsε,f)-A0(s,xsf)|ds+∫0t‍∥A(s,Xsε,f)-A(s,xsf)∥ℝm⊗ℝd|f˙(s)|ds+supτ∈[0,t]|εM(τ)|≤C28,T∫0t‍supτ∈[-r,s]|Xε,f(τ)-xf(τ)|×(1+∑i=1m‍|f˙i(s)|)ds+supτ∈[0,t]|εM(τ)|.
The Gronwall inequality tells us to see that
(54)supτ∈[-r,t]|Xε,f(τ)-xf(τ)|≤(supτ∈[0,t]|εM(τ)|)×exp[C28,T∫0t‍(1+∑i=1m‍|f˙i(s)|)ds]≤C29,T,f(supτ∈[0,t]|εM(τ)|).
For each k=1,…,d, the martingale representation theorem enables us to see that there exists a 1-dimensional Brownian motion {Bk(t);t∈[0,T]} starting at the origin with
(55)Mk(t)=Bk(〈Mk〉(t)),〈Mk〉(t)=∫0t‍∑i=1m‍|Aik(s,Xsε,f)|2ds,
for k=1,…,d. Remark that 〈Mk〉(t)≤C30,T, because of the boundedness of the ℝd-valued functions Ai(i=1,…,m). Since
(56)ℙ~[supτ∈[0,C30,T]|Bk(τ)|>ρC31,T,fε]≤2exp[-ρ24C30,TC31,T,f2ε2],
from the reflection principle on Brownian motions, we have
(57)ℙ[supτ∈[-r,T]|Xε(τ)-xf(τ)|>ρ,supτ∈[0,T]|εW(τ)-f(τ)|≤αρ]=ℙ~[supτ∈[-r,T]|Xε,f(τ)-xf(τ)|>ρ,supτ∈[0,T]|εW~(τ)|≤αρ]≤ℙ~[supτ∈[0,T]|M(τ)|>ρC29,T,fε]≤ℙ~[supτ∈[0,C30,T]|B(τ)|>ρC29,T,fε]≤ℙ~[⋃k=1d‍{supτ∈[0,C30,T]|Bk(τ)|>ρC29,T,fdε}]≤2dexp[-ρ24C30,TC29,T,f2dε2],
which completes the proof.Proposition 14.
It holds that(58)limR→+∞limsupε↘0εlnℙ[supt∈[-r,T]|Xε(t)|>R]=-∞.Proof .
LetN>2 be sufficient large. From the Itô formula, we see that
(59)(1+|Xε(t)|2)N=(1+|η(0)|2)N+∫0t‍N(1+|Xε(s)|2)N-12εXε(s)·A(s,Xsε)dW(s)+∫0t‍{∑i=1m‍N(1+|Xε(s)|2)N-1×(2Xε(s)·A0(s,Xsε)+ε2∑i=1m‍|Ai(s,Xsε)|2)+2N(N-1)ε2(1+|Xε(s)|2)N-2×∑i=1m‍(Xε(s)·Ai(s,Xsε))2}ds.
Define σR=inf{t>0;|Xε(t)|>R}. Then, it holds that
(60)𝔼[(1+|Xε(t∧σR)|2)N]≤(1+∥η∥∞2)N+𝔼[∫0t∧σR‍{∑i=1m‍N(1+|Xε(s)|2)N-1×(2Xε(s)·A0(s,Xsε)+ε2∑i=1m‍|Ai(s,Xsε)|2)+2N(N-1)ε2(1+|Xε(s)|2)N-2×∑i=1m‍(Xε(s)·Ai(s,Xsε))2}ds]≤(1+∥η∥∞)N+C32,T(N+ε2N+ε2N2)×𝔼[∫0t‍(1+|Xε(s∧σR)|2)Nds],
from the linear growth condition on the coefficients Ai(i=0,1,…,m) of (2). Hence, the Gronwall inequality implies that
(61)𝔼[(1+|Xε(t∧σR)|2)N]≤(1+∥η∥∞)Nexp[C32,T(N+ε2N+ε2N2)t].
In particular, taking N=1/ε yields that
(62)𝔼[(1+|Xε(t∧σR)|2)1/ε]≤(1+∥η∥∞)1/εexp[C33,T(1ε+1)t].
