Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems
We study the regularity problems for unbounded spin systems of anharmonic oscillators, that approximate multi-dimensional Euclidean field theories. The main attention is paid to the effect of anharmonism on the C∞-regularity properties of evolutional semigroup. Our approach is based on a new class...
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irk-123456789-1212782017-06-14T03:07:21Z Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems Antoniouk, A.Val. Antoniouk, A.Vict. We study the regularity problems for unbounded spin systems of anharmonic oscillators, that approximate multi-dimensional Euclidean field theories. The main attention is paid to the effect of anharmonism on the C∞-regularity properties of evolutional semigroup. Our approach is based on a new class of nonlinear estimates on variations, that permit to obtain regular properties for essentially nonlinear equations. 2006 Article Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems / A.V. Antoniouk, A.V. Antoniouk // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 5-14. — Бібліогр.: 18 назв. — англ. 1607-324X PACS: 02.30.Jr, 02.30.Sa, 02.30.Xx, 02.50.Eu,03.654.Db, 05.50.+q, 75.10.Hk DOI:10.5488/CMP.9.1.5 http://dspace.nbuv.gov.ua/handle/123456789/121278 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We study the regularity problems for unbounded spin systems of anharmonic oscillators, that approximate
multi-dimensional Euclidean field theories. The main attention is paid to the effect of anharmonism on the
C∞-regularity properties of evolutional semigroup. Our approach is based on a new class of nonlinear estimates
on variations, that permit to obtain regular properties for essentially nonlinear equations. |
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Article |
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Antoniouk, A.Val. Antoniouk, A.Vict. |
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Antoniouk, A.Val. Antoniouk, A.Vict. Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems Condensed Matter Physics |
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Antoniouk, A.Val. Antoniouk, A.Vict. |
| author_sort |
Antoniouk, A.Val. |
| title |
Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems |
| title_short |
Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems |
| title_full |
Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems |
| title_fullStr |
Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems |
| title_full_unstemmed |
Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems |
| title_sort |
nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems |
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Інститут фізики конденсованих систем НАН України |
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Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems / A.V. Antoniouk, A.V. Antoniouk // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 5-14. — Бібліогр.: 18 назв. — англ. |
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Condensed Matter Physics |
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AT antonioukaval nonlineareffectsintheregularityproblemsforinfinitedimensionalevolutionsofunboundedspinsystems AT antonioukavict nonlineareffectsintheregularityproblemsforinfinitedimensionalevolutionsofunboundedspinsystems |
| first_indexed |
2025-07-08T19:31:15Z |
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2025-07-08T19:31:15Z |
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1837108379696234496 |
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Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 5–14
Nonlinear effects in the regularity problems for infinite
dimensional evolutions of unbounded spin systems
Alexander Val. Antoniouk∗, Alexandra Vict. Antoniouk
Department of Nonlinear Analysis,
Institute of Mathematics NAS Ukraine,
Tereschenkivska 3, 01601 MSP, Kiev 4, Ukraine
Received August 15, 2005, in final form January 31, 2006
We study the regularity problems for unbounded spin systems of anharmonic oscillators, that approximate
multi-dimensional Euclidean field theories. The main attention is paid to the effect of anharmonism on the
C∞-regularity properties of evolutional semigroup. Our approach is based on a new class of nonlinear esti-
mates on variations, that permit to obtain regular properties for essentially nonlinear equations.
Key words: anharmonic lattice spin systems, nonlinear regularity problems
PACS: 02.30.Jr, 02.30.Sa, 02.30.Xx, 02.50.Eu,03.654.Db, 05.50.+q, 75.10.Hk
1. Introduction
Presently, a rigorous study of numerous important physical models turns out to be impossible
without elaboration of mathematical tools inherent to these models and without solution of pure
mathematical problems that require the development of adequate calculus.
Many physical processes may be described in terms of infinite dimensional stochastic differential
equations of general form
dξx(t) = −G(ξx(t))dt +D(ξx(t))dW (t), ξx(0) = x, (1.1)
where W is a Brownian motion representing some heat source, D is the diffusion coefficient of
inhomogeneous medium and the drift part G corresponds to the force existing in the system, which
keeps the system at equilibrium. The corresponding semigroup (Ptf)(x) = Ef(ξx(t)), constructed
as a mean E with respect to the Wiener measure, describes the heat evolution of a corresponding
physical medium.
