Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients
Below we demonstrate how the C^∞-regular properties of heat dynamics with non-unit nonlinear diffusion coefficient can be studied. We consider an infinite dimensional model, describing evolution of unbounded lattice spins R^Z^d. As a main step we provide a construction of corresponding variational p...
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| Цитувати: | Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients / A.Val. Antoniouk, A.Vict. Antoniouk // Нелинейные граничные задачи. — 2007. — Т. 17. — С. 101-129. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-101162010-07-26T12:02:19Z Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients Antoniouk, A.Val. Antoniouk, A.Vict. Below we demonstrate how the C^∞-regular properties of heat dynamics with non-unit nonlinear diffusion coefficient can be studied. We consider an infinite dimensional model, describing evolution of unbounded lattice spins R^Z^d. As a main step we provide a construction of corresponding variational processes in ℓp(c) spaces with growing weights ck ~ e^a|k|, k belongs Z^d. Developing the approach of nonlinear estimates on variations, we find sufficient conditions on the nonlinear coefficients of differential equation that lead to C^∞-regularity of solutions with respect to the initial data and C^∞-regularity of corresponding heat semigroup. 2007 Article Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients / A.Val. Antoniouk, A.Vict. Antoniouk // Нелинейные граничные задачи. — 2007. — Т. 17. — С. 101-129. — Бібліогр.: 11 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/10116 en Інститут прикладної математики і механіки НАН України |
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Below we demonstrate how the C^∞-regular properties of heat dynamics with non-unit nonlinear diffusion coefficient can be studied. We consider an infinite dimensional model, describing evolution of unbounded lattice spins R^Z^d. As a main step we provide a construction of corresponding variational processes in ℓp(c) spaces with growing weights ck ~ e^a|k|, k belongs Z^d.
Developing the approach of nonlinear estimates on variations, we find sufficient conditions on the nonlinear coefficients of differential equation that lead to C^∞-regularity of solutions with respect to the initial data and C^∞-regularity of corresponding heat semigroup. |
| format |
Article |
| author |
Antoniouk, A.Val. Antoniouk, A.Vict. |
| spellingShingle |
Antoniouk, A.Val. Antoniouk, A.Vict. Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients |
| author_facet |
Antoniouk, A.Val. Antoniouk, A.Vict. |
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Antoniouk, A.Val. |
| title |
Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients |
| title_short |
Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients |
| title_full |
Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients |
| title_fullStr |
Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients |
| title_full_unstemmed |
Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients |
| title_sort |
regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients |
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Інститут прикладної математики і механіки НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/10116 |
| citation_txt |
Regularity of infinite dimensional heat dynamics of unbounded lattice spins with non-constant diffusion coefficients / A.Val. Antoniouk, A.Vict. Antoniouk // Нелинейные граничные задачи. — 2007. — Т. 17. — С. 101-129. — Бібліогр.: 11 назв. — англ. |
| work_keys_str_mv |
AT antonioukaval regularityofinfinitedimensionalheatdynamicsofunboundedlatticespinswithnonconstantdiffusioncoefficients AT antonioukavict regularityofinfinitedimensionalheatdynamicsofunboundedlatticespinswithnonconstantdiffusioncoefficients |
| first_indexed |
2025-07-02T12:00:15Z |
| last_indexed |
2025-07-02T12:00:15Z |
| _version_ |
1836536421528109056 |
| fulltext |
Нелинейные граничные задачи 17, 88-116 (2007) 101
c©2007. A.Val. Antoniouk, A.Vict. Antoniouk
REGULAITY OF INFINITE DIMENSIONAL
HEAT DYNAMICS OF UNBOUNDED LATTICE SPINS
WITH NON-CONSTANT DIFFUSION COEFFICIENTS
Below we demonstrate how the C∞-regular properties of heat dynamics with
non-unit nonlinear diffusion coefficient can be studied. We consider an infinite
dimensional model, describing evolution of unbounded lattice spins IRZZd
. As a
main step we provide a construction of corresponding variational processes in
ℓp(c) spaces with growing weights ck ∼ ea|k|, k ∈ ZZd.
Developing the approach of nonlinear estimates on variations, we find suffi-
cient conditions on the nonlinear coefficients of differential equation that lead to
C∞-regularity of solutions with respect to the initial data and C∞-regularity of
corresponding heat semigroup.
Keywords and phrases: heat dynamics, nonlinear diffusion, variations, regu-
larity
MSC (2000): 60H15
1. Introduction.
It is already known, e.g. [7, 8], that for the stochastic differential
equations
dy0 = B(y0)dWt − F (y0)dt, y0(0) = x0 (1)
with coefficients, that are globally Lipschitz and have all bounded
derivatives, there is C∞-regularity of solutions y0
t (x
0) with respect to
the initial data x0. Moreover, corresponding heat semigroup, defined
as a mean Ptf(x0) = E f(y0
t (x
0)) with respect to the Wiener measure,
preserves spaces of continuously differentiable functions with bounded
derivatives. These results follow from application of fixed point and
implicit function theorems to variations yj
t (x) =
∂jy0
t (x
0)
∂(x0)j
of process
y0
t (x
0) with respect to the initial data x0.
The consideration of more wide class of stochastic differential
equations with essentially nonlinear non-Lipschitz coefficients leads
Research is partially supported by grants of the National Committee on
Science and Technology
102 A.Val. Antoniouk, A.Vict. Antoniouk
to a monotone conditions of coercitivity and dissipativity: ∀C >
0 ∃M such that
coercitivity < F (x) − F (y), x− y > −
−C‖B(x) − B(y)‖2 ≥ −M‖x− y‖2
dissipativity < F (x), x > −C‖B(x)‖2 ≥ −M(1 + ‖x‖2),
that are sufficient for the existence, uniqueness and continuous de-
pendence of solutions with respect to the initial data [10, 11].
In [2, 4, 5] it was shown that the application of Cauchy-Liouville-
Picard scheme to the problem of C∞-regularity for non-Lipschitz
differential equations meets difficulties. Here we discussed a particular
case of system (1) with constant diffusion coefficient B = 1, that has
important applications to the classical Gibbs lattice systems with
unbounded spins. To be able to work with such nonlinear differential
equations we followed [8, 9], where, after the shift ηt = yt − Wt,
equation (1) becomes ordinary differential equation on variable ηt:
dηt = −F (ηt +Wt)dt
with random control Wt.
In [2, 4, 5] we found that due to the structure of the associated
with (1) variational system
dyi =
∑
j1+...+js=i,
s ≥ 1
B(s)(y0)yj1..yjsdW −
∑
j1+...+js=i,
s ≥ 1
F (s)(y0)yj1..yjsdt
y1(0) = Id, yi(0) = 0, i ≥ 2
(2)
the variation of Nth order is proportional to the Nth power of the
variation of 1st order.
Such proportionality led to nonlinear estimates on variations
ρn(t) =
n∑
j=1
E pj(‖y(t)‖) ‖y
j(t)‖
m/j
Xj
≤ eMtρn(0), (3)
permitting to apply monotone methods to the problem of C∞-regula-
rity. The weights pj and topologies Xj on variations were found to be
related with the order of nonlinearity of coefficients of initial equation
Regularity of infinite dimensional heat dynamics 103
(1). Moreover, the order of nonlinearity also influenced the structure
of topologies in the spaces of differentiable functions, preserved by
heat semigroup Pt.
In [2] it was observed that the variations should be constructed
in spaces ℓp(c) with exponentially growing on lattice ZZd weights, i.e.
ck ∼ ea|k|, k ∈ ZZd. For diffusion coefficient B = I this property
follows from Kato results about the construction of solutions to the
linear ordinary differential equations. For B = I terms with B(s) =
0 for s ≥ 1 in (2) are absent and (2) becomes non-autonomous
inhomogeneous linear equation on variable yi with control y0
t .
The use of process η and application of Kato results becomes
impossible for non-constant diffusion coefficient B 6= I. The solution
of this problem is a main topic of this article.
In Section 2 we describe a model with non-constant nonlinear
diffusion coefficient and state main results about the properties of
variations of diffusion process and regularity of its semigroup. In
Section 3 we define the stochastic integrals
∫ t
0
BsdWs with B ∈ ℓp(c)
and construct the nonlinear diffusion and its variations with respect
to the initial data. In Section 4 we prove nonlinear estimate (3).
Section 5 is devoted to the study of continuity and C∞ regularity of
variations with respect to the initial data. Here we also demonstrate
the regularity of heat semigroup Pt (proof of Theorem 1).
Finally remark, that even the problem of the first order re-
gularity with respect to the initial data is still under question for
more general classes of stochastic differential equations, e.g. [6] and
references therein.