Therefore, the Chebyshev inequality leads us to see that
(63)ℙ[supt∈[-r,T]|Xε(t)|>R]=ℙ[σR≤T]≤ℙ[|Xε(T∧σR)|≥R]≤(1+R2)-1/ε𝔼[(1+|Xε(T∧σR)|2)N]≤(1+∥η∥∞21+R2)1/εexp[C33,T(1ε+1)T],
so we have
(64)limsupε↘0εlnℙ[supt∈[-r,T]|Xε(t)|>R]≤ln(1+∥η∥∞21+R2)+C33,TT,
which completes the proof.LetR≥1. Define that σR=inf{t>0;|Xε(t)|>R} and Xε,R(t)=Xε(t∧σR).Proposition 15.
For anyδ>0, it holds that
(65)limR→+∞limsupε↘0εlnℙ[supt∈[-r,T]|Xε(t)-Xε,R(t)|>δ]=-∞.Proof .
Remark that(66)ℙ[supt∈[-r,T]|Xε(t)-Xε,R(t)|>δ]≤ℙ[supt∈[-r,T]|Xε(t)-Xε,R(t)|>δ,supt∈[-r,T]|Xε(t)|≤R]+ℙ[supt∈[-r,T]|Xε(t)|>R]=ℙ[supt∈[-r,T]|Xε(t)-Xε,R(t)|>δ,σR≥T]+ℙ[σR≤T]=ℙ[σR≤T]≤ℙ[|Xε(T∧σR)|≥R]≤(1+∥η∥∞21+R2)1/εexp[C33,T(1ε+1)T],
as seen in the proof of Proposition 14. So, we can get
(67)limsupε↘0εlnℙ[supt∈[-r,T]|Xε(t)-Xε,R(t)|>δ]≤ln(1+∥η∥∞21+R2)+C33,TT,
which completes the proof.Proof of Theorem10.
We will prove the assertion in two steps of the form: the case whereAi(i=1,…,m) are bounded, and the general case on Ai(i=1,…,m).
Step1. Suppose that the coefficients Ai(i=1,…,m) are bounded. Propositions 12 and 13 are sufficient to our goal (cf. [17, 18]). In fact, the large deviation principle for the family {ℙ∘(Xε)-1;0<ε≤1} comes from the one for {ℙ∘(εW)-1;0<ε≤1} in Lemma 9.
Step2. We will discuss the general case on Ai(i=1,…,m). Let R≥1, and F be a closed set in Cη([-r,T];ℝd). Denote by FR=F∩B(0;R)¯ and by FRδ the closed δ-neighborhood of FR, where B(0;R) is the open ball in Cη([-r,T];ℝd) with radius R centered at 0∈Cη([-r,T];ℝd). Then, it holds that
(68)ℙ[Xε∈F]≤ℙ[Xε∈F,supt∈[-r,T]|Xε(t)|≤R]+ℙ[supt∈[-r,T]|Xε(t)|>R]=ℙ[Xε,R∈FR]+ℙ[supt∈[-r,T]|Xε(t)|>R].
As seen in Step 1, we have already obtained the large deviation principle for {ℙ∘(Xε,R)-1;0<ε≤1} with the good rate function I~R, where I(f) is given in Lemma 9 and
(69)I~R(g)=inf{I(f);f∈H,g=xf,supt∈[-r,T]|xf(t)|≤R}.
So, we have
(70)limsupε↘0εlnℙ[Xε,R∈FR]≤-infg∈FRI~R(g).
Therefore, we can get
(71)limsupε↘0εlnℙ[Xε∈F]≤limR→+∞{(-infg∈FRI~R(g))≤limR→+∞∨(limsupε↘0εlnℙ[supt∈[-r,T]|Xε(t)|>R])}=limR→+∞(-infg∈FRI~(g))≤-infg∈FI~(g),
from Proposition 14, which completes the proof on the upper estimate of the large deviation principle.
Next, we will pay attention to the lower estimate of the large deviation principle. LetG be an open set in Cη([-r,T];ℝd), and take g~ in G∩B(0;R)¯. Then, we can find δ>0 such that B(g~;δ)⊂G. Thus, we have
(72)-I~(g~)=-I~R(g~)≤-infg∈B(g~;δ/2)I~R(g)≤liminfε↘0εlnℙ[Xε,R∈B(g~;δ2)]≤liminfε↘0ε×ln{[supt∈[-r,T]|Xε(t)-Xε,R(t)|>δ2]ℙ[Xε∈G]+ℙ[supt∈[-r,T]|Xε(t)-Xε,R(t)|>δ2]}≤(liminfε↘0εlnℙ[Xε∈G])∨(liminfε↘0εlnℙ[supt∈[-r,T]|Xε(t)-Xε,R(t)|>δ2]).