Traditionally the linear part B of force G(ξx(t)) = [Bξx(t) + F (ξx(t))] is separated from
nonlinear term F . Since in many applications B is given by some unbounded operator, the research
mainly concentrated on the case of unbounded B, which became a topic of the stochastic partial
differential equations theory, e.g. [1,2]. At the same time, the simplified classes of nonlinear F of
the most linear order of growth, the so-called globally Lipschitz nonlinearities (i.e. when ||F (x)−
F (y)|| 6 const ||x − y||), were considered. The case of non-Lipschitz coefficients was successfully
treated only for partial problems, such as existence, uniqueness and ergodic behaviour of solutions,
e.g. [1–4] and references therein.
However, many important problems of modern physics contain essential nonlinearities that
cannot be handled with the tools developed for the models with Lipschitz nonlinearities. The most
striking example is the diverging perturbation series for Euclidean fields with nonlinear interactions
in high dimensions. Similar complications arise in other important physical models, e.g. [5,6].
In this article we discuss regularity properties of evolutions with non-Lipschitz nonlinearities,
i.e. we study how nonlinearity of the problem (1.1) effects the regular dependence of the process
ξx(t) with respect to initial data x and what requirements the topologies of spaces of differentiable
∗
E-mail: antoniouk@imath.kiev.ua
c© Alexander Val. Antoniouk, Alexandra Vict. Antoniouk 5
A.V.Antoniouk, A.V.Antoniouk
functions should meet to permit the construction of semigroup Pt in these spaces. Though the
monotonicity conditions of coefficients of (1.1), which lead to the existence and uniqueness of
nonlinear process ξx(t) and its semigroup Pt, are known long ago [3,4] and the question of regularity
has already been raised in literature, one may consult e.g. [1,2,7] and the most recent [8,9] to see
that the final solution is still far from reach.
A question arises here. Which methods from nonlinear analysis or stochastic theory can be
applied to the investigation of regularity properties in non-Lipschitz case? The application of
the classical tools of nonlinear analysis, such as implicit function theoremes, finite dimensional
Galerkin or Yosida approximations of nonlinearity F (in order to get the regularity properties
of ξx
t and Pt from the regularity properties of approximating problems) would be complicated.
First of all, the nonlinear mappings in infinite dimensional space are mostly non-Lipschitz even
locally on balls. Moreover, as it is discussed in [10] and [11, §1.1], the use of standard topologies of
spaces of continuously differentiable functions for regularity problems is not viable in non-Lipschitz
case. Each kind of nonlinearity requires guessing certain corrections of topologies of the classical
functional spaces and such corrections are not visible at the level of approximations.
One may also attempt the tools of stochastic theory, such as Girsanov transformation, Bismut-
Elworthy formula and the related approach of Malliavin calculus, e.g. [12,13]. Though Girsanov
transformation removes the nonlinear drift from equation (1.1), the developed techniques will not be
adapted to the general diffusion coefficientD in (1.1). The applications of Bismut-Elworthy formula
are possible, e.g [2,11,14–17], but these methods are more adapted to the study of differentiability
of the process ξx(t) with respect to the random parameter and actually require a bit of work in non-
Lipschitz case. So, their development would be more oriented to the mastering of stochastics and
would be indirect to the pure nonlinear analytical problem about the regularity of non-Lipschitz
equations (1.1) and their semigroups.
The question is whether the direct work with regularity properties for nonlinear equations (1.1)
is possible at all? To find an answer we should ask what is actually nonlinearity. One may say that
it is a nonlinear response to linear operations. For example, for the linear differentiation operation
∂x one has by chain rule
∂(n)
x (f ◦ g)(x) =
n∑
j=1
∑
k1+···+kj=n
(f (j) ◦ g)(x) ∂(k1)
x g(x) · · · ∂(kj)
x g(x).