2. Basic model and statement of main results.
We consider the stochastic process on the lattice product of
spin spaces IRZZd
=
∏
k=(k1,...,kd)∈ZZd IR1, described by the following
nonlinear equation
y0(t) = x0 +
∫ t
0
B(y0(s))dW (s) −
∫ t
0
[F (y0(s)) + Ay0(s)]ds (4)
Nonlinear diagonal maps
IRZZd
∋ x = {xk}k∈ZZd −→ B(x) = {B(xk)}k∈ZZd ∈ IRZZd
104 A.Val. Antoniouk, A.Vict. Antoniouk
IRZZd
∋ x = {xk}k∈ZZd −→ F (x) = {F (xk)}k∈ZZd ∈ IRZZd
are generated by smooth functions B,F ∈ C∞(IR1) of polynomial
with derivatives behaviour and the linear finite diagonal map A :
IRZZd
→ IRZZd
is defined by
∃r0 (Ax)k =
∑
j: |j−k|≤r0
A(k − j)xj , k ∈ ZZd
and is bounded in any space ℓp(c), sup|k−j|=1 |ck/cj| <∞.
The cylinder Wiener process W = {Wk(t)}k∈ZZd with values in
ℓ2(a),
∑
k∈ZZd ak = 1, a ∈ IP is canonically realized on measurable
space (Ω = C0([0, T ], ℓ2(a)),F ,Ft,P) with canonical filtration Ft =
σ{W (s)|0 ≤ s ≤ t} and cylinder Wiener measure P. Processes
Wk, k ∈ ZZd are independent IR1-valued Wiener processes. Henceforth
we denote by E the expectation with respect to measure P and by IP
the set of all vectors a = {ak}k∈ZZd such that δa = sup
|k−j|=1
|ak/aj| <∞.
Let us impose the following conditions on the coefficients {F,B}.
1. Coercitivity and dissipativity: ∀M ∃KM , K1, K2 such that
(x− y)(F (x) − F (y))−M(B(x) − B(y))2 ≥ KM(x− y)2 (5)
xF (x) −M B2(x) ≥ −K1x
2 −K2 (6)
Inequality (5) implies in particular that ∀M ∃KM
−F ′(x) +M [B′(x)]2 ≤ KM (7)
2. Nonlinear parameters: Function F : IR1 → IR1 is monotone
and ∃kF ,kB ≥ −1 with 2kB ≤ kF such that ∀n ∈ IN ∃Cn
∀i = 0, ..., n ∀x, y ∈ IR1
|F (i)(x) − F (i)(y)| ≤ Cn|x− y|(1 + |x| + |y|)kF (8)
|B(i)(x) −B(i)(y)| ≤ Cn|x− y|(1 + |x| + |y|)kB (9)
Main result is that under the above conditions the heat diffusion
semigroup
(Ptf)(x) = E f(y0(t, x0)) (10)
Regularity of infinite dimensional heat dynamics 105
preserves spaces of continuously differentiable functions, which topo-
logies depend on the order of nonlinearity kF . This result generalizes
[1, 3, 2], where the unit diffusion case B(x) = 1 was considered.
Let us say that array Θ = Θ1∪ ...∪Θn, n ∈ IN with Θm be a set
of pairs of mth-order (p,G = G1 ⊗ ...⊗Gm), Gi ∈ IP , i = 1, ..., m, is
quasi-contractive with parameter kF if ∀m = 2, ..., n ∀(p,G) ∈ Θm and
∀i, j ∈ {2, ..., m}, i < j there is a pair (p̃, G̃ = G̃1⊗...⊗G̃m−1) ∈ Θm−1
such that ∃K ∈ IR+
∀z ∈ IR+ (1 + z)
kF +1
2 p̃(z) ≤ K p(z) (11)
(Ĝ{i,j})ℓ ≤ K G̃ℓ, ℓ = 1, ..., m− 1 (12)
Above p, p̃ are smooth functions of polynomial behaviour (27) and
inequality (12) is understood as a coordinate inequality between (m−
1)th order tensors for (m− 1)-tensor
Ĝ{i,j} = G1⊗...⊗Gi−1⊗Gi+1⊗...⊗Gj−1⊗a−(kF +1)GiGj⊗Gj+1⊗...⊗Gm
constructed by m-tensor G = G1 ⊗ ...⊗Gm.
Definition 1. Function f ∈ DΘ,r(ℓ2(a)), r ≥ 0, iff
1. There is a set of Borel measurable partial derivatives
ℓ2(a) ∋ x→ ∂τf(x) ∈ IR1 ∀τ = {j1, ..., js}, |τ | ≤ n (13)
such that ∀x0 ∈ ℓ2(a), ∀h ∈ AC([a, b])
f(x0 + h(·))
b
a
=
∫ b
a
ds
∑
k∈ZZd
∂kf(x0 + h(s))h′k(s) (14)
and ∀τ |τ | ≤ n− 1
∂τf(x0 + h(·))
b
a
=
∫ b
a
ds
∑
k∈ZZd
∂τ∪{k}f(x0 + h(s))h′k(s) (15)
Here we used notation
AC([a, b]) = ∩
p≥1, c∈IP
AC([a, b], ℓp(c)) (16)
106 A.Val. Antoniouk, A.Vict. Antoniouk
for
AC([a, b], X) = {h ∈ C([a, b], X) : ∃h′ ∈ L1([a, b], X)}
2. The norm is finite
‖f‖DΘ,r
= ‖f‖Lipr + max
m=1,...,n
‖∂(m)f‖Θm <∞ (17)
where
‖f‖Lipr = sup
x∈ℓ2(a)
|f(x)|
(1 + ‖x‖ℓ2(a))r+1
+ (18)
+ sup
x,y∈ℓ2(a)
|f(x) − f(y)|
‖x− y‖ℓ2(a)(1 + ‖x‖ℓ2(a) + ‖y‖ℓ2(a))r
and for multifunction of mth order ∂(m)f(x) = {∂τf(x), |τ | = m}
‖∂(m)f‖Θm = sup
x∈ℓ2(a)
max
(p,G)∈Θm
|||∂(m)f(x)|||G
p(1 + ‖x‖2
ℓ2(a))
(19)
with |||∂(m)f(x)|||2G =
∑
τ={j1,...,jm}⊂ZZd
G1
j1...G
m
jm
|∂τf(x)|2 for G = G1 ⊗
...⊗Gm.
Theorem 1. Let F,B satisfy conditions (5)-(9) and Θ = Θ1∪...∪Θn,
n ∈ IN be quasi-contractive array with parameter kF . Suppose that
function f ∈ DΘ,r(ℓ2(a)), r ≥ 0, i.e.
Then ∀ ≥> 0 semigroup Pt preserves scale of spaces DΘ,r(ℓ2(a)), r >
0 and there are KΘ,r, MΘ,r such that
∀f ∈ DΘ,r(ℓ2(a)) ‖Ptf‖DΘ,r
≤ KΘ,re
MΘ,rt‖f‖DΘ,r
(20)
The formal differentiation of (10) with respect to x0 shows that
the derivatives of semigroup is related with the variations of process
y0
t with respect to the initial data x0. Let τ = {j1, ..., jn}, js ∈ ZZd
be any ordered array of points from ZZd. To the set τ we associate
vector yτ = {yk,τ}k∈ZZd, which satisfies equation
yk,τ = x̃k,τ +
∫ t
0
(B′(y0
k)yk,τ + ϕB
k,τ)dWk−
∫ t
0
(F ′(y0
k)yk,τ + (Ayτ)k + ϕF
k,τ)ds, k ∈ ZZd,
(21)
Regularity of infinite dimensional heat dynamics 107
derived by differentiation of (4) with respect to variables {x0
jn
, ..., x0
j1
}.
Above the inhomogeneous parts ϕB
τ and ϕF
τ are constructed from
functions B and F by the following rule
ϕD
k,τ =
∑
γ1∪...∪γs=τ, s≥2
D(s)(y0
k)yk,γ1
...yk,γs, (22)
where yγ1
, ..., yγs are the solutions of lower rank variational equations.
Summation in (22) runs on all possible subdivisions of set τ={j1,..,jn}
on the nonintersecting subsets γ1, ..., γs ⊂ τ, |γ1|+...+|γs| = |τ |, s ≥
2, |γi| ≥ 1.
