Some regularity results on stochastic convolutions in point delay differential equations perturbed by noise

Statistics and Probability Letters 156 (2020) 108624

Contents lists available at ScienceDirect

Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Some regularity results on stochastic convolutions in point delay differential equations perturbed by noise Kai Liu Department of Mathematical Sciences, School of Physical Sciences, The University of Liverpool Peach Street, Liverpool, L69 7ZL, U.K.

article

info

a b s t r a c t

Article history: Received 15 January 2019 Received in revised form 4 June 2019 Accepted 9 September 2019 Available online 18 September 2019 MSC: 60H15 60G15 60H05

In this work, we obtain some regularity results on stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded coefficient operators driven by white noise. We first establish some estimates on fundamental solutions which play an important role in the subsequent theory. Then we apply these estimates to some stochastic convolutions arising in stochastic delay differential equations to obtain the desired regularity property. Crown Copyright © 2019 Published by Elsevier B.V. All rights reserved.

Keywords: Regularity property Fundamental solution Stochastic convolution

1. Introduction Let X be a Banach space and consider a linear functional differential equation with delay in X ,

⎧ ⎨

0

∫

a(θ )A2 y(t + θ )dθ dt + f (t)dt , t ≥ 0,

dy(t) = Ay(t)dt + A1 y(t − r)dt + −r

⎩

(1.1)

y(0) = φ0 , y(θ ) = φ1 (θ ), θ ∈ [−r , 0], φ = (φ0 , φ1 ),

where r > 0 is some constant incurring the system delay, a(·) ∈ L2 ([−r , 0], R) and φ = (φ0 , φ1 ) is an appropriate initial datum. Here A : D (A) ⊂ X → X is the infinitesimal generator of an analytic semigroup etA , t ≥ 0, A1 , A2 are two closed linear operators with domains D (Ai ) ⊃ D (A), i = 1, 2, and f is a continuous function taking values in X . For simplicity, we assume that the analytic semigroup etA is negative type (see, Section A.4 in Da Prato and Zabczyk (2014)) so as that there exist constants M ≥ 1 and µ > 0 such that

∥etA ∥ ≤ Me−µt ,

∥AetA ∥ ≤ M /t

for all

t > 0.

δ

(1.2)

For any δ ∈ (0, 1), let (−A) denote the usual fractional power of operator A, which is defined as the inverse of (−A)−δ where (−A)−δ :=

1

Γ (δ )

∞

∫

t δ−1 etA dt .

0

Here Γ (·) is the standard Gamma function. E-mail address: [email protected]. https://doi.org/10.1016/j.spl.2019.108624 0167-7152/Crown Copyright © 2019 Published by Elsevier B.V. All rights reserved.

2

K. Liu / Statistics and Probability Letters 156 (2020) 108624

Equations of the type (1.1) were investigated by Di Blasio et al. (1985), Sinestrari (1984) and the fundamental solution to (1.1) was introduced by Jeong et al. (1993). In particular, it is known that the fundamental solution G(·) : R → L (X ) to (1.1) is an operator-valued function which is strongly continuous in X and satisfies the integral equation

G(t) =

⎧ ⎨

etA +

⎩

t

∫

e(t −s)A A1 G(s − r)ds + 0

O,

0

∫ t∫ 0

a(θ )e(t −s)A A2 G(s + θ )dθ ds, −r

t ≥ 0,

(1.3)

t < 0,

where O is the null operator in X . The fundamental solution G(·) enables us to solve the initial value problem (1.1). In fact, it may be shown that under some appropriate conditions on f and φ = (φ0 , φ1 ), the unique mild solution y to (1.1) is represented as y(t) = G(t)φ0 +

0

∫

Ut (θ )φ1 (θ )dθ +

t

∫

−r

G(t − s)f (s)ds,

t ≥ 0,

(1.4)

0

where Ut (θ ) = G(t − θ − r)A1 +

θ

∫

G(t − θ + τ )a(τ )A2 dτ ,

θ ∈ [−r , 0],

−r

with the initial condition y(0) = φ0 and y(θ ) = φ1 (θ ), θ ∈ [−r , 0]. This is a time delay version of the usual variation of constants formula for abstract differential equations. We introduce the following condition: (H) a(·) ∈ L∞ ([−r , 0], R) and D ((−A)µ ) ⊂ D (A1 ),

D ((−A)ν ) ⊂ D (A2 )

(1.5)

for some µ, ν ∈ (0, 1). Theorem 1.1 (Liu, 2019). Assume that condition (H) holds, then (a) for any γ ∈ [ν, 1), it is true that

∥(−A)γ G(t)∥ ≤

Cn,γ (t − nr)γ

for all

t ∈ (nr , (n + 1)r ],

(1.6)

where Cn,γ > 0, n ∈ N0 := {0, 1, . . .}, are constants depending on n and γ ; (b) for any γ ∈ [ν, 1) and 0 < β < 1 − γ , it is true that

∥(−A)γ (G(t) − G(s))∥ ≤ Cn,β,γ

(t − s)β (s − nr)β+γ

for all

nr < s < t < (n + 1)r ,

(1.7)

where Cn,β,γ > 0, n ∈ N0 , are constants depending on n, β and γ . The condition (H) is somewhat restrictive. For example, it excludes the important case: A = A1 or A2 . We shall show in this work that (H) can be actually removed if we focus only on those systems with point delay. Precisely, let A2 = 0 in (1.1) and consider a linear functional differential equation with delay in X , dy(t) = Ay(t)dt + A1 y(t − r)dt + f (t)dt , t ≥ 0,

{

y(0) = φ0 , y(θ ) = φ1 (θ ), θ ∈ [−r , 0], φ = (φ0 , φ1 ),

(1.8)

and the fundamental solution G(·) : R → L (X ) to (1.8) is an operator-valued function which is strongly continuous in X and satisfies G(t) =

⎧ ⎨

etA + O,

t

∫

e(t −s)A A1 G(s − r)ds, 0

t ≥ 0,

(1.9)

t < 0.