The consideration of terms with j = 1 and j = n displays symmetry
∂(n)
x g(x) ∼ [∂xg(x)]
n (1.2)
that holds for all n ∈ IN and is present in all intermediate terms due to
∂(k1)
x g(x) . . . ∂(kj)
x g(x) ∼ [∂xg(x)]
k1 . . . [∂xg(x)]
kj ∼ [∂xg(x)]
k1+···+kj ∼ [∂xg(x)]
n.
In this article we discuss the consequences of symmetry (1.2) for nonlinear equations (1.1). We
develop the results of [10,16,17] and consider an infinite dimensional model of interacting particles
with unbounded spins, that, in particular, approximates the high dimensional Euclidean fields with
nonlinear interaction [5,6].
2. Description of model and nonlinear estimate on variation s
Consider a particular case of stochastic differential equation (1.1):
{
dξx
k (t) = dWk(t) − {F (ξx
k (t)) + (Bξx(t))k}dt;
ξx
k (0) = xk, k ∈ Z
d.
(2.1)
Here k = (k1, . . . , kd) is a point of d-dimensional lattice Z
d, coordinate ξx
k takes values in space
R
1, called the spin space of kth particle.
6
Regularity properties of evolutions of unbounded spin systems
Process W (t) = {Wk(t)}k∈Zd , t > 0 is formed from independent Wiener processes, running at
spin spaces of each particle k ∈ Z
d. Its canonical realizations (Ω,F ,Ft,WZd) may be described by
fixing some vector {ak > 0}k∈Zd ,
∑
k∈Zd ak = 1, and probability space Ω = C0(R+, `2(a)) with
Borel σ-algebra F . The flow of σ-algebras Ft is formed by events till the time t and WZd is a
product of Wiener measures at each point of lattice k ∈ Z
d [1]. The linear finite-diagonal map
B : R
Z
d
→ R
Z
d
introduces the interaction between particles of finite radius r0. It is defined by a
finite set of real numbers {bi : i ∈ Z
d, |i| 6 r0}:
(Bx)k =
∑
j∈Zd: |j−k|6r0
bk−jxj
and represents a bounded mapping in any space
`p(c) = `p(c,Z
d) =
x ∈ R
Z
d
: ||x||`p(c) =
∑
k∈Zd
ck|xk|
p
1/p
<∞
defined by vectors c = {ck > 0}k∈Zd , such that δc = sup|k−j|=1 |ck/cj| <∞. Henceforth we denote
a set of such vectors c by P.
The mapping F : R
Z
d
3 {xk}k∈Zd = x −→ F (x) = {F (xk)}k∈Zd ∈ R
Z
d
introduces nonlinearity
in the model, i.e gives each particle some potential. It is generated by monotonous increasing
C∞-function F, F (0) = 0, which satisfies the condition of no more than polynomial growth
∃ k > −1 ∀ i > 1 |F (i)(x) − F (i)(y)| 6 Ci|x− y|(1 + |x| + |y|)k. (2.2)
We are going to demonstrate that nonlinearity directly effects the regularity properties of ξx(t), Pt
and the structure of topologies in the spaces of their regularity.
Let us remark that the above conditions guarantee the solvability of equation (2.1) for the
initial data x ∈ `2(a) [1]. Therefore, the associated Feller semigroup is constructed as a mean with
respect to the product Wiener measure
(Ptf)(x) = E(f(ξx(t))). (2.3)
Its generator may be calculated on the C∞-function f with compact support, that depend on the
finite number of variables xk by formula:
[Hf ](x) =
∑
k∈Zd
−
1
2
∂2
k +
F (xk) +
∑
j∈Zd
bk−jxj
∂k
f(x), (2.4)
where we introduced a notation ∂k = ∂/∂xk for partial derivative. Since each coordinate of solu-
tion ξx(t) fulfills the equation (2.1), the representation of generator (2.4) follows from the finite
dimensional Ito formula, applied to the finite number of coordinate processes ξx
k .