To prove Theorem 1 it is necessary to find the joint topologies
for solvability of system in variations (21), and to check that at the
special choice of initial data in (21)
x̃k,τ = δkj for τ = {j}, |τ | = 1 and x̃k,τ = 0 for |τ | ≥ 2 (23)
the variation yτ is interpreted as a derivative of y0 with respect to x0
∂|τ |y0
k(t, x
0)
∂x0
jn
...∂x0
j1
= yk,τ (24)
Equation (21) possesses a certain nonlinear symmetry with res-
pect to the lower rank variations, where the ith order variation and
the ith degree of the first order variation appear simultaneously. Like
in [2] introduce the following nonlinear object
ρτ (y; t) = E
n∑
i=1
pi(zt)
∑
γ⊂τ, |γ|=i
‖yγ‖
mγ
ℓmγ (cγ) (25)
where the set τ = {j1, ..., jn}, ji ∈ ZZd, zt = 1 + ‖y0(t, x0)‖2
ℓ2(a) and
mγ = m1/|γ|.
Impose the following hierarchy of weights pi, cτ . It is dictated by
the unbounded operator coefficients with control y0 in (21), (22) and
depends on the order of nonlinearity kF ≥ 2kB:
1. The vectors cγ = {ck,γ}k∈ZZd ⊂ IP fulfill
∀α ⊂ τ ∀γ1∪ ...∪γs = α ∀s ≥ 2 ∃Kγ1,...,γs;α such that ∀k ∈ ZZd
[ck,α]|α|a
−
kF +1
2
m1
k ≤ Kγ1,...,γs;α[ck,γ1
]|γ1|...[ck,γs]
|γs| (26)
108 A.Val. Antoniouk, A.Vict. Antoniouk
2. Positive monotone functions pi ∈ C∞(IR+) of polynomial be-
haviour
∃ε > 0 ∀z ∈ IR+ pi(z) ≥ ε p′i(z) ≥ ε
∃C (1 + z)|p′′i (z)| ≤ Cp′i(z) (1 + z)p′i(z) ≤ Cpi(z) (27)
satisfy condition
∃Kp ∀j ∈ {2, ..., n} ∀i1, ..., is, s ≥ 2 i1 + ...+ is = j
[pj(z)]
jz
kF +1
2
m1 ≤ Kp[pi1(z)]
i1 ...[pis(z)]
is , z ∈ IR+ (28)
Theorem 2. Let F,B satisfy conditions (5)-(9) and y0, yτ be solutions
to (4) and (21) for x0 ∈ ℓ2(a) and zero-one initial data x̃γ (23).
Suppose that hierarchies (26) and (28) are valid.
Then the nonlinear quasi-contractive estimate holds
∃M = Mτ ∀t ≥ 0 ρτ (y; t) ≤ eMtρτ (y; 0) (29)
3. ℓp(c)-valued stochastic integrals and construc-
tion of diffusion process and its variations.
In the following Lemma we construct ℓp(c)-valued stochastic
integral, appearing in (21), and prove Ito formula for the norm of
ℓp(c)-valued continuous processes. This result will permit to work
correctly with variations yτ in ℓmτ (cτ ) scales, arising in nonlinear
expression (25).
Lemma 1. Let Φ(t), Ψ(t) be Ft-adapted processes with values in
ℓp(c), c ∈ IP , p ≥ 1 such that
∀q ≥ 1, sup
t∈[0,T ]
E (‖Φ(t)‖q
ℓp(c) + ‖Ψ(t)‖q
ℓp(c)) <∞
Then the process, defined by coordinates
ηk(t) = ηk(0) +
∫ t
0
Φk(s)dWk(s) +
∫ t
0
Ψk(s)ds
Regularity of infinite dimensional heat dynamics 109
for η(0) ∈ Lq(Ω,P, ℓp(c)), belongs to the space of continuous Ft-
adapted processes, equipped with the norm (E sup
t∈[0,T ]
‖ · ‖q
ℓp(c))
1/q and
Ito formula is fulfilled
‖η(t)‖q
ℓp(c) = ‖η(0)‖q
ℓp(c)+
+q
∫ t
0
‖η(s)‖q−p
ℓp(c) < η⋆(s), η(s)Φ(s)dW (s) >ℓp(c) +
+q
∫ t
0
‖η(s)‖q−p
ℓp(c) < η⋆(s), η(s)Ψ(s) + p−1
2
Φ2(s) >ℓp(c) ds+
+ q(q−p)
2
∫ t
0
‖η(s)‖q−2p
ℓp(c)
∑
k∈ZZd
c2k|ηk(s)|
2p−2Φ2
k(s)ds
(30)
where we used notation
< η⋆, y >ℓp(c)=
∑
k∈ZZd
ck|ηk|
p−2yk (31)
Moreover ∀q ≥ p ≥ 2 ∀T > 0 ∃Kq,T such that
E sup
t∈[0,T ]
‖
∫ t
0
Φ(s)dW (s)‖q
ℓp(c) ≤ Kq,T
∫ T
0
E ‖Φ(t)‖q
ℓp(c)dt (32)
Remark 1. First note that the coefficients of diffusion process B(xk)
and F (xk) are transition invariant. Therefore the required by Lemma
1 inclusions {B(xk)}k∈ZZd, {F (xk)}k∈ZZd ∈ ℓp(a) lead to the require-
ment
∑
k∈ZZd ak <∞ on topologies of spaces ℓp(a), where the initial
diffusion process (4) can be constructed.
On the contrary, we do not have restrictions on the weights in
spaces ℓp(c) for variational processes yτ . Indeed, the principal part of
variational equations has form {B′(xk)yk,τ}k∈ZZd, {F ′(xk)yk,τ}k∈ZZd ,
i.e. has additional factor yτ . Due to the zero-one initial data for
variational equations (23), there is an inclusion yτ (0) ∈ ℓp(c) for
any c ∈ IP . Therefore, it becomes possible to construct variations in
any space ℓp(c).
This is also important for the study of regularity properties of
semigroup, because in Lemma 2 we need the estimates on variations,
which grow exponentially fast ck ∼ ea|k|, k ∈ ZZd.
110 A.Val. Antoniouk, A.Vict. Antoniouk
Proof. First of all note that for any vector h ∈ ℓp(c), c ∈ IP
the process {hkWk(t, ω)}k∈ZZd = hW (t, ω) has P a.e. ω ∈ Ω ℓp(c)-
valued continuous on t ∈ [0,∞) paths. This fact follows from the
Kolmogorov theorem and estimates
E ‖hW (t)‖q
ℓp(c) ≤ (
∑
k∈ZZd
ckh
p
k)
(q−p)/p
E (
∑
k∈ZZd
ckh
p
k|Wk(t)|
q) =
= ‖h‖q
ℓp(c)t
q/2
E |W0(1)|q <∞
E ‖h(W (t) −W (s))‖q
ℓp(c) = E (
∑
k∈ZZd
ckh
p
k|W (t) −W (s)|p)q/p ≤
≤ (
∑
k∈ZZd
ckh
p
k)
(q−p)/p
E (
∑
k∈ZZd
ckh
p
k|Wk(t) −Wk(s)|
q) =
= ‖h‖q
ℓp(c)(t− s)q/2
E |W0(1)|q <∞
where we used Hölder inequality and the properties of cylinder Wiener
process, W0 is a Wiener process at point 0 ∈ ZZd of lattice.
Now consider the Ft-adapted process
H̃(t) = H i, for t ∈ (ti, ti+1], i ≥ 0, and H̃(t0) = H0 for t0 = 0,
where all H i are Fti-measurable and H i ∈ L∞(Ω,P; ℓp(c)). Then due
to the continuity of terms H i(ω)(W (t, ω) −W (ti, ω)) the stochastic
integral, defined by
Z̃k(t) = {
∫ t
0
H̃(s)dW (s)}k =
=
i−1∑
j=0
Hj
k(Wk(tj+1) −Wk(tj)) +H i
k(Wk(t) −Wk(ti)), t ∈ (ti, ti+1]
and Z̃k(0) = 0 has ℓp(c) pathwise continuous version and is a martin-
gale.
Therefore for ℓp(c)-valued continuous martingale Z̃(t) due to [8,
Th.3.8] we have inequality
E sup
t∈[0,T ]
‖Z̃(t)‖q ≤ (
q
q − 1
)q sup
t∈[0,T ]
E ‖Z̃(t)‖q (33)
where the r.h.s. norm is finite by assumptions on H i ∈ L∞.