⎩

In particular, we shall establish in this work the following result. Theorem 1.2. Let G(·) be the fundamental solution to (1.8), i.e., G(·) satisfies (1.9), then (a) for any γ ∈ (0, 1), it is true that

∥(−A)γ G(t)∥ ≤

Cn,γ (t − nr)γ

for all

t ∈ (nr , (n + 1)r ],

where Cn,γ > 0, n ∈ N0 , are constants depending on n and γ ;

(1.10)

K. Liu / Statistics and Probability Letters 156 (2020) 108624

3

(b) for any γ ∈ (0, 1) and 0 < β < 1 − γ , it is true that

∥(−A)γ (G(t) − G(s))∥ ≤ Cn,β,γ

(t − s)β

for all

(s − nr)β+γ

nr < s < t < (n + 1)r ,

(1.11)

where Cn,β,γ > 0, n ∈ N0 , are constants depending on n, β and γ . Let {Ω , F , P} be a complete probability space, equipped with some filtration {Ft }t ≥0 . Let {W (t), t ≥ 0} be a Q -Wiener process with respect to {Ft }t ≥0 in a separable Hilbert space (H , ⟨·, ·⟩H ), defined on {Ω , F , P}, with covariance operator Q , i.e.,

E⟨W (t), x⟩H ⟨W (s), y⟩H = (t ∧ s)⟨Qx, y⟩H for all x, y ∈ H , where Q is some positive, self-adjoint operator on H. We frequently call W (t), t ≥ 0, an H-valued, Q -Wiener process with respect to {Ft }t ≥0 if the trace Tr(Q ) < ∞, whereas if Q = I, W (·) is called a cylindrical process. We introduce a subspace HQ = R (Q 1/2 ) ⊂ H, the range of Q 1/2 , which is a Hilbert space endowed with the inner product

⟨u, v⟩HQ := ⟨Q −1/2 u, Q −1/2 v⟩H

for any

u, v ∈ HQ .

Let L2 (HQ , H) denote the space of all Hilbert–Schmidt operators from HQ into H. For arbitrarily given T ≥ 0, let J(t , ω), t ∈ [0, T ], be a measurable, L2 (HQ , H)-valued process, and we define the following norm for arbitrary t ∈ [0, T ],

{ ∫ t [ ] } 12 Tr J(s, ω)Q 1/2 (J(s, ω)Q 1/2 )∗ ds . |J |t := E

(1.12)

0

In particular, we denote by U 2 [0, T ]; L2 (HQ , H) all the L2 (HQ , H)-valued measurable processes J, adapted to the filtration {Ft }t ≤T , satisfying |J |T < ∞. Suppose that W (·) is a Q -Wiener process in H such that

(

Qek = λk ek ,

k ≥ 1,

)

λk > 0,

(1.13)

where {ek } is a complete orthonormal basis in H, then it is immediate that W (t) =

∞ ∑ √

λk wk (t)ek ,

t ≥ 0,

(1.14)

k=1 t

∫

J(s)dW (s) ∈ H,

where {wk (·)} is a group of independent, standard real-valued Brownian motions. The stochastic integral 0

t ≥ 0, may be defined for all J ∈ U 2 ([0, T ] × Ω ; L2 (HQ , H)) by t

∫

2

J(s)dW (s) = L − lim

n ∫ t ∑ √

n→∞

0

k=1

λk J(s)ek dwk (s),

t ∈ [0, T ].

(1.15)

0

The reader is referred to Da Prato and Zabczyk (2014) for more details on this topic. In this work, we are concerned about the following linear stochastic retarded functional differential equation in H,

{

dy(t) = Ay(t)dt + A1 y(t − r)dt + dW (t), t ≥ 0,

(1.16)

y(0) = φ0 , y(θ ) = φ1 (θ ), θ ∈ [−r , 0], φ = (φ0 , φ1 ),

where A, A1 are given as in (1.1), W (·) is a Q -Wiener process in H and φ = (φ0 , φ1 ) is an appropriate initial datum. Throughout this work, we do not assume that Tr(Q ) < ∞. It is well known that the unique mild solution y of (1.16) is represented as y(t) = G(t)φ0 +

0

∫

G(t − θ − r)A1 φ1 (θ )dθ +

t

∫

−r

G(t − s)dW (s),

t ≥ 0,

(1.17)

0

with the initial condition y(0) = φ0 and y(θ ) = φ1 (θ ), θ ∈ [−r , 0]. In particular, if φ = (0, 0), then the unique mild solution (1.17) is the so-called stochastic convolution process t

∫

G(t − s)dW (s),

y(t) = WG (t) :=

t ≥ 0.

(1.18)

0

In this work, we obtain some regularity results for the stochastic convolution process (1.18). 2. Proof of Theorem 1.2 The proof of the following lemma is referred to Liu (2019). Lemma 2.1. Let γ ∈ (0, 1) and β > 0. For any 0 < s < t < ∞, there exists a number Cβ,γ > 0 such that

∥(−A)γ etA − (−A)γ esA ∥ ≤ Cβ,γ (t − s)β · s−β−γ .