It is important that operator H also arises as energy operator
(Hu, u)L2(µ) =
1
2
∫
RZd
∑
k∈Zd
|∂ku(x)|
2dµ(x)
of Gibbs lattice measure µ of the form:
dµ(x) =
1
Z
exp
−
∑
i,j∈Zd: |i−j|6r0
bi−jxixj
∏
k∈Zd
e−Φ(xk)dxk, Φ(xk) = 2
∫ xk
0
F (z)dz. (2.5)
Measure µ describes the model of anharmonic crystal with a finite radius of interaction r0. In
particular, measure µ represents one of the possible lattice approximations to the Euclidean field
models with interaction.
7
A.V.Antoniouk, A.V.Antoniouk
To obtain regularity properties of process ξx(t) (2.1) and semigroup Pt (2.3) let us find the
representation for partial derivatives of semigroup ∂τPtf , where τ = {k1, . . . , kn} and ∂τ =
∂|τ |/∂xk1
. . . ∂xkn
. The formal successive differentiation of (2.3) leads to
∂τ (Ptf)(x) =
m∑
s=1
∑
γ1∪···∪γs=τ
E
〈
∂(s)f(ξx(t)), ξx
γ1
(t) ⊗ · · · ⊗ ξx
γs
(t)
〉
, (2.6)
where ∂(s)f denotes a set of all sth order partial derivatives of function f : ∂(s)f = {∂γf}|γ|=s for
∂γf = ∂j1 . . . ∂js
f with γ = {j1, . . . js}, and we used the notation
〈
∂(s)f(ξx(t)), ξx
γ1
⊗ · · · ⊗ ξx
γs
〉
=
∑
i1,..,is∈Zd
(
∂{i1,...,is}f
)
(ξx(t)) ξx
i1,γ1
. . . ξx
is,γs
.
In (2.6) summation
∑
γ1∪···∪γs=τ over on all possible subdivisions of the set τ = {k1, . . . , kn}, ki ∈
Z
d over the nonempty nonintersecting subsets γ1, . . . , γs ⊂ τ , with |γ1|+ · · ·+ |γs| = |τ |, s > 2. In
Theorem 4 we will precise a class of functions f for which representation (2.6) becomes rigorous.
Vector ξx
τ = {ξx
i,τ}i∈Zd in (2.6) is derivative of ξx(t) with respect to the initial data x = {xk}k∈Zd
ξx
i,τ =
∂|τ |ξx
i (t)
∂xkn
. . . ∂xk1
(2.7)
and is called hereinafter a τ th variation of ξx(t).
The equation for ξτ is derived by the formal successive differentiation of (2.1) with respect to
x:
dξx
i,τ
dt
= −F ′(ξx
i )ξx
i,τ −
∑
j: |j−i|6r0
bj−iξ
x
j,τ − ϕx
i,τ ;
ξk,τ (0) = xk,τ .
(2.8)
where ϕx
i,τ = ϕx
i,τ (ξx, ξx
·,γ , γ ⊂ τ, γ 6= τ).
ϕx
i,τ =
∑
γ1∪···∪γs=τ, s>2
F (s)(ξx)ξi,γ1
. . . ξi,γs
. (2.9)
A precise sense to expression (2.7) as a solution to (2.8) can be given only under the special
choice of initial data
x̃k,τ =
{
δk,j , |τ | = 1, τ = {j} ⊂ Z
d;
0, |τ | > 1.
(2.10)
Let us turn the attention of the reader to the equation (2.8) which is a linear nonautonomous and
inhomogeneous equation with respect to variation ξτ . Its inhomogeneous part depends on the lower
order variations ξγ , γ ⊂ τ and, displays symmetry (1.2) just like the r.h.s. of (2.6).
Representation (2.6) gives the relation between the partial derivatives of semigroup (2.3) and
the behaviour of variations ξτ with respect to the initial data x, {x̃γ}γ⊆τ . Therefore, to construct a
semigroup Pt in the spaces of continuously differentiable functions we have to study the variations
ξx
τ of process ξx(t), i.e. its differentiability with respect to the initial data x.
The key idea is that for variational equation (2.8) symmetry (1.2) becomes proportionality:
variation ξτ , τ = {k1, . . . , kn}, in the r.h.s. of (2.8) is proportional to the product of first order
variations
n∏
i=1
ξ{ki} in the r.h.s. of (2.8).