Regularity of infinite dimensional heat dynamics 111
By Ito formula for f(Z̃(t)) = ‖Z̃(t)‖q
ℓp(c)
f(Z̃(t)) = f(Z̃(0))+ q
∫ t
0
‖Z̃(s)‖q−p
ℓp(c)
∑
k∈ZZd
ck|Z̃k(s)|
p−1H̃k(s)dWk(s)+
+
q(p− 1)
2
∫ t
0
‖Z̃(s)‖q−p
ℓp(c)
∑
k∈ZZd
ck|Z̃k(s)|
p−2H̃2
kds+
+
q(q − p)
2
∫ t
0
‖Z̃(s)‖q−2p
ℓp(c)
∑
k∈ZZd
c2k|Z̃k(s)|
2(p−1)H̃2
k(s)ds
and due to
∑
|dkbk| ≤
∑
|dk|
∑
|bk| one has
E ‖Z̃(t)‖q
ℓp(c) ≤
q(q − 2)
2
E
∫ t
0
‖Z̃(s)‖q−2
ℓp(c)‖H̃(s)‖2
ℓp(c)ds
Finally, using (33), we obtain
E sup
t∈[0,T ]
‖Z̃(t)‖q
ℓp(c) ≤
≤ ( q
q−1
)q q(q−1)
2
E sup
t∈[0,T ]
‖Z̃(t)‖q−2
ℓp(c)
∫ T
0
‖H̃(s)‖2
ℓp(c)ds ≤
≤ Kq(E sup
t∈[0,T ]
‖Z̃(t)‖q
ℓp(c))
(q−2)/q(E (
∫ T
0
‖H̃(s)‖2
ℓp(c)ds)
q/2)2/q
This leads to
E sup
t∈[0,T ]
‖Z̃(t)‖q
ℓp(c) ≤ K
q/2
q E (
∫ T
0
‖H̃(s)‖2
ℓp(c)ds)
q/2 ≤
≤ K
q/2
q T (q−2)/q
∫ T
0
E ‖H̃(s)‖q
ℓp(c)ds
and gives the statement of theorem for all functions of H̃ type. Due
to their density, closing inequality (32) we have the definition of
stochastic integral and inequality (32) for all Φ. Moreover, the martin-
gale property of Z(t) and its P a.e. continuity is a simple consequence
of estimate (32)
E sup
t∈[0,T ]
‖
∫ t
0
H̃1dW −
∫ t
0
H̃2dW‖q
ℓp(c) ≤ Kq,T
∫ T
0
E ‖H̃1 − H̃2‖
q
ℓp(c)dt
112 A.Val. Antoniouk, A.Vict. Antoniouk
which gives uniform on [0, T ] convergence on measure and therefore
P a.e. convergence on subsequence.
To prove Ito formula, first note that
|ηk(s)|
p = |ηk(0)|p + p
∫ t
0
|ηk(s)|
p−1{Φk(s)dWk(s)+
+Ψk(s)ds} + p(p−1)
2
∫ t
0
|ηk(s)|
p−2Φ2
k(s)ds
Summing up on k ∈ ZZd with weights ck we have Ito formula for
‖ηk(t)‖
p
ℓp(c) which immediately gives (30). 2
Theorem 3. For x0 ∈ ℓ
p(kF +1)2+ε
(a), ε > 0, p ≥ 2, equation (4)
has a unique strong solution, i.e. Ft-adapted continuous ℓp(a)-valued
process y0, which satisfies (4) in the sense of (E sup
t∈[0,T ]
‖ · ‖q
ℓp(a))
1/q
topology, q ≥ 2. It admits a representation as a sum of ℓp(a)-valued
continuous martingale M0(t) =
∫ t
0
B(y0)dW and ℓp(a)-valued conti-
nuous finite variation process V0(t) = −
∫ t
0
(F (y0)+Ay0)ds and fulfills
estimate
∀q ≥ 2 sup
t∈[0,T ]
E ‖y0‖q
ℓ
p(kF +1)
(a) <∞ (34)
For x0 ∈ ℓp(a) there is a unique generalized solution y0(t, x0),
i.e. a limit of strong solutions in the sense of ( sup
t∈[0,T ]
E ‖ · ‖q
ℓp(a))
1/q
topology, q ≥ 2 and the following estimate holds
∀q ∃Cq,p, Dq,p :
sup
t∈[0,T ]
E ‖y0(t, x0)‖q
ℓp(a) ≤ eCq,pT (‖x0‖q
ℓp(a) +Dq,p) (35)
Moreover
∃C ′
q,p ∀x0, y0 ∈ ℓp(a) :
sup
t∈[0,T ]
E ‖y0(t, x0) − y0(t, y0)‖q
ℓp(a) ≤ eC′
q,pT‖x0 − y0‖q
ℓp(a)
(36)
Remark, that the construction of solution y0(t, x0) in the ℓp(a),
p ≥ 2 spaces is required for the proof of differentiability with respect
to the initial data.
Proof is quite standard. It uses some infinite-dimensional Lipschitz
approximations of equation (4) with a successive application of mo-
notone methods, like in [10, 11]. Being a little technical result, it is
ommitted. 2
Regularity of infinite dimensional heat dynamics 113
Theorem 4. Let m1 > |τ |, mγ = m1/|γ| and vectors {cτ} ⊂ IP fulfill
(26). Then ∀x0 ∈ ℓ2(a) and zero-one initial data x̃γ (23) the equation
(21) has a unique strong solution yτ in space ℓmτ (cτ ), i.e. there is Ft-
adapted ℓmτ (cτ )-valued continuous process yτ(t, x
0; x̃γ , γ ⊂ τ) such
that it fulfills equation (21) in the sense of (E sup
t∈[0,T ]
‖ · ‖q
ℓmτ (cτ ))
1/q
topology, q ≥ mτ .
It is represented as a sum of ℓmτ (cτ ) continuous martingale
Mτ (t) =
∫ t
0
(B′(y0)yτ + ϕB
τ )dW and ℓmτ (cτ ) continuous finite va-
riation process V0(t) = −
∫ t
0
(F ′(y0)yτ + Ayτ + ϕF
τ )ds. Moreover, the
following estimate holds: ∀q ≥ mτ ∀R > 0 ∃Kτ (R) such that
sup
t∈[0,T ]
E ‖yτ(t, x
0; x̃γ , γ ⊂ τ)‖q
ℓmτ (cτ ) ≤ Kτ (R) (37)
for R = max(‖x0‖ℓ2(a); ‖x̃γ‖ℓmγ (cγ), γ ⊂ τ).
Proof. The solvability of equations (21) is obtained inductively with
respect to the number of points in set τ = {j1, ..., jm}, ji ∈ ZZd. First
of all note that at |τ | = 1 the inhomogeneous parts ϕB
τ ≡ ϕF
τ ≡ 0
and the proof of inductive base coincides with the proof of inductive
step.
We prove more general result: if for any γ ⊂ τ, |γ| < |τ | the
statement of Theorem 4 holds in scale {ℓmγ (d
icγ)}γ⊂τ for any i ≥
0, then the same is true for τ . Vector d ∈ IP is such that dk ≥
a
−(
kF +1
2
+ε)m1
k for some ε > 0.
Introduce notations F ′
λ(x) = λ(x)F ′(x) and B′
λ(x) = λ(x)B′(x)
for λ ∈ C∞(IR1, [0, 1]) such that for some Nλ > 0
λ(x) = 0 for |x| ≥ Nλ + 1 and λ(x) = 1 for |x| ≤ Nλ (38)
and consider the approximating equation to (21)
yλ
k,τ(t) = x̃k,τ +
∫ t
0
{B′
λ(y
0
k)y
λ
k,τ + ϕB
k,τ}dWk−
−
∫ t
0
{F ′
λ(y
0
k)y
λ
k,τ + (Ayλ
τ )k + ϕF
k,τ}ds
(39)
Remark that hierarchy (26) holds for vectors {dicγ} at any fixed i ≥ 0
and that the zero-one initial data x̃γ ∈ ℓmγ (dicγ) at any i ≥ 0.
114 A.Val. Antoniouk, A.Vict. Antoniouk
Step 1. Equation (39) has a unique strong solution yλ
τ in space
ℓmτ (d
icτ ), i.e. there is Ft-adapted ℓmτ (d
icτ )-valued pathwise conti-
nuous process
yλ
τ (t, x0; x̃γ, γ ⊂ τ)
such that it fulfills equation (39) in the sense of (E sup
t∈[0,T ]
‖·‖q
ℓmτ (dicτ ))
1/q
- topology, q ≥ mτ , and admits a representation as a sum of conti-
nuous martingale Mλ
τ (t) =
∫ t
0
{B′
λ(y
0)yλ
τ + ϕB
τ }dW and continuous
finite variation process V λ
τ (t) = −
∫ t
0
{F ′
λ(y
0)yλ
τ + Ayλ
τ + ϕF
τ }ds.