(2.1)

4

K. Liu / Statistics and Probability Letters 156 (2020) 108624

Proof of Theorem 1.2 (a). To this end, we intend to develop an induction scheme. We first consider the case that n = 0 and in this case (−A)γ G(t) = (−A)γ etA ,

t ∈ [0, r ].

Hence, for t ∈ (0, r ] we have

∥(−A)γ G(t)∥ ≤ ∥(−A)γ etA ∥ ≤ Mγ /t γ ,

Mγ > 0,

(2.2)

which is (1.10) with n = 0. Now suppose that the fundamental solution G(·) satisfies all the estimates in Theorem 1.2(a) on the intervals [0, r ], · · ·, [(n − 1)r , nr ]. In the interval [nr , (n + 1)r ], we clearly have the relation (−A)γ G(t) = (−A)γ etA + (−A)γ

nr

∫

e(t −s)A A1 G(s − r)ds + (−A)γ

t

∫

e(t −s)A A1 G(s − r)ds nr

r

(2.3)

=: I1 (t) + I2 (t) + I3 (t). We estimate each term on the right hand side of (2.3). First, note that Mγ

∥I1 (t)∥ ≤

≤ Mγ (t − nr)−γ ,

tγ

t ∈ (nr , (n + 1)r ],

(2.4)

and I2 (t) =

n−1 ∑

(−A)γ

(j+1)r

∫

e(t −s)A A1 G(s − r)ds,

t ∈ (nr , (n + 1)r ].

(2.5)

jr

j=1

It is known by Theorem 1 in Tanabe (1988) that for i = 0, 1, A0 = A, Cn′

∥Ai G(t)∥ ≤

t − nr

 ∫ t   Ai G(u)du ≤ Cn′ , 

,

nr ≤ s < t ≤ (n + 1)r ,

(2.6)

s

for some Cn′ > 0, n ∈ N0 . Hence, for t ∈ (nr , (n + 1)r ] and j = 0, 1, . . . , n − 2, it is easy to see by (2.6) and Lemma 2.1 that

∫ (j+1)r     e(t −s)A A1 G(s − r)ds (−A)γ jr ∫ (j+1)r ∥(−A)γ e(t −s)A − (−A)γ e(t −jr)A ∥ · ∥A1 A−1 ∥ · ∥AG(s − r)∥ds ≤ jr

∫  + ∥(−A)γ e(t −jr)A ∥ · ∥A1 A−1 ∥ ·  (j+1)r

∫ ≤ Cγ ,µ

(t −

jr

′

≤

Cγ ,µ Cj r 2γ

(s − jr)γ

·

r

γ

·

s)2γ

Cj′ s − jr

(j+1)r

 

AG(s − r)ds

jr

ds + Mγ ,µ Cj′ ·

(2.7)

1 (t − jr)γ

′

+

γ

Mγ ,µ Cj rγ

,

Cγ ,µ , Cj′ , Mγ ,µ > 0.

On the other hand, for j = n − 1 and t ∈ (nr , (n + 1)r ], we have that for 0 < β < 1 − γ ,

∫ nr     e(t −s)A A1 G(s − r)ds (−A)γ (n−1)r ∫ nr ≤ ∥(−A)γ e(t −s)A − (−A)γ e(t −(n−1)r)A ∥ · ∥A1 A−1 ∥ · ∥AG(s − r)∥ds (n−1)r  ∫ nr    + ∥(−A)γ e(t −(n−1)r)A ∥ · ∥A1 A−1 ∥ ·  AG(s − r)ds (n−1)r ∫ nr Cn′ −1 Mγ (s − (n − 1)r)β ≤ Cβ,γ ,n · ds + C′ (t − s)β+γ s − (n − 1)r (t − (n − 1)r)γ n−1 (n−1)r ∫ nr Cn′ −1 Mγ · Cn′ −1 (s − (n − 1)r)β ≤ Cβ,γ ,n · ds + (nr − s)β+γ s − (n − 1)r rγ (n−1)r ′ Mγ · Cn−1 = Cβ,γ ,n · Cn′ −1 · B(1 − β − γ , β ) + , γ

(2.8)

r

where Cβ,γ ,n , Mγ , Cn−1 > 0 and B(·, ·) is the standard Beta function. Combining (2.5), (2.7) and (2.8), we have for t ∈ (nr , (n + 1)r ] that ′

K. Liu / Statistics and Probability Letters 156 (2020) 108624

∥I2 (t)∥ ≤

n−2 ( ∑ Cγ ,µ Cj′ r γ

γ

j=1

+

Mγ ,µ Cj′ ) rγ

+ Cβ,γ ,n · Cn′ −1 · B(1 − β − γ , β ) +

5

Mγ · Cn′ −1 rγ

.

(2.9)

Finally, for nr < t ≤ (n + 1)r, we have by (2.6) and Lemma 2.1 that

∫ t    ∥I3 (t)∥ ≤  (−A)γ (e(t −s)A − e(t −nr)A )A1 A−1 AG(s − r)ds nr ∫ t   γ (t −nr)A −1  + ∥(−A) e ∥ · ∥A1 A ∥ AG(s − r)ds nr ∫ t C ′ ∥A1 A−1 ∥Mγ −β−γ ′ −1 · (s − nr)β−1 ds + n ≤ Cβ,γ ,n ∥A1 A ∥ (t − s) (t − nr)γ nr −1 −1 ′ ′ ≤ {Cβ,γ ,n B(1 − β − γ , β )∥A1 A ∥ + Cn ∥A1 A ∥Mγ }(t − nr)−γ ,

(2.10)

′ ′′ ′ where Cβ,γ ,n , Cβ,γ ,n , Cn , Mγ > 0 and 0 < β < 1 − γ . Now the inequality (1.10) for t ∈ (nr , (n + 1)r ] follows from (2.4), (2.9) and (2.10). The proof is complete now.