Taking into account this observation we introduce a special nonlinear expression
ρτ (ξ; t) = E
n∑
s=1
ps(zt)
∑
γ⊆τ, |γ|=s
||ξγ ||
mγ
`mγ (cγ)
(2.11)
that reflects this symmetry and, in nonlinear manner, takes into consideration the regularity of the
process ξx(t) with respect to the initial data. Above τ = {k1, . . . , kn}, ki ∈ Z
d, zt = ||ξx(t)||2`2(a),
8
Regularity properties of evolutions of unbounded spin systems
mγ = m1/|γ|, |γ| is a number of points in set γ ⊆ Z
d and each ps is increasing C∞ polynomial
that fulfills the condition: ∃ ε > 0, ∃K > 0, such that ∀ z ∈ R+
ps(z) > ε & (1 + z) (|p′s(z)| + |p′′s (z)|) 6 Kps(z). (2.12)
It should be noted that in (2.11) we use notations `p(a) and `mτ
(cτ ) for the spaces, where initial
process ξx(t) and variational processes ξτ (t) are considered. The main reason is that the process
ξx(t) can be constructed only in space `p(a) with
∑
k∈Zd ak <∞. On the other hand, due to initial
data (2.10), the choice of weights cτ ∈ P is quite arbitrary.
The following theorem states an a priori estimate on any order regularity of the process ξx
t .
Theorem 1 Let F satisfy (2.2), m1 > |τ | be fixed, mγ = m1/|γ| and ξx, {ξτ}γ⊆τ be strong
solutions to systems (2.1), (2.8). Suppose that functions ps(z), s = 1, . . . , n and vectors {cγ}γ⊆τ ⊂
P in (2.11) fulfill:
(1) ∃Kp ∀ j = 2, . . . , n ∀ i1, . . . , is : i1 + · · · + is = j, s > 2
[pj(z)]
j(1 + z)
k+1
2
m1 6 Kp · [pi1(z)]
i1 . . . [pis
(z)]is ; (2.13)
(2) for any subdivision of the set γ = α1 ∪ · · · ∪ αs, γ ⊂ τ on nonempty nonintersecting subsets
α1, . . . , αs, s > 2 there are constants Rγ,α1,...,αs
such that ∀ k ∈ Z
d
[ck,γ ]|γ|a
−k+1
2
m1
k 6 Rγ,α1,...,αs
[ck,α1
]|α1| . . . [ck,αs
]|αs|. (2.14)
Upper indexes outside the brackets [· · · ] mean powers and parameter k is introduced in (2.2).
Then there is a constant M ∈ R
1, such that the nonlinear estimate of exponential type on
the a priori regularity of process ξx
t holds
ρτ (ξ; t) 6 eMtρτ (ξ; 0). (2.15)
Let us remark that the set of functions pi and vectors cτ , which satisfy the conditions (2.13),
(2.14), is sufficiently large. First of all, for pi and cτ that fulfill (2.13), (2.14) function q · pi and
vector d · cτ = {dkck,τ}k∈Zd again fulfill (2.13) and (2.14), where d ∈ P and q fulfills (2.12). An
example may be given by
p̃i = q(z)(1 + z)
k+1
2
(m1/i−m1/|τ |),
c̃k,γ = a
k+1
2
m1
|γ|−1
|γ|
k
∏
j∈τ
ψ
m1/|γ|
k−j , γ = {j1, . . . , js} (2.16)
with some polynomial q and vector ψ = {ψk}k∈Zd ∈ P. They fulfill (2.13) and (2.14) with constants
Kp = Rγ;α1,...αs
= 1. Indeed, due to ak 6 1
a
−k+1
2
m1
k [c̃k,τ ]|τ | = a
k+1
2
m1(|τ |−2)
k
∏
j∈τ
ψm1
k−j 6 a
k+1
2
m1(|τ |−s)
k
∏
j∈τ
ψm1
k−j
=
s∏
i=1
[a
k+1
2
m1
|γi|−1
|γi|
k
∏
b∈γi
ψ
m1/|γi|
k−b ]|γi| = [c̃k,γ1
]|γ1| . . . [c̃k,γs
]|γs|, (2.17)
where τ = γ1 ∪ · · · ∪ γs, |γ1| + · · · + |γs| = |τ |, s > 2. Similar calculation holds for p̃i.