Indeed, in the Banach space of Ft-adapted ℓmτ (d
icτ )-valued path-
wise continuous processes η(t) equipped with a norm
‖η‖τ,i = (E sup
t∈[0,T ]
‖η(t)‖q
ℓmτ (dicτ ))
1/q
introduce a map
(Uη)k(t) = x̃k,τ +
∫ t
0
ϕB
k,τdWk −
∫ t
0
ϕF
k,τds+ (40)
+
∫ t
0
B′
λ(y
0
k)ηk(s)dWk(s) −
∫ t
0
{F ′
λ(y
0
k)ηk(s) + (Aη)k(s)}ds
By Lemma 1 and due to the boundedness of coefficients F ′
λ, B
′
λ
and
‖A‖L(ℓmτ (dicτ )) <∞ we have
ρT (Uη1,Uη2) ≡ E sup
t∈[0,T ]
‖Uη1 − Uη2‖q
ℓmτ (dicτ )
≤
≤Mτ,λ,T
∫ T
0
E ‖η1(s) − η2(s)‖q
ℓmτ (dicτ )
ds ≤Mτ,λ,T
∫ T
0
ρs(η
1, η2)ds
Therefore ρT (Umη1,Umη2) ≤
Mm
τ,λ,T
m!
TmρT (η1, η2) and there is m0
such that the map Um0 is a strict contraction in ‖ · ‖τ,i. For η0 ≡ 0
by Lemma 1 we have
‖Uη0‖τ,i ≤ ‖x̃τ‖ℓmτ (dicτ )+
+C1 sup
t∈[0,T ]
(E ‖ϕB
τ ‖
q
ℓmτ (dicτ )
)1/q + C2 sup
t∈[0,T ]
(E ‖ϕF
τ ‖
q
ℓmτ (dicτ )
)1/q
Regularity of infinite dimensional heat dynamics 115
Above we used inequality
[
E (
∫ T
0
‖Zs‖ds)
q
]1/q
≤
≤ T (q−1)/q
(
E
∫ T
0
‖Zs‖
qds
)1/q
≤ T
(
sup
t∈[0,T ]
E ‖Zt‖
q
)1/q (41)
for any Ft-adapted Banach space valued process Zt.
By [2, Theorem 4.15] with Q(·) = F (s)(·) or B(s)(·), ζ0 = ζγ1
=
... = ζγs = 0, s = ℓ and Hölder inequality with ri = |τ |+1
|γi|
, i = 1, ..., s,
r0 = |τ | + 1 imply for ϕD = ϕF or ϕB (22)
( sup
t∈[0,T ]
E ‖ϕD
τ ‖
q
ℓmτ (dicτ )
)1/q ≤
≤ K
∑
γ1,...,γs
( sup
t∈[0,T ]
E (1 + ‖y0‖ℓ2(a))
q(kF +1)r0)1/qr0×
×
s∏
j=1
( sup
t∈[0,T ]
E (1 + ‖yγj
‖ℓmγj
(dicγj ))
qrj )1/qrj
(42)
which gives ‖Uη0‖τ,i <∞ by (35) and inductive assumption. Therefore
the sequence {Umη0}m≥1 converges in ‖ · ‖τ,i to some Ft-adapted
ℓmτ (d
icτ )-valued pathwise continuous process yλ
τ . By Lemma 1 se-
quence (40) converges to (39) with corresponding martingale and
finite variation parts.
Step 2. ∀i ≥ 0 ∀q ≥ 1 ∃Cτ such that
sup
λ
sup
t∈[0,T ]
E ‖yλ
τ ‖
q
ℓmτ (dicτ ) ≤ Cτ (43)
where supremum is taken over all functions λ ∈ C∞(IR1, [0, 1]), which
fulfill (38).
Indeed, by Ito formula for q ≥ 2mτ
h(t) = E ‖yλ
τ ‖
q
ℓmτ (dicτ )
= h(0)+
+q
∫ t
0
E ‖yλ
τ ‖
q−mτ
ℓmτ (dicτ )
< (yλ
τ )⋆, yλ
τ (−F ′
λy
λ
τ −Ayλ
τ − ϕF
τ ) >ℓmτ (dicτ ) ds+
+
q(mτ − 1)
2
∫ t
0
E ‖yλ
τ ‖
q−mτ
ℓmτ (dicτ ) < (yλ
τ )⋆, (B′
λy
λ
τ + ϕB
τ )2 >ℓmτ (dicτ ) ds+
116 A.Val. Antoniouk, A.Vict. Antoniouk
+
q(q −mτ )
2
∫ t
0
E ‖yλ
τ ‖
q−2mτ
ℓmτ (dicτ )
∑
k∈ZZd
d2i
k c
2
k,τ |y
λ
k,τ |
2(mτ−1)(B′
λy
λ
k,τ +ϕB
k,τ)
2ds
Inequality (7) and property 0 ≤ λ(·) ≤ 1 give that ∀M ∃KM
−F ′
λ(x) +M [B′
λ(x)]
2 = −λ(x)F ′(x) +Mλ2(x)[B′(x)]2 ≤
≤ −λ(x)F ′(x) +Mλ(x)[B′(x)]2 ≤ λ(x)KM ≤ KM
Using boundedness of ‖A‖L(ℓmτ (dicτ )) and inequalities
∑
|ukvk| ≤
∑
|uk|
∑
|vk|, |x|
m−p|y|p ≤
m− p
m
|x|m +
p
m
|y|m (44)
| < ζ⋆, xy >ℓm(c) | ≤
m− 2
m
‖ζ‖m
ℓm(c) +
1
m
‖x‖m
ℓm(c) +
1
m
‖y‖m
ℓm(c)
we obtain
h(t) ≤ h(0) + (q‖A‖ + qKq−1 + (q − 1)2)
∫ t
0
h(s)ds+
+
∫ t
0
E ‖ϕF
τ ‖
q
ℓmτ (dicτ )ds+ 2(q − 1)
∫ t
0
E ‖ϕB
τ ‖
q
ℓmτ (dicτ )ds (45)
For inductive base ϕF
τ ≡ ϕB
τ ≡ 0, |τ | = 1, therefore by Gronwall-
Bellmann inequality the statement of Step 2 holds for any i ≥ 0.
Inductive assumption (37) in any ℓmγ (d
icγ), |γ| < |τ |, (35) and
(42) give the boundedness of the last two terms in (45). Then the
application of Gronwall-Bellmann inequality finishes the proof of
(43).
Step 3. ∀i ≥ 0 ∀q ≥ 1 for functions λ, µ which fulfill (38) we have
sup
t∈[0,T ]
E ‖yλ
τ − yµ
τ ‖
q
ℓmτ (dicτ ) → 0, Nλ, Nµ → ∞ (46)
Regularity of infinite dimensional heat dynamics 117
Like in Step 2 by Ito formula for q ≥ 2mτ
h(t) = E ‖yλ
τ − yµ
τ ‖
q
ℓmτ (dicτ )
= −q
∫ t
0
E ‖yλ
τ − yµ
τ ‖
q−mτ
ℓmτ (dicτ )
×
×〈(yλ
τ − yµ
τ )⋆, (yλ
τ − yµ
τ ){F ′
λy
λ
τ − F ′
µy
µ
τ + A(yλ
τ − yµ
τ )}〉ds+
+ q(mτ−1)
2
∫ t
0
E ‖yλ
τ − yµ
τ ‖
q−mτ
ℓmτ (dicτ )
×
×〈(yλ
τ − yµ
τ )⋆, (B′
λy
λ
τ − B′
µy
µ
τ )2〉ℓmτ (dicτ )ds+
+ q(q−mτ )
2
∫ t
0
E ‖yλ
τ − yµ
τ ‖
q−2mτ
ℓmτ (dicτ )
×
×
∑
k∈ZZd
d2i
k c
2
k,τ |y
λ
k,τ − yµ
k,τ |
2(mτ−1)(B′
λy
λ
k,τ − B′
µy
µ
k,τ)
2ds
Using inequalities (44) and coordinate relations
F ′
λ(y
0)yλ
τ −F
′
µ(y0)yµ
τ = (λ(y0)−µ(y0))F ′(y0)yλ
τ +µ(y0)F ′(y0)(yλ
τ −y
µ
τ )
(B′
λ(y
0)yλ
τ −B′
µ(y0)yµ
τ )2 ≤
≤ 2µ2(y0)[B′(y0)]2(yλ
τ − yµ
τ )2 + 2(λ(y0) − µ(y0))2[B′(y0)]2(yλ
τ )2 ≤
≤ 2µ(y0)[B′(y0)]2(yλ
τ − yµ
τ )2 + 2(λ(y0) − µ(y0))2[B′(y0)]2(yλ
τ )2
we obtain
h(t) ≤ (q‖A‖ + (q − 1)2)
∫ t
0
h(s)ds+ q
∫ t
0
E ‖yλ
τ − yµ
τ ‖
q−mτ
ℓmτ (dicτ )
×
×〈(yλ
τ − yµ
τ )⋆, (yλ
τ − yµ
τ )2µ(y0){−F ′(y0) + (q − 1)[B′(y0)]2}〉 + ds
+
∫ t
0
E ‖(λ(y0) − µ(y0))F ′(y0)yλ
τ ‖
q
ℓmτ (dicτ )ds+ (47)
+2(q − 1)
∫ t
0
E ‖(λ(y0) − µ(y0))B′(y0)yλ
τ ‖
q
ℓmτ (dicτ )ds (48)
118 A.Val. Antoniouk, A.Vict. Antoniouk
Due to conditions (8)-(9) for 0 ≤ λ(·) ≤ µ(·) ≤ 1
|F ′(y0
k)(λ(y0
k) − µ(y0
k))| ≤ Kχ{|y0
k| ≥ Nλ}(1 + |y0
k|
2)
kF +1
2 ≤
≤ Ka
−(
kF +1
2
+ε)
k aε
kχ
2ε{|y0
k| ≥ Nλ}(ak + ak|y
0
k|
2)
kF +1
2 ≤
≤ Ka
−(
kF +1
2
+ε)
k [ak
|y0
k|
2
N2
λ
]ε(1 + ‖y0‖2
ℓ2(a))
kF +1
2 ≤
≤
Ka
−(
kF +1
2
+ε)
k
N2ε
λ
(1 + ‖y0‖2
ℓ2(a))
kF +1
2
+ε
(49)
where χ{A} denotes the characteristic function of set A.