The first equality is right, because of g~∈B(0;R)¯, while the third inequality is the consequence of the large deviation principle for Xε,R as seen in Step 1. The forth inequality is right, because Xε∈B(g~;δ/2)c under Xε,R∈B(g~;δ)c and supt∈[-r,T]|Xε(t)-Xε,R(t)|≤δ/2. Taking the limit as R→+∞ leads us to see that
(73)-I~(g~)≤liminfε↘0εlnℙ[Xε∈G],
from Proposition 15, which completes the proof on the lower estimate of the large deviation principle. The proof of Theorem 10 is complete.


## 5. Density Estimates

In this section, we will consider the estimate of the densitypε(t,y) for the solution Xε(t), from the viewpoint of the Malliavin calculus.Theorem 16 (Upper estimate).
Suppose that theℝd-valued functions Ai(i=1,…,m) satisfy the uniformly elliptic condition (24). Then, it holds that
(74)limsupε↘0ε2lnpε(t,y)≤-I-(y),
where the function I- is given in Corollary 11.Proof.
Let0<σ<1 be sufficiently small, and Λσ∈C0∞(ℝd;[0,1]) such that
(75)Λσ(z)={1(|z-y|≤σ),0(|z-y|>2σ).
Take U=∏j=1d‍[aj,bj]⊂ℝd such that U⊂Supp[Λσ]. Then, the integration by parts formula tells us to see that
(76)ℙ[Xε(t)∈U]=𝔼[𝕀U(Xε(t))Λσ(Xε(t))]=𝔼[∫-∞Xε,1(t)‍⋯∫-∞Xε,d(t)‍𝕀U(y1,…,yd)dy1⋯dydΓ(1,…,d)(Xε(t),Λσ(Xε(t)))∏j=1d‍]=∫U‍𝔼[∏j=1d‍𝕀(yj,+∞)(Xε,j(t))×Γ(1,…,d)(Xε(t),Λσ(Xε(t)))∏j=1d‍]dy1⋯dyd,
where
(77)Γj(Xε(t),Λσ(Xε(t)))=δ(Λσ(Xε(t))∑k=1d‍[(Vε(t))-1]jkDXε,k(t)),Γ(1,…,d)(Xε(t),Λσ(Xε(t)))=Γd(Xε(t),Γ(1,…,d-1)(Xε(t),Λσ(Xε(t)))),
and δ is the Skorokhod integral operator. Remark that, under the uniformly elliptic condition (24) on the ℝd-valued functions Ai(i=1,…,m),
(78)∥Γ(1,…,d)(Xε(t),Λσ(Xε(t)))∥𝕃p(Ω)≤C34,p,T,η∥(Vε(t))-1∥𝕃α(Ω)∥Xε(t)∥β,γ∥Λσ(Xε(t))∥κ,σ≤C35,p,T,ηε-2d,
where α,γ,σ>1 and β,κ∈ℤ+, by using Proposition 5 and the proof of Lemma 7. Hence, the density pε(t,y) can be estimated from the above as follows:
(79)pε(t,y)=𝔼[∏j=1d‍𝕀(yj,+∞)(Xε,j(t))Γ(1,…,d)(Xε(t),Λσ(Xε(t)))]≤𝔼[|Γ(1,…,d)(Xε(t),Λσ(Xε(t)))|𝕀Supp[Λσ](Xε(t))]≤C35,p,T,ηε-2dℙ[Xε(t)∈Supp[Λσ]]1/q,
where q>1 such that 1/p+1/q=1. From Corollary 11, we have
(80)limsupε↘0ε2lnℙ[Xε(t)∈Supp[Λσ]]≤-infz∈Supp[Λσ]I-(z).
Since the function I- is a lower semicontinuous, taking the limit as σ↘0 and q↘1 enables us to see that
(81)limsupε↘0ε2lnpε(t,y)≤-I-(y),
which is the conclusion of Theorem 16.Remark 17.