Proof. We apply Ito formula to the expression ρτ (2.11), then we use symmetries (1.2) and
hierarchies (2.13), (2.14) with further application of Gronwall-Bellmann inequality. Introduce no-
tations
gγ(t) = E
[
p|γ|(zt)||ξγ(t)||
mγ
`mγ (cγ)
]
,
9
A.V.Antoniouk, A.V.Antoniouk
where zt = ||ξx(t)||2`2(a), and
hi
τ (ξ; t) =
i∑
s=1
∑
γ⊆τ, |γ|=s
gγ(t)
for i > 1, h0
τ (ξ; t) = 0 for i = 0. We prove inductively that
∀ i = 1, .., n ∃Mi ∈ R hi
τ (ξ; t) 6 eMithi
τ (ξ; 0), (2.18)
which at i = n gives the statement of theorem. If for any γ ⊆ τ, |γ| = i we prove
dgγ(t)
dt
6 K1gγ(t) +K2h
i−1
τ (ξ; t), (2.19)
then Gronwall-Bellmann inequality implies (2.18):
hi
τ (ξ; t) 6 eMi−1thi−1
τ (ξ; 0) +
∑
γ⊆τ, |γ|=i
{
eK1tgγ(0) +K2
∫ t
0
eK1(t−s)eMi−1shi−1
τ (ξ; 0)ds
}
6 e(Mi−1+K1)t
(
1 + 2|τ |K2t
)
hi
τ (ξ; 0) 6 e(Mi−1+K1+2|τ|K2)thi
τ (ξ; 0).
To prove (2.19) let us assume that processes ξx(t), ξγ(t), γ ⊆ τ are strong solutions to equations
(2.1) and (2.8). Therefore ξx(t) is a sum of Wiener process and finite variation part and ξγ(t), γ ⊆ τ
are processes of finite variation. As a consequence, Ito formula can be applied to the expression
gγ(t), e.g. [1,3,4,18]. It gives
pi(zt)||ξγ(t)||
mγ
`mγ (cγ) = pi(z0)||ξγ(0)||
mγ
`mγ (cγ) + 2
∫ t
0
p′i(zs)||ξγ(s)||
mγ
`mγ (cγ) (ξx(s), dW (s))
+
∫ t
0
{
mγpi(zs)
〈
dξγ(s)
ds
, [ξγ(s)]
#
〉
`mγ (cγ)
− ||ξγ ||
mγ
`mγ (cγ)(Hp)(zs)
}
ds. (2.20)
Here
(x, y) =
∑
k∈Zd
akxkyk, 〈u, v#〉`m(c) =
∑
k∈Zd
ckukvk · |vk|
m−2
for v# = ||v||m−2
`m(c)Fv with duality map F in space `m(c) and operator H is defined in (2.4).
Inequality
|〈f, ξ#〉| 6
1
m
||f ||m`m(c) +
m− 1
m
||ξ||m`m(c),
property F ′(x) > 0, x ∈ R, the boundedness of map B in any space `p(c), p > 1, δc < ∞ and
inequality Hpi(zt) > −Mpi
pi(zt) (see [10, Hint 9]) imply
dgγ(t)
dt
6 const gγ(t) +
∑
α1∪···∪αs=γ, s>2
Epi(||ξ
x||2`2(a)) ||F (s)(ξx)ξα1
. . . ξαs
||
mγ
`mγ (cγ). (2.21)
Due to (2.2) we have
|F (s)(ξx
k )| 6 C(1 + |ξx
k |)
k+1
6 C · a
−k+1
2
k (1 + ||ξx(t)||2`2(a))
k+1
2 .
As mγ = mα · |α|/|γ| we get
|ξk,α1
|mγ · · · |ξk,αs
|mγ = [|ξk,α1
|mα1 ]
|α1|/|γ| · · · [|ξk,αs
|mαs ]|αs|/|γ|.