Proof of Theorem 1.2 (b). Once again, we want to develop an induction scheme here. By virtue of (2.1), we first obtain the estimate in Theorem 1.2(b) for 0 < s < t < r, 0 < β < 1 − γ :

∥(−A)γ (G(t) − G(s))∥ = ∥(−A)γ (etA − esA )∥ ≤ Mβ,γ (t − s)β · s−β−γ ,

Mβ,γ > 0.

Now suppose that G(t) satisfies those estimates in Theorem 1.2 (b) on the intervals [0, r ], · · ·, [(n − 1)r , nr ]. Let nr < s < t < (n + 1)r and we clearly have (−A)γ (G(t) − G(s)) = (−A)γ (etA − esA )

+ (−A)γ

nr

(∫

e(t −u)A A1 G(u − r)du −

∫

r

γ

+ (−A)

t

(∫

(t −u)A

e

∫

nr

e(s−u)A A1 G(u − r)du

r s

(2.11) e(s−u)A A1 G(u − r)du

A1 G(u − r)du −

)

)

nr

nr

=: K1 (t , s) + K2 (t , s) + K3 (t , s). First, by virtue of Lemma 2.1 we have for 0 < β < 1 − γ that

∥K1 (t , s)∥ ≤ Mγ (t − s)β s−β−γ ≤

Mγ (nr)γ

(t − s)β · (s − nr)−β−γ ,

Mγ > 0.

(2.12)

Also, it is well known (see, e.g., (48) in Liu (2019)) that

∥etA − esA ∥ ≤ Cα (t − s)α · s−α ,

Cα > 0,

(2.13)

for any 0 < s < t < ∞ and α ∈ (0, 1]. For K2 (t , s), it is easy to see by virtue of (2.7), (2.8) and (2.13) that

∥K2 (s, t)∥ ≤ ∥e(t −nr)A − e(s−nr)A ∥ ·

∫ n  ∑  (−A)γ k=2

β

≤ Cn,β,γ (t − s) · (s − nr)

kr

 

(k−1)r

e(nr −u)A A1 G(u − r)du

−β

≤ Cn,β,γ r γ (t − s)β · (s − nr)−β−γ ,

(2.14)

0 < β < 1 − γ,

for some Cn,β,γ > 0. On the other hand, we have by assumption that K3 (t , s) =

t

∫

(−A)γ e(t −u)A A1 G(u − r)du

s

∫

s

+ (−A)γ (e(t −u)A − e(s−u)A )A1 G(u − r)du nr ∫ t ∫ t = (−A)γ e(t −u)A A1 (G(u − r) − G(t − r))du + (−A)γ e(t −u)A A1 G(t − r)du s s ∫ s + (−A)γ (e(t −u)A − e(s−u)A )A1 (G(u − r) − G(s − r))du nr ∫ s + (−A)γ (e(t −u)A − e(s−u)A )A1 G(s − r)du nr

=: J1 + J2 + J3 + J4 .

(2.15)

6

K. Liu / Statistics and Probability Letters 156 (2020) 108624

for any nr < s < t < (n + 1)r. It is known (see, Theorem 1 in Tanabe, 1992) that

∥A1 (G(t) − G(s))∥ ≤ Cn,δ (t − s)δ (s − nr)−δ−1 ,

nr < s < t ≤ (n + 1)r ,

(2.16)

for any δ ∈ (0, 1) and some Cn,δ > 0. Hence, by Theorem 1.1 we have for sufficiently small δ > 0 with 1 − β − γ + δ > 0, β + γ − δ > 0 that

∥J1 ∥ ≤

′ Cβ,γ ,δ,n (t

′ ≤ Cβ,γ ,δ,n

β

t

∫

− s)

1 (t − s)β (t − u)γ

s

(t − s)β

t

∫

(s − nr)β+γ

(t − u)δ (u − nr)−δ−1 (s − nr)β+γ (s − nr)−(β+γ ) du (2.17)

(t − u)δ−γ −β · (u − nr)β+γ −δ−1 du

nr

β −β−γ ′ , ≤ Cβ,γ ,δ,n B(1 − γ − β + δ, β + γ − δ )(t − s) (s − nr)

′ Cβ,γ ,δ,n > 0.

′′ For term J2 , there exists a number Cβ,γ ,n > 0 such that β −β−γ ′′ ∥J2 ∥ ≤ Cβ,γ ,n (t − s) (s − nr)

(s − nr)β+γ

t

∫

(t − s)β (t − u)γ

s

·

1 t − nr

du

′′ −1 β −β−γ ≤ Cβ,γ (t − s)1−β−γ · (t − nr)β+γ −1 ,n (1 − γ ) (t − s) (s − nr)

(2.18)

′′ −1 β −β−γ ≤ Cβ,γ . ,n (1 − γ ) (t − s) (s − nr)

On the other hand, by virtue of (2.16) we have for sufficiently small δ > 0 such that 1 − β − γ + δ > 0, β + γ − δ > 0 and

∥J3 ∥ ≤

′′′ Cβ,γ ,δ,n

s

∫

(t − s)β · (s − u)−β−γ (s − u)δ (u − nr)−δ−1 du

nr s

∫

′′′ β −β−γ ≤ Cβ,γ ,δ,n (t − s) (s − nr)

(s − u)−β−γ +δ · (u − nr)β+γ −δ−1 du

(2.19)

nr

′′′ β −β−γ ≤ Cβ,γ , ,δ,n B(1 − β − γ + δ, β + γ − δ )(t − s) · (s − nr)

′′′ Cβ,γ ,δ,n > 0.