As stated in Remark8, a similar problem can be also studied under the hypoelliptic condition, in the case
(82)Ai(t,f)=A~i(t,f(0))(i=1,…,m),
where A~i:[0,T]×ℝd→ℝd with the good conditions on the boundedness and the regularity (cf. [16]).Now, we will study the lower estimate of the densitypε(t,y) for the solution process Xε to (2). Before doing it, we will prepare some arguments.Proposition 18.
Letf∈H, and assume the uniformly elliptic condition (24) on the functions Ai(i=1,…,m). Then, it holds that (83)detvf(t)>0,
for each t∈(0,T], where vf(t) is the Gram matrix for xf(t).Proof.
Letu∈[0,T], and {Z-(t,u);t∈[-r,T]} be the ℝd⊗ℝd-valued mappings given by the following functional differential equation:
(84)Z-(t,u)=0(t∈[-r,0]ort∈(0,u)),dZ-(t,u)=∇A0(t,xtf)Z-t(·,u)dt+∇A(t,xtf)Z-t(·,u)f˙(t)dt(t∈[u,T]),
where Z-t(·,u)={Z-(t+τ,u);τ∈[-r,0]}. From the condition on the coefficients Ai(i=0,1,…,m), we see that
(85)supτ∈[u,t]∥Z-(τ,u)-Id∥ℝd⊗ℝd≤supτ∈[u,t]∥∫uτ‍∇A0(s,xsf)Z-s(·,u)ds∥ℝd⊗ℝd+supτ∈[u,t]∥∫uτ‍∇A(s,xsf)Z-s(·,u)f˙(s)ds∥ℝd⊗ℝd≤∫ut‍(∥∇A0(s,xsf)∥C([-r,0];ℝd)⊗ℝd+∥∇A(s,xsf)∥C([-r,0];ℝd)⊗ℝm⊗ℝd|f˙(s)|)ds+∫ut‍(∥∇A0(s,xsf)∥C([-r,0];ℝd)⊗ℝd+∥∇A(s,xsf)∥C([-r,0];ℝd)⊗ℝm⊗ℝd|f˙(s)|)×supτ∈[s-r,s]∥Z-(τ,u)-Id∥ℝd⊗ℝdds≤C36,T∫ut‍(1+∑i=1m‍|f˙i(s)|)ds+C37,T∫ut‍(1+∑i=1m‍|f˙i(s)|)×supτ∈[u,s]∥Z-(τ,u)-Id∥ℝd⊗ℝdds.
Remark that
(86)∫ut‍(1+∑i=1m‍|f˙i(s)|)ds≤C38,f,T(t-u)1/2.
Hence, the Gronwall inequality tells us to see that
(87)supτ∈[u,t]∥Z-(τ,u)-Id∥ℝd⊗ℝd≤C39,f,T(t-u)1/2.
On the other hand, remark that we have already seen in the proof of Proposition 12 that
(88)supτ∈[-r,t]|xf(τ)|≤C22,T,η,f.
Now, we will pay attention to the lower estimate ofdetvf(t). Since, for each u∈[0,T], {Duxf(t);t∈[-r,T]} satisfies the equation
(89)Duxf(t)=0(t∈[-r,0]),Duxf(t)=∫0u‍A(s,xsf)𝕀(s≤t)ds+∫0t‍∇A0(s,xsf)Duxsfds+∫0t‍∇A(s,xsf)Duxsff˙(s)ds(t∈(0,T]),
we have
(90)Duxf(t)=∫0u∧t‍Z-(t,s)A(s,xsf)ds,
similarly to Corollary 6. Hence, the Gram matrix vf(t) can be expressed as follows:
(91)vf(t)=∫0t‍Z-(t,u)A(u,xuf)A(xufu,xufxuf)*Z-(t,u)*du.