Therefore for dk ≥ a
−(
kF +1
2
+ε)mτ
k we have estimate on (47)
sup
t∈[0,T ]
E ‖(λ(y0) − µ(y0))F ′(y0)yλ
τ ‖
q
ℓmτ (dicτ )
≤
≤ 1
N2εq
λ
Kq sup
t∈[0,T ]
E (1 + ‖y0‖2
ℓ2(a))
(
kF +1
2
+ε)q‖yλ
τ ‖
q
ℓmτ (di+1cτ ) → 0,
Nλ, Nµ → ∞
(50)
where we applied (35) and statement of Step 2. The analogous con-
vergence holds for term (48). Using 0 ≤ µ(·) ≤ 1 and (7) we have
h(t) ≤ (q‖A‖ + qKq−1 + (q − 1)2)
∫ t
0
h(s)ds+ δλ,µ
with δλ,µ → 0, Nλ, Nµ → ∞. By Gronwall-Bellmann inequality we
obtain (46).
Step 4. End of the proof: Theorem 4 is fulfilled for yτ in any space
ℓmτ (d
icτ ), i ≥ 0.
By Step 3 there is Ft-adapted ℓmτ (d
icτ )-valued process y#(t, x0;
x̃γ , γ ⊂ τ) such that ∀q ≥ mτ
sup
t∈[0,T ]
E ‖y#
τ − yλ
τ ‖
q
ℓmτ (dicτ )
→ 0, Nλ → ∞ (51)
To construct the strong solution yτ it is sufficient to prove that the
equation (39) converges to (21) in the topology (E sup
t∈[0,T ]
‖·‖q
ℓmτ (dicτ )
)1/q
Regularity of infinite dimensional heat dynamics 119
when Nλ → ∞. By Lemma 1 and choice B′
λ(x) = λ(x)B′(x)
(E sup
t∈[0,T ]
‖
∫ t
0
{B′
λ(y
0)yλ
τ − B′(y0)y#
τ }dW‖q
ℓmτ (dicτ ))
1/q ≤
≤ K
1/q
q,T T
1/q sup
t∈[0,T ]
(E ‖(λ(y0) − 1)B′(y0)yλ
τ ‖
q
ℓmτ (dicτ )
)1/q (52)
+K
1/q
q,T T
1/q sup
t∈[0,T ]
(E ‖B′(y0)(yλ
τ − y#
τ )‖q
ℓmτ (dicτ )
)1/q (53)
Like in (50) the term (52) tends to zero at Nλ → ∞. To the
second term we apply [2, Theorem 4.15]
(53) ≤ C sup
t∈[0,T ]
[E (1 + ‖y0‖ℓ2(a))
q(kF +1)‖yλ
τ − y#
τ ‖
q
ℓmτ (di+1cτ )]
1/q → 0,
Nλ → ∞.
Above we also used (51) and (35). Therefore the stochastic integral in
(39) converges to the stochastic integral in (21) and gives ℓmτ (d
icτ )-
pathwise continuous martingale. The convergence of continuous finite
variation part of (39) to the corresponding part of (21) is checked in
a similar way.
We obtain, that the r.h.s. of (39) converges in topology (E sup
t∈[0,T ]
‖·
‖q
ℓmτ (dicτ )
)1/q, thus the l.h.s. yλ
τ of (39) also has a limit in the same
topology: ∃ yτ such that yλ
τ → yτ , Nλ → ∞. Such convergence
improves (51) and provides a necessary strong solution yτ as ℓmτ (d
icτ )
pathwise continuous modification of y#
τ .
The uniqueness of strong solution yτ is proved by induction on
|τ |. Suppose that we have shown the uniqueness for all |γ| < |τ |. By
Ito formula for two different solutions y1
τ and y2
τ we have in analogue
to Step 3
h(t) = E ‖y1
τ − y2
τ‖
q
ℓmτ (dicτ ) ≤ q‖A‖
∫ t
0
h(s)ds+
+q
∫ t
0
E ‖y1
τ − y2
τ‖
q−mτ
ℓmτ (dicτ )
∑
k∈ZZd
di
kck,τ |y
1
k,τ − y2
k,τ |
mτ×
×{−F ′(y0
k) + (q − 1)[B′(y0
k)]
2} ≤ (q‖A‖ + qKq−1)
∫ t
0
h(s)ds
120 A.Val. Antoniouk, A.Vict. Antoniouk
where we used (7). By h(0) = 0 we obtain h(t) ≡ 0 which gives the
uniqueness.
It remains to show estimate (37). By Ito formula for strong
solution yτ to (21) and by (44)
h(t) = E ‖yτ(t)‖
q
ℓmτ (dicτ ) ≤ ‖x̃τ‖
q
ℓmτ (dicτ )+
(q‖A‖ + (q − 1)2)
∫ t
0
h(s)ds+ q
∫ t
0
E ‖yτ(t)‖
q−mτ
ℓmτ (dicτ )
×
×
∑
k∈ZZd
di
kck,τ |yk,τ |
mτ{−F ′(y0
k) + (q − 1)[B′(y0
k)]
2}ds+
+
∫ t
0
E ‖ϕF
τ ‖
q
ℓmτ (dicτ )
ds+ 2(q − 1)
∫ t
0
E ‖ϕB
τ ‖
q
ℓmτ (dicτ )
ds
We use (35), (7) and inequality (42) to obtain
h(t) ≤ ‖x̃τ‖
q
ℓmτ (dicτ )
+K(R)+
+(q‖A‖ + qKq−1 + (q − 1)2)
∫ t
0
h(s)ds
(54)
and therefore (37), which ends the proof of Theorem 4. 2
4. Nonlinear estimate on variations (Proof of
Theorem 2).
First we restrict to the case x0 ∈ ℓ
2(kF +1)2+ε
(a), ε > 0, i.e. when
y0 is a strong solution in the sense of Theorem 3. Introduce notations
hi
τ (y; t) = E
i∑
s=1
[ps(zt)
∑
γ⊂τ, |γ|=s
‖yγ‖
mγ
ℓmγ (cγ)], i = 1, ..., |τ |
gγ(t) = E pi(zt)‖yγ(t)‖
mγ
ℓmγ (cγ), |γ| = i (55)
If we prove that for all γ ⊂ τ, |γ| = i and i = 1, ..., |τ |
gγ(t) ≤ eD1tgγ(0) +D2
∫ t
0
eD1(t−s)hi−1
τ (y; s)ds (56)
then we will have the recurrence base and step for the statement of
Theorem at i = |τ |.