In a similar way, for term J4 we have

∥J4 ∥ ≤

′′′′ Cβ,γ ,n (t

β

−β−γ

s

∫

− s) (s − nr)

(s − u)−β−γ (s − nr)β+γ ·

nr

′′′′ β −β−γ ≤ Cβ,γ · ,n (t − s) (s − nr)

(s − nr)1−β−γ 1−β −γ

·

1 s − nr

1

du (2.20)

(s − nr)1−β−γ

′′′′ −1 β −β−γ ≤ Cβ,γ , ,n (1 − β − γ ) (t − s) · (s − nr)

′′′′ Cβ,γ ,n > 0.

With the aid of (2.15)–(2.20), we thus obtain

∫ t ∫ s     e(t −u)A A1 G(u − r)du − (−A)γ e(s−u)A A1 G(u − r)du (−A)γ nr

nr

(2.21)

≤ Cn,β,γ · (t − s)β (s − nr)−β−γ

for some Cn,β,γ > 0. Now combining (2.11)–(2.14) and (2.21), we finally obtain (1.11) on the interval (nr , (n + 1)r ]. The proof is thus complete. 3. Fundamental solution In the remainder of this work, we assume that the operators A and A1 in (1.16) are diagonal with respect to the complete orthonormal basis {ek } given in (1.13) such that Aek = −µk ek ,

A 1 ek = µ ˜ k ek ,

(3.1)

for some numbers µk > 0, µ ˜ k ∈ R, k ∈ N = {1, 2, . . .}. For α, β ∈ R, let g(t), t ≥ 0, denote the unique solution of the following differential difference equation g ′ (t) + α g(t) = β g(t − r), t ≥ 0,

{

(3.2)

g(0) = 1, g(θ ) = 0, θ ∈ [−r , 0), and the fundamental solution G is determined by G(t)ek = g(µk , t)ek ,

k ∈ N,

(3.3)

where g(µk , ·) is the unique solution to (3.2) with α = µk and β = µ ˜ k , k ∈ N. It is easy to see that the stochastic convolution WG (·) is given at present by WG (t) =

∞ ∑ √ k=1

λk

t

∫

g(µk , t − s)ek dwk (s), 0

t ≥ 0.

(3.4)

K. Liu / Statistics and Probability Letters 156 (2020) 108624

7

Proposition 3.1. For any constant γ ∈ (0, 1), it holds true for each n ∈ {0, 1, . . .} that t

∫

−γ

g 2 (µk , u)du ≤ Cn,γ · µk (t − s)1−γ

for all

nr ≤ s < t ≤ (n + 1)r

(3.5)

s

where Cn,γ > 0 are some constants depending on n and γ . Proof. By virtue of (3.1) and (3.3), it is immediate that for each k ∈ N and t ≥ 0, γ /2

(−A)γ /2 G(t)ek = g(µk , t)(−A)γ /2 ek = µk g(µk , t)ek . This implies by virtue of (1.10) in Theorem 1.1(a) that for each k ∈ N, γ

γ /2

µk g 2 (µk , t) = ∥µk g(µk , t)ek ∥2H = ∥(−A)γ /2 G(t)ek ∥2H ≤ ∥(−A)γ /2 G(t)∥2 · ∥ek ∥2H ≤

Mn2,γ (t − nr)γ

,

∀ t ∈ (nr , (n + 1)r ],

which immediately implies for each k ∈ N that −γ

g 2 (µk , t) ≤ Mn2,γ µk (t − nr)−γ ,

∀ t ∈ (nr , (n + 1)r ].

(3.6)

Hence, we further have for any nr < s < t ≤ (n + 1)r, n ∈ {0, 1, . . .}, that t

∫

−γ

g 2 (µk , u)du ≤ Mn2,γ · µk

s

≤

t

∫

(u − nr)−γ du s

−γ Mn2,γ · µk [

1−γ

(3.7)

]

(t − nr)1−γ − (s − nr)1−γ .

Note that for any real numbers a ≥ b ≥ 0 and 0 < δ ≤ 1, we have aδ − bδ ≤ (a − b)δ .

(3.8)

It thus follows from (3.7), (3.8) that for any nr ≤ s < t ≤ (n + 1)r, n ∈ {0, 1, . . .}, t

∫

g 2 (µk , u)du ≤ s

−γ Mn2,γ · µk [

1−γ

(t − nr)1−γ − (s − nr)1−γ

] (3.9)

−γ

≤

Mn2,γ · µk 1−γ

(t − s)

1−γ

as desired. The proof is thus complete. □ Proposition 3.2. Let γ ∈ (0, 1) and 0 < β < 1 − γ . For any nr ≤ s < t ≤ (n + 1)r, n ∈ {0, 1, . . .}, there exists Cγ ,β,n > 0 such that s

∫

g(µk , t − u) − g(µk , s − u) du ≤ Cγ ,β,n · µk (t − s)β .