Let tα∈[0,T] be sufficiently close to t. So, we see that
(92)detvf(t)=det[∫0t‍Z-(t,u)A(u,xuf)A(xufu,xufxuf)*Z-(t,u)*du]≥{infζ∈𝕊d-1∫0t‍∑i=1m‍(ζ·Z-(t,u)Ai(u,xuf))2du}d={infζ∈𝕊d-1∫0t‍∑i=1m‍(ζ·Z-(t,u)Ai(u,xuf))2du}d×𝕀(supt∈[-r,T]|xf(t)|≤C22,T,η,f)×𝕀(supu∈[tα,t]∥Z-(t,u)-Id∥ℝd⊗ℝd≤C39,T,f(t-tα)1/2)≥{12infζ∈𝕊d-1∫tαt‍∑i=1m‍(ζ·Ai(u,xuf))2du-∫tαt‍∑i=1m‍∥Z-(t,u)-Id∥ℝd⊗ℝd2|Ai(u,xuf)|2du}d×𝕀(supt∈[-r,T]|xf(t)|≤C22,T,η,f)×𝕀(supu∈[tα,t]∥Z-(t,u)-Id∥ℝd⊗ℝd≤C39,T,f(t-tα)1/2)≥{t-tα2infζ,t,g∑i=1m‍(ζ·Ai(t,g))2-C40,T,η,f(t-tα)2}d×𝕀(supt∈[-r,T]|xf(t)|≤C22,T,η,f)×𝕀(supu∈[tα,t]∥Z-(t,u)-Id∥ℝd⊗ℝd≤C39,T,f(t-tα)1/2)≥C41,T,η,f(t-tα)d×𝕀(supt∈[-r,T]|xf(t)|≤C22,T,η,f)×𝕀(supu∈[t,tα]∥Z-(t,u)-∥ℝd⊗ℝd≤C39,T,f(t-tα)1/2)=C41,T,η,f(t-tα)d,
which is strictly positive. Here, we will remark that there exists the constant C41,T,η,f>0 with
(93)12infζ,t,g∑i=1m‍(ζ·Ai(t,g))2-C40,T,η,f(t-tα)≥C41,T,η,f,
because the functions Ai(i=1,…,m) satisfy the uniformly elliptic condition (24), and tα is sufficiently close to t, which justifies the sixth inequality.Forf∈H, let {Xε,f(t);t∈[-r,T]} be the ℝd-valued process determined by the following equation:
(94)Xε,f(t)=η(t)(t∈[-r,0]),dXε,f(t)=A0(t,Xtε,f)dt+εA(t,Xtε,f)dW(t)+A(t,Xtε,f)f˙(t)dt(t∈(0,T]).
Let {Z~f(t);t∈[-r,T]} be the ℝd-valued process determined by the following equation:
(95)Z~f(t)=0(t∈[-r,0]),dZ~f(t)=A(t,xtf)dW(t)+∇A0(t,xtf)Z~tfdt+∇A(t,xtf)Z~tff˙(t)dt(t∈(0,T]).Lemma 19.
Lett∈(0,T]. It holds that
(96)limε↘0∥Yε,f(t)-Z~f(t)∥k,p=0,
for any p>1 and k∈ℤ+, where Yε,f(t)=(Xε,f(t)-xf(t))/ε.Proof.
We will prove the statement along the following procedure.
Step1. For any p>1,
(97)limε↘0𝔼[supt∈[-r,T]|Xε,f(t)-xf(t)|p]=0.
In fact, since
(98)Xε,f(t)-xf(t)=∫0t‍{A0(s,Xsε,f)-A0(s,xsf)}ds+∫0t‍{A(s,Xsε,f)-A(s,xsf)}f˙(s)ds+ε∫0t‍A(s,Xsε,f)dW(s),
for t∈[0,T], and the coefficients Ai(i=0,1,…,m) satisfy the Lipschitz condition and the linear growth condition, we can get the assertion of Step 1 by using the Hölder inequality, the Burkholder inequality, and the Gronwall inequality.
Step2. For any p>1,
(99)limε↘0𝔼[supt∈[-r,T]|Yε,f(t)-Z~f(t)|p]=0,
which tells us to see that the assertion of Lemma 19 holds in the case of k=0.
In fact, we will remark that(100)Ai(s,Xsε,f)-Ai(s,xsf)ε-∇Ai(s,xsf)Z~sf=∇Ai(s,xsf)(Ysε,f-Z~sf)+12∇2Ai(s,σXsε,f+(1-σ)xsf)[Xsε,f-xsf,Ysε,f],
from the Taylor theorem for i=0,1,…,m, where 0<σ<1 is the constant. Here, for each t∈[0,T] and φ∈C([-r,0];ℝd),∇2Ai(t,φ)[·,·] is the bilinear mapping on C([-r,0];ℝd)×C([-r,0];ℝd). Since
(101)Yε,f(t)-Z~f(t)=∫0t‍[A0(s,Xsε,f)-A0(s,xsf)ε-∇A0(s,xsf)Z~sf]ds+∫0t‍[A(s,Xsε,f)-A(s,xsf)ε-∇A(s,xsf)Z~sf]f˙(s)ds+∫0t‍{A(s,Xsε,f)-A(s,xsf)}dW(s),
for t∈[0,T], and the coefficients Ai(t,·)(i=0,1,…,m) are in C1+,b∞(C([-r,0];ℝd);ℝd) with respect to the second variable in C([-r,0];ℝd) for each t∈[0,T], we can get the assertion in Step 2 via the Hölder inequality, the Burkholder inequality, and the Gronwall inequality.