10
Regularity properties of evolutions of unbounded spin systems
Therefore, properties (2.13) and (2.14) imply the estimate on each term in (2.21):
(2.21) 6 C KpRγ,α1..αs
E
∑
k∈Zd
s∏
i=1
{
p|αi|(||ξ
x||2`2(a))ck,αi
|ξk,αi
|mαi
}|αi|/|γ|
. (2.22)
Inequality |x1 . . . xs| 6
|x1|
q1
q1
+ · · · + |xs|
qs
qs
with qj = |γ|/|αj| implies
(2.22) 6 C KpRγ,α1..αs
E
s∑
j=1
|αj |
|γ|
p|αj|(||ξ
x||2`2(a)) ||ξαj
||
mαj
`mαj
(cαj
)
6 C KpRγ,α1,..,αs
hi−1
τ (ξ; t). (2.23)
In the last inequality we assumed that for subdivision α1 ∪ · · · ∪ αs = γ, |γ| = i at s > 2 we have
|αj | 6 i− 1. Therefore (2.19) is proved.
3. C
∞-regularity of semigroup Pt
Now we can discuss the structure of topologies in spaces, in which the differentiability properties
of semigroup Pt hold. It is determined by nonlinear estimate (2.15) and essentially depends on the
order of nonlinearity k of map F , which is reflected in special hierarchy of weights in seminorms. In
[10] it is demonstrated that such hierarchy of weights is non-void, if the semigroup is constructed
in the spaces of differentiable functions.
Introduce Banach space Lipr(`2(a)), r > 0, equipped with norm
||f ||Lipr
= sup
x∈`2(a)
|f(x)|
(1 + ||x||`2(a))r+1
+ sup
x,y∈`2(a)
|f(x) − f(y)|
||x− y||`2(a)(1 + ||x||`2(a) + ||y||`2(a))r
<∞. (3.1)
For m ∈ IN , we denote a finite array of weights {(q,G) : (q,G) ∈ Θm} by Θm, where G =
G1 ⊗ · · · ⊗ Gm is m-tensor constructed from vectors Gi ∈ P, i = 1, . . . ,m and q is a smooth
function that fulfills (2.12).
Definition 2 Let r > 0, n > 1 and Θ = Θ1 ∪ · · · ∪ Θn, Θi 6= ∅, i = 1, . . . , n be a finite array
of weights. Function f belongs to the space of continuously differentiable functions CΘ,r(`2(a)) iff
f ∈ Lipr(`2(a)) and
1) for any m ∈ {1, . . . , n} and τ = {j1, . . . , jm}, ji ∈ Z
d, there is a continuous partial derivative
∂τf ∈ C(`2(a),R
1). These derivatives fulfill integral relations: ∀x ∈ `2(a), ∀h ∈ X∞([a, b])
f(x+ h(·))
b
a
=
∫ b
a
ds
∑
k∈Zd
∂kf(x+ h(s))h′k(s) (3.2)
and ∀ τ = {j1, . . . , j`}, |τ | = ` 6 n− 1
∂τf(x+ h(·))
b
a
=
∫ b
a
ds
∑
k∈Zd
∂τ∪{k}f(x+ h(s))h′k(s). (3.3)
Here we used the notation
X∞([a, b]) = ∩
p>1,c∈P
AC∞([a, b], `p(c)) (3.4)
where AC∞([a, b], X) = {h ∈ C([a, b], X) : ∃h′ ∈ L∞([a, b], X} for Banach space X.
11
A.V.Antoniouk, A.V.Antoniouk
2) The norm is finite
||f ||CΘ,r
= ||f ||Lipr
+ max
m=1,n
||∂(m)f ||Θm <∞, (3.5)
where
||∂(m)f ||Θm = sup
x∈`2(a)
max
(qm,Gm)∈Θm
|||∂(m)f(x)|||Gm
qm(||x||2`2(a))
(3.6)
with |||∂(m)f(x)|||2Gm =
∑
τ={j1...jm}⊂Zd
G1
j1 . . . G
m
jm
|∂τf(x)|2 for Gm = G1 ⊗ · · · ⊗Gm.