Regularity of infinite dimensional heat dynamics 121
By Ito formula
gγ(t) = gγ(0) −
∫ t
0
E ‖yγ‖
mγ
ℓmγ (cγ)(H
F,Bpi)(zs)ds−
−mγ
∫ t
0
E pi(zs) < y⋆γ , yγ[F
′(y0)yγ + Ayγ + ϕF
γ ] >ℓmγ (cγ) ds+
+
mτ (mτ − 1)
2
∫ t
0
E pi(zs) < y⋆γ , [B
′(y0)yγ + ϕB
γ ]2 >ℓmγ (cγ) ds+
+2mγ
∫ t
0
E p′i(zs)
∑
k∈ZZd
akck,γy
0
kB(y0
k)|yk,γ|
mγ−2yk,γ{B
′(y0
k)yk,γ +ϕB
k,γ}ds
where we used notation (31) and operator HF,B acts on smooth
function f(·) by rule
(HF,Bf)(x) =
∑
k∈ZZd
{−
1
2
B2(xk)
∂2
∂x2
k
+ (F (xk) + (Ax)k)
∂
∂xk
}f(x)
Immediately remark that for functions p which fulfills (27) the fol-
lowing property takes place
∃C1 ∈ IR HF,Bp(z) ≥ −C1p(z) (57)
for z = 1 + ‖x‖2
ℓ2(a). Indeed,
HF,Bp(z) =
∑
k∈ZZd
ak{2F (xk)xk −B2(xk) − 2(Ax)kxk}p
′(z)−
−
∑
k∈ZZd
2a2
kB
2(xk)x
2
kp
′′
i (z) ≥ −2‖A‖L(ℓ2(a))zp
′(z)+
+
∑
k∈ZZd
ak{2F (xk)xk − B2(xk)}p
′(z) − 2z|p′′i (z)|
∑
k∈ZZd
akB
2(xk) ≥
≥ −2‖A‖Cp(z) +
∑
k∈ZZd
ak{2F (xk)xk − (1 + 2C)B2(xk)}p
′(z) ≥
≥ −2‖A‖Cp(z) +
∑
k∈ZZd
ak{−K1x
2
k −K2}p
′(z) ≥
≥ −(2‖A‖C + (K1 +K2)C)p(z) ≡ −C1p(z)
122 A.Val. Antoniouk, A.Vict. Antoniouk
where we successively applied
∑
|ukvk| ≤
∑
|uk|
∑
|vk|, (27), (6) and∑
ak = 1.
Using (44) and (57) we obtain
gγ(t) ≤ gγ(0) + (C1 +mγ‖A‖ + (mγ − 1)2)
∫ t
0
gγ(s)ds+
+mγ
∫ t
0
E pi(zs)〈y
⋆
γ , y
2
γ{−F
′(y0
k) + (mγ − 1)[B′(y0
k)]
2}〉ℓmγ (cγ)ds+
+
∫ t
0
Epi(zs)‖ϕ
F
γ ‖
mγ
ℓmγ (cγ)ds+
+2(mγ − 1)
∫ t
0
Epi(zs)‖ϕ
B
γ ‖
mγ
ℓmγ (cγ)ds+
(58)
+2mγK4
∫ t
0
E zsp
′
i(zs) < y⋆γ , (1 + [B′(y0)]2)y2
γ >ℓmγ (cγ) ds+
+2mγK3
∫ t
0
E zsp
′
i(zs) < y⋆γ , (1 + |B′(y0)|)yγϕ
B
γ >ℓmγ (cγ) ds
Assumption (27), applied to (58), (27) and (7) lead to
gγ(t) ≤ gγ(0) + (C1 +mγ‖A‖ + (mγ − 1)2 + 2mγK4C+
+2K3C(mγ − 1) +mγKmγ−1+2K4C)
∫ t
0
gγ(s)ds+
+
∫ t
0
Epi(zs)‖ϕ
F
γ ‖
mγ
ℓmγ (cγ)ds+2(mγ−1)
∫ t
0
E pi(zs)‖ϕ
B
γ ‖
mγ
ℓmγ (cγ)ds+
+2K3C
∫ t
0
E pi(zs)‖(1 + |B′(y0)|)ϕB
γ ‖
mγ
ℓmγ (cγ)ds
(59)
All terms in (59) have the same structure
∫ t
0
E pi(zs)‖
∑
α1∪...∪αs=γ, s≥2
D
s(y0)yα1
...yαs‖
mγ
ℓmγ (cγ)ds (60)
where function D
s(·) = F (s)(·), B(s)(·) or (1 + |B′(·)|)B(s)(·). Using
condition (8)-(9) and property 2kB ≤ kF we estimate (60) by
(60) ≤ K1
∑
α1∪..∪αs = γ,
s ≥ 2
∫ t
0
E pi(zs) ×
×
∑
k∈ZZd
ck,γ[D
s(y0
k)]
mγ |yk,α1
|mγ ...|yk,γs|
mγds ≤
Regularity of infinite dimensional heat dynamics 123
≤ K1
∑
...
∫ t
0
E pi(zs)
∑
k∈ZZd
ck,γa
−
kF +1
2
mγ
k ×
×(ak + ak|y
0
k|
2)
kF +1
2
mγ |yk,α1
|mγ ...|yk,αs|
mγds ≤
(61)
≤ K1
∑
α1∪..∪αs =γ,
s ≥ 2
∫ t
0
E pi(zs)z
kF +1
2
mγ
s ×
×
∑
k∈ZZd
ck,γa
−
kF +1
2
mγ
k |yk,α1
|mγ ...|yk,αs|
mγds
By hierarchies (26), (28) we obtain
(61) ≤ K1K
1/|γ|
p ×
×
∑
α1∪..∪αs = γ,
s ≥ 2
K
1/|γ|
α1,...,αs;γ
∫ t
0
E
∑
k∈ZZd
{p|αi|(zs)ck,αi
|yk,αi
|mαi}|αi|/|γ|ds ≤
≤ K1K
1/|γ|
p
∑
...
K1/|γ|
α1,...,αs;γ
s∑
i=1
E
∫ t
0
p|αi|(zs)‖yαi
‖
mαi
ℓmαi
(cαi)
ds ≤
≤ K1K
1/|γ|
p 2|τ | max
α1∪...∪αs=γ⊂τ
K
1/|γ|
α1∪...∪αs;γ h
i−1
τ (y; t)
Here we used ∀j = 1, ..., s |αi| < |γ| and inequality |x1...xs| ≤
|x1|
q1/q1 + ... + |xs|
qs/qs with qj = |γ|/|αj|. Finally we have
gγ(t) ≤ gγ(0) +D1
∫ t
0
gγ(s)ds+D2
∫ t
0
hi−1(y; s)ds
which leads to (56) and proves the quasi-contractive nonlinear estimate
for x0 ∈ ℓ
2(kF +1)2+ε
(a), ε > 0. The closure to x0 ∈ ℓ2(a) is done with
application of estimates (36), (62) and polynomiality of pi. 2
5. Regularity of variations and Proof of Theorem
1.
124 A.Val. Antoniouk, A.Vict. Antoniouk
Before the study the differentiability of y0(t, x0) on variable x0
we obtain the continuity of variations with respect to initial data
x0. This result will be applied to close the nonlinear estimate on
variations from x0 ∈ ℓ
p(k+1)2+ε
(a) to x0 ∈ ℓ2(a) and to prove C∞-
differentiability of y0
t (x
0) with respect to the initial data x0.
Theorem 5. Let m1 > |τ |, mγ = m1/|γ|, vectors {cτ} ⊂ IP fulfill
(26) and x̃γ be zero-one initial data (23). Then ∀q ≥ mτ ∀R >
0 ∃Kτ (R) such that ∀x0, y0 ∈ ℓ2(a) the variations fulfill
sup
t∈[0,T ]
E ‖yτ(t, x
0; x̃γ , γ ⊂ τ) − yτ(t, y
0; x̃γ, γ ⊂ τ)‖q
ℓmτ (cτ ) ≤
≤ Kτ (R)‖x0 − y0‖q
ℓ2(a)
(62)
with R = max(‖x0‖ℓ2(a), ‖y
0‖ℓ2(a), ‖x̃γ‖ℓmγ (dcγ)) for dk ≥ a
−
kF +1
2
m1
k ,
k ∈ ZZd.
Proof is similar to the proof of nonlinear estimate on variations and
proceeds with application of Ito formula instead of pathwise estimates
of [2, Th.4.18]. 2
To obtain the integral representation of Theorem 6, we need
the following Lemma, which gives uniform on |τ | ≤ n0 estimates on
variations. This result is also required for the study the high order
differentiability of the stochastic flow and heat semigroup Pt.