Step3. Let u∈[0,T]. Then, for any p>1,
(102)sup0<ε≤1𝔼[supt∈[-r,T]|DuYε,f(t)|p]<+∞.
Remark that
(103)DuYε,f(t)=∫0u∧t‍{A(s,Xsε,f)+A(s,Xsε,f)-A(s,xsf)ε}ds+∫0t‍1ε{∇A0(s,Xsε,f)DuXsε,f-∇A0(s,xsf)Duxsf}ds+∫0t‍1ε{∇A(s,Xsε,f)DuXsε,f-∇A(s,xsf)Duxsf}f˙(s)ds+∫0t‍∇A(s,Xsε,f)DuXsε,fdW(s),
for t∈[u,T]. Since
(104)Duxf(t)=∫0u∧t‍A(s,xsf)ds+∫0t‍∇A0(s,xsf)Duxsfds+∫0t‍∇A(s,xsf)Duxsff˙(s)ds,
as seen in Proposition 18, we have
(105)supt∈[-r,T]|Duxf(t)|≤C42,T,η,f.
Moreover, similarly to Proposition 5, we have
(106)𝔼[supt∈[-r,T]|DuXε,f(t)|p]≤C43,p,T,η,f,
for any p>1. Then, the assertion in Step 3 can be justified by using the Hölder inequality, the Burkholder inequality, and the Gronwall inequality.
Step4. Let u∈[0,T]. Then, for any p>1,
(107)limε↘0𝔼[supt∈[-r,T]|DuYε,f(t)-DuZ~f(t)|p]=0.
In fact, since
(108)DuYε,f(t)=∫0u‍{A(s,Xsε,f)+A(s,Xsε,f)-A(s,xsf)ε}𝕀(s≤t)ds+∫0t‍1ε{∇A0(s,Xsε,f)DuXsε,f-∇A0(s,xsf)Duxsf}ds+∫0t‍1ε{∇A(s,Xsε,f)DuXsε,f-∇A(s,xsf)Duxsf}f˙(s)ds+∫0t‍∇A(s,Xsε,f)DuXsε,fdW(s),DuZ~f(t)=∫0u‍{A(s,xsf)+∇A(s,xsf)Z~sf}𝕀(s≤t)ds+∫0t‍∇A0(s,xsf)DuZ~sfds+∫0t‍∇2A0(s,xsf)[Duxsf,Z~sf]ds+∫0t‍∇A(s,xsf)DuZ~sff˙(s)ds+∫0t‍∇2A(s,xsf)[Duxsf,Z~sf]f˙(s)ds+∫0t‍∇A(s,xsf)DuxsfdW(s),
for t∈[u,T], and the coefficients Ai(t,·)(i=0,1,…,m) are in C1+,b∞(C([-r,0];ℝd);ℝd) with respect to the second variable in C([-r,0];ℝd) for each t∈[0,T], the assertion can be obtained via the Hölder inequality, the Burkholder inequality, and the Gronwall inequality. Here ∇2A(s,xsf)=(∇2A1(s,xsf),…,∇2Am(s,xsf)).
Step5. Let k∈ℕ be arbitrary, and u1,…,uk∈[0,T]. Then, for any p>1,
(109)limε↘0𝔼[supt∈[-r,T]|Du1,…,ukkYε,f(t)-Du1,…,ukkZ~f(t)|p]=0.
We have already proved the case of k=1 in Step 4. Remark that
(110)Du1,…,ukk(∫0t‍φ(s)dW(s))=∫0t‍Du1,…,ukk(φ(s))dW(s)+∑j=1k‍Du1,…,uj-1,uj+1,…,ukk-1(∫0uj∧t‍φ(s)ds),Du1,…,ukk(∫0t‍ψ(s)ds)=∫0t‍Du1,…,ukk(ψ(s))ds,
for adapted processes φ and ψ with nice properties. Then, we can get the assertion by induction on k∈ℕ.