Remark. Definition of CΘ,r is not transparent at the first glance and we would like to give some
comments.
For fixed ω ∈ Ω, t ∈ [0.T ] the map `2(a) 3 x → ξx(ω, t) ∈ `2(a) and its variations {ξτ} have
nonlinear responses with respect to initial data in representation of ∂τPtf (2.6). This circumstance
motivated us to give a sense (3.2), (3.3) to the derivatives of function f ∈ CΘ,r. It may be considered
as the existence of Frechet derivatives on some projective limit of spaces.
In particular, properties (3.5) and (3.2), (3.3) establish that for function f ∈ CΘ,r there exist
continuous partial derivatives up to the order n. To show this one should take h(t) = tek, t ∈ [0, 1]
with vector ek = {δk
j }j∈Zd in (3.2) and (3.3). Due to the finiteness of norm ||f ||CΘ,r
the r.h.s. of
(3.2) and (3.3) are well-defined for such h.
In the next definition we introduce a special hierarchy of weights in topology of space CΘ,r that
guarantees the regularity properties of semigroup Pt.
Definition 3 Finite array Θ = Θ1 ∪ · · · ∪ Θn, n ∈ IN is subordinated to the nonlinearity of order
k iff ∀m = 2, . . . , n, for any pair (q,G = G1 ⊗ · · ·⊗Gm) ∈ Θm and ∀ i, j ∈ {1, . . . ,m}, i 6= j, there
is a pair (q̃, G̃ = G̃1 ⊗ · · · ⊗ G̃m−1) ∈ Θm−1 such that
∃K : ∀ z ∈ R
1
+ (1 + z)
k+1
2 q̃(z) 6 Kq(z); ∀ ` = 1, . . . ,m− 1 (Ĝ{i,j})`
6 K G̃`. (3.7)
In (3.7) (m− 1)-tensor Ĝ{i,j} is constructed from m-tensor G by rule
Ĝ{i,j} = G1 ⊗ . . . .
î
⊗A−(k+1)GiGj
↑j
⊗..⊗Gm,
with vector A−(k+1) = {a−k+1
k }k∈Zd. Notation G1 ⊗ . . . .
î
⊗Gs means that in tensor product, the
ith – vector is omitted and G1 ⊗ .. ⊗ B
↑j
⊗.. ⊗ Gs means that on the jth place in tensor product,
there is an inserted vector B. For each `, inequality (3.7) is understood as a coordinate inequality
between two vectors (i.e. c = {ck} 6 d = {dk} iff ∀ k ck 6 dk).
Let us remark that the structure of seminorm || · ||Θm and condition (3.7) on Θ is dictated by
nonlinear estimate and guarantees that the exponential estimate on semigroup Pt is scale CΘ,r. In
particular, in [10] it was demonstrated that the hierarchies (3.7) are unavoidable if one wishes to
have a property (3.8) for nonlinear F .
The next Theorem states the continuous differentiability of semigroup Pt on the functions from
scale CΘ,r. Let us note that this result holds for multiparticle interactions in Gibbs measure and
this result will appear elsewhere.
Theorem 4 Let F fulfill (2.2) with parameter k, r > 0 and finite array Θ = Θ1 ∪ · · · ∪ Θn be
subordinated to the nonlinearity of order k. Then ∀ t > 0 : Pt : CΘ,r → CΘ,r and ∃KΘ,r, MΘ,r
∀ f ∈ CΘ,r ||Ptf ||CΘ,r
6 KΘ,re
MΘ,rt||f ||CΘ,r
. (3.8)
In particular, ∀ f ∈ CΘ,r derivatives of semigroup fulfill representation (2.6).
Proof. A detailed proof will appear in [11, Ch.4].
12
Regularity properties of evolutions of unbounded spin systems
Acknowledgements
Authors are grateful to the referee for careful reading the manuscript as well as for making
several suggestions for its improvement. A warm hospitality of organizers during the conference in
Lviv is much appreciated.
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13
A.V.Antoniouk, A.V.Antoniouk
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