[

]2

−γ

0

Proof. First note that for any k ∈ N, nr < s < t < (n + 1)r and jr < v < (j + 1)r, j ∈ {0, 1, . . . , n − 1}, (−A)γ /2 (G(t − s + v ) − G(v ))ek = (g(µk , t − s + v ) − g(µk , v ))(−A)γ /2 ek γ /2

= µk (g(µk , t − s + v ) − g(µk , v ))ek , where {ek } is the orthonormal basis of H in (3.1). Therefore, by virtue of (1.11) in Theorem 1.1(b), we have for any nr < s < t < (n + 1)r and jr < v < (j + 1)r, j ∈ {0, 1, . . . , n − 1} that

]2 ] γ[ γ /2 [ µk g(µk , t − s + v ) − g(µk , v ) = ∥µk g(µk , t − s + v ) − g(µk , v ) ek ∥2H = ∥(−A)γ /2 (G(t − s + v ) − G(v ))ek ∥2H ≤ ∥(−A)γ /2 (G(t − s + v ) − G(v ))∥2 · ∥ek ∥2H (t − s)β ≤ Mj2,β,γ , Mj,β,γ > 0, (v − jr)β+γ

(3.10)

8

K. Liu / Statistics and Probability Letters 156 (2020) 108624

which immediately implies

∫

s

]2

g(µk , t − u) − g(µk , s − u) du

[

0 s

∫

]2

g(µk , t − s + v ) − g(µk , v ) dv

[

= 0

≤

n−1 ∫ ∑

(j+1)r

]2

g(µk , t − s + v ) − g(µk , v ) dv +

[

jr

j=0

∑

µk (t − s)β Mj2,β,γ −γ

n (∑

−γ

µk

1−β −γ

(j+1)r

∫

]2

g(µk , t − s + v ) − g(µk , v ) dv

Mj2,β,γ

(v − jr)−β−γ dv + µk (t − s)β Mn2,β,γ −γ

jr

j=0

=

[ nr

n−1

≤

s

∫

)

s

∫

(v − nr)−β−γ dv nr

· r 1−β−γ (t − s)β

j=0

=: Cγ ,β,n · µk (t − s)β , −γ

nr ≤ s < t ≤ (n + 1)r ,

where Cγ ,β,n =

n (∑

1 1−β −γ

Mj2,β,γ

)

· r 1−β−γ > 0.

j=0

The proof is thus complete.

□

4. Stochastic convolution Now we are in a position to establish our regularity properties for the stochastic convolution (1.18). Recall that Qek = λk ek , λk > 0, in (1.13) and Aek = −µk ek , µk > 0, in (3.1), k ≥ 1. Theorem 4.1. Suppose that γ ∈ (0, 1) and ∞ ∑ λk γ < ∞. µ k k=1

Then the trajectories of WG (t), t ≥ 0, in (1.18) are almost surely α -Hölder continuous with respect to t ∈ [0, ∞) for 0<α<

1−γ 2

.

Proof. It suffices to show the theorem for any 0 ≤ s < t < ∞ with 0 < t − s < r. For any nr ≤ s < t ≤ (n + 1)r, n ∈ N, we have t

∫ WG (t) − WG (s) =

s

∫

[G(t − u) − G(s − u)]dW (u).

G(t − u)dW (u) + s

0

Since the integrals are independent, it follows that 2 H)

E(∥WG (t) − WG (s)∥

≤

∞ ∑

λk

k=1

t

∫

g (µk , u)du + 2

s

∞ ∑ k=1

λk

∫

s

(g(µk , t − u) − g(µk , s − u))2 du

0

=: J1 (t , s) + J2 (t , s). By virtue of Propositions 3.1 and 3.2, we have for nr ≤ s < t ≤ (n + 1)r that J1 (t , s) ≤

∞ ∑

∞ ( ∑ λk ) −γ β λk · Cn,γ · µk (t − s)1−γ ≤ Cn,γ r 1−γ −β γ (t − s) , µ k k=1 k=1

Cn,γ > 0,

and

(

J2 (t , s) ≤ Cγ ,β,n

∞ ∑ λk ) β γ (t − s) , µ k k=1

Cγ ,β,n > 0,

where 0 < β < 1 − γ . Since WG (t) − WG (s) is Gaussian, then for any m ∈ N, there exists a constant Cm > 0 such that

[

1−γ −β E(∥WG (t) − WG (s)∥2m + Cγ ,β,n ) H ) ≤ Cm (Cn,γ r

∞ ∑ λk ]m (t − s)mβ . γ µ k k=1

K. Liu / Statistics and Probability Letters 156 (2020) 108624

9

Choosing m such that mβ > 1 and applying the well-known Kolmogorov test, one can find that WG (t) is α -Hölder continuous for

α=

β 2

1

−

2m

.

Since m is arbitrary, one can obtain the desired result by letting β ↑ 1 − γ .

□

5. Regularity in space of continuous functions In this section, we consider the continuity of the stochastic convolution WG (t), t ≥ 0, in a smaller Banach space of H. To this end, we assume that H = L2 (O) where O is a bounded open subset of Rn with smooth boundary ∂ O. Suppose that Aek = −µk ek , k ∈ N, where {ek }∞ k=1 ⊂ D (A) is the complete orthonormal basis of H in (1.13) and the sequence of positive numbers {µk }∞ k=1 satisfies

µk ≥ µ0 > 0

µ0 > 0,

for some

and ∞ ∑ λk < ∞. µk k=1

We set WG (t)(ξ ) = WG (t , ξ ) and write the stochastic convolution as WG (t , ξ ) =

∞ ∑ √

λk

∫

t

g(µk , t − s)ek (ξ )dwk (s),

t ≥ 0,

ξ ∈ O.