Then, the assertion is the direct consequences of Step 2 and Step 5. The proof of Lemma19 is complete.Theorem 20 (Lower estimate).
Suppose that theℝd-valued functions Ai(i=1,…,m) satisfy the uniformly elliptic condition (24). Then, it holds that
(111)liminfε↘0ε2lnpε(t,y)≥-I-(y),
where the function I- is given in Corollary 11.Proof.
Since the assertion of Theorem20 is trivial in the case of I-(y)=+∞, we will suppose that I-(y)<+∞. Let Φ∈C0∞(ℝd;ℝ) be nonnegative. For sufficiently small 0<σ<1, recall the function Λσ as introduced in the proof of Theorem 16: Λσ∈C0∞(ℝd;[0,1]) such that
(112)Λσ(z)={1(|z-y|≤σ),0(|z-y|>2σ).
Then, the Girsanov theorem tells us to see that
(113)𝔼[Φ(Xε(t))]=𝔼[Φ(Xε,f(t))exp(-∫0t‍∑i=1m‍f˙i(s)εdWi(s)-12ε2∥f∥H2)]=exp(-∥f∥H2+4σ2ε2)×𝔼[Φ(Xε,f(t))exp(-∫0t‍∑i=1m‍f˙i(s)εdWi(s)+2σε2)]≥exp(-∥f∥H2+4σ2ε2)×𝔼[Φ(Xε,f(t))𝕀(ε∫0t‍∑i=1m‍f˙i(s)dWi(s)≤2σ)]≥exp(-∥f∥H2+4σ2ε2)×𝔼[Φ(Xε,f(t))Λσ(ε∫0t‍∑i=1m‍f˙i(s)dWi(s))].
Here, the third inequality comes from the nonnegativity in the exponent
(114)-1ε∫0t‍f˙(s)dW(s)+2σε2≥0,
while the forth inequality holds because of 0≤Λσ≤1 and Λσ≠0 on the complement of [-2σ,2σ]. Thus, the limiting argument enables us to see that
(115)pε(t,y)≥exp(-∥f∥H2+4σ2ε2)×𝔼[δy(Xε,f(t))Λσ(ε∫0t‍∑i=1m‍f˙i(s)dWi(s))]=ε-dexp(-∥f∥H2+4σ2ε2)×𝔼[δ0(Yε,f(t))Λσ(ε∫0t‍∑i=1m‍f˙i(s)dWi(s))],
where δy is the Dirac delta function. Since
(116)limε↘0𝔼[δ0(Yε,f(t))Λσ(ε∫0t‍∑i=1m‍f˙i(s)dWi(s))]≤1,
from Lemma 19, we have
(117)limε↘0ε2ln(ε-d𝔼[δ0(Yε,f(t))Λσ(ε∫0t‍∑i=1m‍f˙i(s)dWi(s))])=0.
Moreover, from the definition of the function I-(y), we can find f∈H with y=xf(t) such that
(118)∥f∥H22≤I-(y)+σ.
Hence, it holds that
(119)liminfε↘0ε2lnpε(t,y)≥-(∥f∥H22+2σ)≥-I-(y)-3σ.
Taking the limit as σ↘0 completes the proof.Corollary 21.
Suppose that theℝd-valued functions Ai(i=1,…,m) satisfy the uniformly elliptic condition (24). Then, it holds that
(120)pε(t,y)~exp[-I-(y)ε2],
as ε↘0, where the function I- is given in Corollary 11.Proof.
Direct consequences of Theorems16 and 20.Finally, we will study the asymptotic behavior of the densityp(t,y) for X(t) in a short time. Let 0<r0≤r be a constant, and x∈ℝd. We will consider the case
(121)η(t)=x(t∈[-r,0]),A0(t,f)≡0,Ai(t,f)=A~i(f~,f(0))(i=1,…,m),
where f~∈C([-r,0];ℝd) such that f~(t)=f(t)(t∈[-r,-r0]), and f~(t)=f~(-r0)(t∈[-r0,0]), for f∈C([-r,0];ℝd). Suppose that the functions A~i(i=1,…,m) satisfy the uniformly elliptic condition of the form:
(122)infζ∈𝕊d-1inff∈C([-r,-r0];ℝd)infy∈ℝd∑i=1m‍(ζ·A~i(f,y))2>0.