Lemma 2. Under conditions (5)-(9) for zero-one initial data x̃γ (23)
we have
∀ψ ∈ IP ∀n ≥ 1 ∀q ≥ 1 ∃Kn(R,ψ, q) such that
sup
t∈[0,T ]
E |yk,τ(t, x
0, x̃γ)|
q ≤ Kn(R,ψ, q)a
−
kF +1
2
q(|τ |−1)
k
∏
j∈τ
ψ−1
k−j (63)
sup
t∈[0,T ]
E |yk,τ(t, x
0, x̃γ) − yk,τ(t, y
0, x̃γ)|
q ≤
≤ Kn(R,ψ, q)a
−
kF +1
2
q(2|τ |−1)
k
∏
j∈τ
ψ−1
k−j‖x
0 − y0‖q
ℓ2(a)
(64)
with R = max(‖x0‖ℓ2(a), ‖y
0‖ℓ2(a)).
Regularity of infinite dimensional heat dynamics 125
Proof uses a special choice of weights c̃k,γ = a
kF +1
2
m1
|γ|−1
|γ|
k
∏
j∈γ
ψk−j ,
γ ⊂ τ with m1
def
= q|τ | and coincides with proof of [2, Corollary 4.19].
It can be omitted. 2
Now we turn to the differentiability of process y0 (4) with respect
to the initial data.
Theorem 6. Let F,B satisfy conditions (5)-(9). Then ∀x0 ∈ ℓ2(a),
zero-one initial data x̃γ (23) and h ∈ AC([a, b]) for all t ∈ [0, T ] and
P a.e. ω ∈ Ω the path
χ0(·) = y0(t, x0 + h(·)) − y0(t, x0 + h(a)) ∈ AC([a, b])
In particular, in any space ℓp(c), c ∈ IP , p ≥ 1 its derivative is given
by first order variation
y0(t, x0 + h(·))
b
a
= ℓp(c)
∫ b
a
∑
j∈ZZd
y{j}(t, x
0 + h(s))h′j(s)ds (65)
Space AC([a, b]) was introduced in (16).
Proof. First we prove representation (65) for initial data x0 ∈
ℓ
m1(kF +1)2+ε
(a), ε > 0, in space Lq(Ω,P, ℓm1
(c1)), q ≥ 1, with vector
c1 ∈ IP such that dkck,1 ≤ ak for dk ≥ a
−
kF +1
2
m1
k . Due to Theorem
3 for x0 ∈ ℓ
m1(kF +1)2+ε
(a), ε > 0, there is a strong solution y0 to
equation (4) in space with topology E sup
t∈[0,T ]
‖ · ‖q
ℓm1
(a) and estimate
holds E ‖y0(t, x0) − y0(t, y0)‖q
ℓm1
(a) ≤ eCqt‖x0 − y0‖q
ℓm1
(a). Inequality
‖ · ‖ℓm1
(c1) ≤ ‖ · ‖ℓm1
(a) implies that for function h ∈ AC([a, b]) the
map [a, b] ∋ s → y0(t, x0 + h(s)) ∈ Lq(Ω,P, ℓm1
(c1)) is absolutely
continuous. The theory of absolutely continuous functions in reflexive
Banach space gives that for a.e. s ∈ [a, b] there is Lq(Ω,P, ℓm1
(c1))
strong derivative
d
ds
y0(t, x0 + h(s)) and representation holds
y0(t, x0 + h(·))
b
a
= Lq(Ω,P, ℓm1
(c1))
∫ b
a
d
ds
y0(t, x0 + h(s))ds (66)
126 A.Val. Antoniouk, A.Vict. Antoniouk
To reconstruct the strong derivative let us show that for h ∈
AC([a, b]) and a.e s ∈ [a, b] such that
lim
α→0
‖
h(s+ α) − h(s)
α
− h′(s)‖ℓm1
(a) = 0
the convergence holds
sup
t∈[0,T ]
E
∥∥∥∥
y0
k(t, x
0 + h(s+ α)) − y0
k(t, x
0 + h(s))
α
−
−
∑
j∈ZZd
yk,{j}(t, y
0)h′j(s)
∥∥∥∥∥
q
ℓm1
(c1)
→ 0, α→ 0
Further proof coincides with the proof of [2, Th.4.20] with use
of Ito formula instead of pathwise estimates. 2
Next Theorem states any order differentiability of process y0(t, x0).
Theorem 7. Let F,B fulfill conditions (5)-(9). Then ∀x0 ∈ ℓ2(a),
zero-one initial data x̃γ (23) and h ∈ AC([a, b]) (16) we have for all
t ∈ [0, T ], P a.e. ω ∈ Ω and ∀k ∈ ZZd, ∀τ the path
χk,τ(·) = yk,τ(t, x
0 + h(·)) − yk,τ(t, x
0 + h(a)) ∈ AC([a, b], IR1)
In particular different order variations are related by
yk,τ(t, x
0 + h(·))
b
a
=
∫ b
a
∑
j∈ZZd
yk,τ∪{j}(t, x
0 + h(s))h′j(s)ds
Proof. Like in the proof of Theorem 6 we first consider initial data
x0 ∈ ℓ
m1(kF +1)2+ε
(a), ε > 0, for some m1 > |τ |. Choose vectors
{cn}n≥1 so that
∀k ∈ ZZd ck,n+1dk ≤ ck,n, ck,1dk ≤ ak (67)
with dk ≥ a
−
kF +1
2
m1
k . These vectors obviously satisfy condition (26).
Introduce notation X|τ | = ℓmτ (c|τ |). Applying Theorem 5 in scale
{X|τ |} and inequality ‖ · ‖X
|τ |+1
≤ const‖ · ‖X
|τ |
we have the absolute
continuity of the map
[a, b] ∋ s→ yτ(t, x
0 + h(s)) ∈ Lq(Ω,P, X|τ |+1)
Regularity of infinite dimensional heat dynamics 127
for any t ∈ [0, T ] and h ∈ AC([a, b]). The theory of absolutely conti-
nuous functions implies the existence of strong derivative
Lq(Ω,P, X|τ |+1)
d
ds
yτ(t, x
0 + h(s)) for a.e. s ∈ [a, b]
and gives representation
yτ (t, x
0 + h(·))
b
a
= Lq(Ω,P, X|τ |+1)
∫ b
a
d
ds
yτ (t, x
0 + h(s))ds (68)
If we prove by induction on |τ | that for a.e. s ∈ [a, b] such that
∃ lim
α→0
‖
h(s+ α) − h(s)
α
− h′(s)‖ℓm1
(a) = 0 (69)
the convergence holds
sup
t∈[0,T ]
E
∥∥∥∥
yk,τ(t, x
0 + h(s+ α)) − yk,τ(t, x
0 + h(s))
α
−
−
∑
j∈ZZd
yk,τ∪{j}h
′
j(s)
∥∥∥∥∥
q
X
|τ |+1
→ 0
(70)
for α → 0, then the representation (68) will lead to
yτ (t, x
0+h(·))
b
a
= Lq(Ω,P, X|τ |+1)
∫ b
a
∑
j∈ZZd
yτ∪{j}(t, x
0+h(s))h′j(s)ds
This gives the P a.e. coordinate equality: ∀k ∈ ZZd
yk,τ(t, x
0 + h(·))
b
a
=
∫ b
a
∑
j∈ZZd
yk,τ∪{j}(t, x
0 + h(s))h′j(s)ds (71)
with integrable for P a.e. ω ∈ Ω right hand side
∑
j∈ZZd
yk,τ∪{j}(t, x
0 + h(·))h′j(·) ∈ L1([a, b], IR
1) (72)
Further proof proceeds similar to [2, Th.4.21], with the use of
Ito formula for convergence (70) instead of pathwise estimates.
128 A.Val. Antoniouk, A.Vict. Antoniouk
The developed above technique is sufficient for the study of
differentiable properties of Feller semigroup Pt (10).
Proof of Theorem 1. It completely coincides with one, conducted in
[2, § 4.6] for the unit diffusion case. The only difference is that, using
representation
∂τPtf(x0) =
|τ |∑
σ=1
∑
γ1∪...∪γσ=τ
E < ∂(σ)f(y0), yγ1
⊗ ...⊗ yγσ > (t, x0) (73)
with variations yγ (21) and
< ∂(σ)f(y0), yγ1
⊗ ...⊗ yγσ > (t, x0) =
=
∑
j1,...,jσ∈ZZd
∂{j1,...,jσ}f(y0(t, x0))yj1,γ1
(t, x0)...yjσ,γσ(t, x0)
one should use existence of majorant to show the measurability of
derivatives ∂τPtf(x). 2
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Department of Nonlinear Analysis, Institute of
Mathematics NAS Ukraine,
Tereschenkivska str. 3, 01601 MSP Kiev-4, Ukraine
antoniouk@imath.kiev.ua
Received 15.05.07
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