0

k=1

For the basis {ek }∞ k=1 , we suppose that there exists a constant M > 0 such that

{ek } ⊂ C (O),

|ek (ξ )| ≤ M , 1/2 k

∥∇ ek (ξ )∥Rn ≤ M µ

,

k ∈ N,

k ∈ N,

ξ ∈ O,

(5.1)

ξ ∈ O.

Further, it is known (see (5.46) in Da Prato and Zabczyk, 2014) that for each κ ∈ [0, 1], there exists a constant Mκ > 0 such that

|ek (ξ ) − ek (η)|2 ≤ Mκ µκk ∥ξ − η∥2Rκn ,

ξ , η ∈ O,

k ∈ N.

(5.2)

Theorem 5.1. If there exists a number δ ∈ (0, 1) such that ∞ ∑ λk < ∞, µk1−δ k=1

(5.3)

then the process WG (t) has a version WG (t , ξ ), t ≥ 0, ξ ∈ O, whose trajectories are almost surely α -Hölder continuous with respect to (t , ξ ) for 0<α<

δ 2

.

In particular, the process WG (t) has a C (O)-valued version with α -Hölder continuous paths for 0 < α < δ/2. Proof. Once again, it suffices to show the theorem for any 0 ≤ s < t < ∞ with 0 < t − s < r. First, for nr < t ≤ (n + 1)r, we have by virtue of (3.5) and (5.3) that for ε > 0 small enough with δ − ε > 0, ∞ ∑ k=1

λk

∫

t

µδ−ε g 2 (µk , s)ds ≤ k

0

∞ ∑

λk

n ∫ ∑

k=1

≤

j=0

∞ ∑ n−1 ∑

(j+1)r

µkδ−ε g 2 (µk , s)ds +

jr

k=1

δ−ε−(1− 2ε )

Cj,δ,ε λk µk

· r ε/2 +

k=1 j=0

≤

∞ ∑

n ∞ ∑ Cj,δ,ε r ε/2 ∑ λk < ∞. ε/2 µ1k −δ µ0 j=0 k=1

∞ ∑ k=1

λk

∫

t

µδ−ε g 2 (µk , s)ds k

nr

δ−ε−(1− 2ε )

Cn,δ,ε λk µk

· r ε/2

(5.4)

10

K. Liu / Statistics and Probability Letters 156 (2020) 108624

Since 0 < δ < 1, it thus follows from (5.2)–(5.4) that for any nr < t ≤ (n + 1)r,

E|WG (t , ξ ) − WG (t , η)|2 =

∞ ∑

λk

≤

g 2 (µk , t − s)|ek (ξ ) − ek (η)|2 ds 0

k=1

∞ ∑

t

∫

λk

0

k=1

≤ Mδ,ε

t

∫

2(δ−ε )

g 2 (µk , t − s)Mδ,ε µδ−ε · ∥ξ − η∥Rn k

ds

n ∞ {( r )ε/2 ∑ ∑ λk } δ−ε ) ∥ξ − η∥2( , Cj,δ,ε Rn µ0 µ1k −δ j=0

(5.5) Mδ,ε > 0.

k=1

On the other hand, for any nr ≤ s < t ≤ (n + 1)r, we have

E|WG (t , ξ ) − WG (s, ξ )|2

=

∞ ∑

λk

t

∫

g 2 (µk , t − u)|ek (ξ )|2 du +

s

k=1

∞ ∑

λk

s

∫

|[g(µk , t − u) − g(µk , s − u)]ek (ξ )|2 du

(5.6)

0

k=1

=: I1 (t , s, ξ ) + I2 (t , s, ξ ). To estimate I1 (t , s, ξ ), we have, by analogy with (3.6), that for each k ∈ N and the above ε > 0 small enough, 1 g 2 (µk , t) ≤ Mn2,δ,ε µδ−ε− (t − nr)δ−ε−1 , k

∀ t ∈ (nr , (n + 1)r ].

Hence, it follows that for nr ≤ s < t ≤ (n + 1)r that t

∫ s

g 2 (µk , u)du ≤ Mn2,δ,ε · µkδ−ε−1 Mn2,δ,ε 1

≤

δ − ε µε0

·

t

∫

(u − nr)δ−ε−1 du

s

(5.7)

1

(t − s)δ−ε 1−δ

µk

Finally, we have for nr ≤ s < t ≤ (n + 1)r, that I1 (t , s, ξ ) =

∞ ∑

λk

t −s

∫

g 2 (µk , v )dv 0

k=1

(5.8)

∞ ∑ λk ≤ · (t − s)δ−ε . 1−δ (δ − ε )µε0 µ k k=1

M0,δ,ε

To estimate I2 (t , s, ξ ), since 0 < δ < 1, by analogy with (3.10), one can get that for each k ∈ N, nr < s < t < (n + 1)r and jr < u < (j + 1)r, j ∈ {1, . . . , n} with ε > 0 small enough, 1−δ+ 2ε

µk

g(µk , t − s + u) − g(µk , u)

[

]2

≤ Mj2,δ,ε

(t − s)δ−ε ε

(u − jr)1−δ+ 2 +δ−ε

,

(5.9)

which immediately implies I2 (t , s, ξ ) ≤

∞ ∑

λk

≤

λk

k=1

+

≤

[

]2

g(µk , t − s + v ) − g(µk , v ) dv

0

k=1

∞ ∑

s

∫

∞ ∑

n ∫ ∑

λk

[

(5.10)

s

[

∑

ε/2

εµ0

]2

g(µk , t − s + v ) − g(µk , v ) dv

nr k=1 n 2 Mj,δ,ε nr ε/2

j=0

]2

g(µk , t − s + v ) − g(µk , v ) dv

(j−1)r

j=1

∫

jr

∞ ∑ λk · (t − s)δ−ε , 1−δ µ k k=1

nr ≤ s < t ≤ (n + 1)r .