For 0<ε≤1, let X={X(t);t∈[-r,T]}, and let Xε={Xε(t);t∈[-r,T]} be the ℝd-valued processes determined by the equations of the form:
(123)X(t)=x(t∈[-r,0]),dX(t)=A~(X~t,X(t))dW(t)(t∈(0,T]),Xε(t)=x(t∈[-r,0]),dXε(t)=εA~(X~tε,Xε(t))dW(t)(t∈(0,T]),
where A~=(A~1,…,A~m). Remark that X=Xε|ε=1. Denote by p(t,y) (or, pε(t,y)) the density for the probability law of X(t)(Xε(t), resp.), whose existence can be justified under the uniformly elliptic condition (122) on the coefficients A~i(i=1,…,m). Then, we have the following.Corollary 22.
Suppose that the functionsA~i(i=1,…,m) satisfy the uniformly elliptic condition (122). Then, it holds that
(124)p(t,y)~exp[-r0I-(y)t](t↘0).Proof.
Recall that(125)X(ε2r0)=x+∫0ε2r0‍A~(X~s,X(s))dW(s)=x+ε∫0r0‍A~(X~ε2s,X(ε2s))dW(s)=x+ε∫0r0‍A~(xid,X(ε2s))dW(s),
where id∈C([-r,-r0];ℝd) such that id(t)=1(t∈[-r,-r0]). Here, the second equality holds from the scaling property on the Brownian motion W, while the third equality follows from ε2s-r0≤0. On the other hand, recall that
(126)Xε(r0)=x+ε∫0r0‍A~(X~sε,Xε(s))dW(s)=x+ε∫0r0‍A~(xid,Xε(s))dW(s),
because of s-r0≤0. From the uniqueness of the solutions, we have X(ε2r0)=Xε(r0) in the sense of the probability law. Hence, we can get
(127)p(ε2r0,y)=pε(r0,y).
As for the density pε(r0,y), we have already obtained the asymptotic behavior of the form:
(128)pε(r0,y)~exp[-I-(y)ε2],
as ε↘0, in Corollary 21. Taking t=ε2r0 completes the proof.Remark 23.
In particular, consider the case of(129)A0(s,f)=0,Ai(s,f)=A-i(f(0))(i=1,…,m),η(t)=x(t∈[-r,0]),
where A-i∈C1+,b∞(ℝd;ℝd) such that the functions A-i(i=1,…,m) satisfy the uniformly elliptic condition of the form:
(130)infζ∈𝕊d-1infy∈ℝd∑i=1m‍(ζ·A-i(y))2>0.
Then, our equation can be written as follows:
(131)X(t)=x(t∈[-r,0]),dX(t)=A-(X(t))dW(t)(t∈(0,T]),
where A-=(A-1,…,A-m). Although our settings include the effect of the time-delay parameter r, the effect of the parameter r in (131) can be ignored. Hence, the solution {X(t);t∈[-r,T]} is the diffusion process, so we have only to choose r=1 in the starting point of our study. Moreover, the choice of r=1 tells us to see that Corollary 22 is the well-known fact, that is, the Varadhan-type estimate, on the asymptotic behavior of the density function for diffusion processes. Hence, Corollary 22 can be also regarded as the generalization of the short-time estimate of the density for diffusion processes.Remark 24.
Ferrante et al. in [10] discussed the large deviation principle for the solution process Xε and the asymptotic estimate of the density, in the case of
(132)Ai(s,f)=A~i(s,f(s-r))(i=1,…,m),
where A~i:[0,T]×ℝd→ℝd with A~i(t,·)∈Cb∞(ℝd;ℝd) for each t∈[0,T]. Moreover, suppose that the functions A~i(i=1,…,m) satisfy the uniformly elliptic condition of the form:
(133)infζ∈𝕊d-1inft∈[0,T]infy∈ℝd∑i=1m‍(ζ·Ai(t,y))2>0.On the other hand, Mohammed and Zhang in [11] studied the large deviation principle for the solution process Xε, in the case of
(134)Ai(t,f)=A~i(t,f(t-r),f(t))(i=1,…,m),
where A~i:[0,T]×ℝd×ℝd→ℝd with A~i(t,·,·)∈C1+,b∞(ℝd×ℝd;ℝd).Since the special forms of the coefficients on the diffusion terms are quite essential in their arguments [10, 11], our situation cannot be included in their frameworks at all.



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