Combining (5.8) and (5.10), we have the conclusion that there exists Cn′ ,δ,ε > 0 such that

E|WG (t , ξ ) − WG (s, ξ )|2 ≤ Cn′ ,δ,ε (t − s)δ−ε ,

for any

Further, for any nr ≤ s < t ≤ (n + 1)r, there exists some

Cn′′,δ,ε

[ ]δ−ε E|WG (t , ξ ) − WG (s, η)|2 ≤ Cn′′,δ,ε ∥ξ − η∥2Rn + (t − s) .

nr ≤ s < t ≤ (n + 1)r .

> 0 such that

K. Liu / Statistics and Probability Letters 156 (2020) 108624

11

Since WG (t , ξ ) − WG (s, η) is a Gaussian random variable, we thus have for any m ∈ N and some associated Cn,m,δ,ε > 0 that

E|WG (t , ξ ) − WG (s, η)|2m ≤ Cn,m,δ,ε ∥ξ − η∥2Rn + (t − s)

[

]m(δ−ε)

.

Choosing m such that m(δ − ε ) > 1 and applying the Kolmogorov test for random field (see Theorem 3.5 in Da Prato and Zabczyk (2014)), we can find that WG (·, ·) is α -Hölder continuous for

α=

δ−ε

−

1

.

2 2m Let m → ∞ and ε → 0, then we have the desired result. □ Example 5.1. Consider the following stochastic delay partial differential equation

⎧ ∂ 2 y(t − r , ξ ) ∂ 2 y(t , ξ ) ⎪ ⎪ dt + a · dt + dw (t , ξ ), ⎨dy(t , ξ ) = 2 ∂ξ ∂ξ 2 ⎪y(0, ·) = 0, y(θ, ·) = 0, θ ∈ [−r , 0], ⎪ ⎩ y(t , ξ ) = 0, ξ ∈ ∂ O, t ∈ (0, ∞),

t ≥ 0,

ξ ∈ O = [0, π]d , (5.11)

where r > 0, a ∈ R and w (·, ·) is a standard Wiener random field on R+ × O. To apply our theory in Section 4 to (5.11), we assume at present that H = L2 (O) in (1.16) and W = w . Let A be the linear operator

{

D (A) = H 2 (O) ∩ H01 (O),

(5.12)

Au = ∆u, ∀ u ∈ D (A),

where ∆ represents the Laplace operator and A1 = aA. Then it is easy to see that en1 ,...,nd (ξ ) =

∥∇ en1 ,...,nd (ξ )∥Rd ≤ ≤

µn1 ,..., nd

( 2 )d/2 π ( 2 )d/2 π

( 2 )d/2

sin(n1 ξ1 ) · · · sin(nd ξd ), (|n1 | + · · · + |nd |) d1/2

√

n21 + · · · + n2d

π = n21 + · · · + n2d ,

and the conditions (5.1) clearly hold. Note that if Q = I, the identity operator, in L2 (O), then (5.3) holds if and only if d

δ <1− .

(5.13) 2 Clearly, if d = 1, (5.13) becomes δ < 1/2. If d > 1, there is not such a value δ > 0 satisfying (5.13). In particular, for (5.11) with d = 1, we have λk = 1, µk = k2 , k ∈ N so as that (5.3) holds for δ ∈ (0, 1/2). The mild solution y(t , ξ ), t ≥ 0, ξ ∈ [0, π], is thus α -Hölder continuous with α ∈ (0, 1/4). References Da Prato, G., Zabczyk, J., 2014. Stochastic Equations in Infinite Dimensions, second ed. In: Encyclopedia of Mathematics and its Applications, Cambridge University Press. Di Blasio, G., Kunisch, K., Sinestrari, E., 1985. Stability for abstract linear functional differential equations. Israel J. Math. 50, 231–263. Jeong, J., Nakagiri, S.I., Tanabe, H., 1993. Structural operators and semigroups associated with functional differential equations in Hilbert spaces. Osaka J. Math. 30, 365–395. Liu, K., 2019. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete Contin. Dyn. Syst. B arXiv:1906.00605, in press. Sinestrari, E., 1984. A noncompact differentiable semigroup arising from an abstract delay equation. C. R. Math. Rep. Acad. Sci. Can. 6, 43–48. Tanabe, H., 1988. On fundamental solution of differential equation with time delay in Banach space. Proc. Japan Acad. 64, 131–180. Tanabe, H., 1992. Fundamental solutions for linear retarded functional differential equations in Banach spaces. Funkcial. Ekvac. 35, 149–177.

Further reading Jeong, J., 1991. Stabilizability of retarded functional differential equation in Hilbert space. Osaka J. Math. 28, 347–365. Pazy, A., 1983. Semigroups of Linear Operators and Applications to Partial Differential Equations. In: Applied Mathematical Sciences, vol. 44, Springer Verlag, New York.