THE HEAT EQUATION WITH RANDOM INITIAL CONDITIONS · Teor Imovr. ta Matem. Statist. Theor.ProbabilityandMath.Statist. Vip. 80, 2009 No.80,2010,Pages71–84 S0094-9000(2010)00795-2

Teor�� Imov�r. ta Matem. Statist. Theor. Probability and Math. Statist.Vip. 80, 2009 No. 80, 2010, Pages 71–84

S 0094-9000(2010)00795-2Article electronically published on August 19, 2010

THE HEAT EQUATION WITH RANDOM INITIAL CONDITIONS

FROM ORLICZ SPACES

UDC 519.21

YU. V. KOZACHENKO AND K. I. VERESH

Abstract. Conditions for justification of the Fourier method for parabolic equationswith random initial conditions from Orlicz spaces of random variables are obtained.Bounds for the distribution of the supremum of solutions of such equations are found.

We study conditions justifying the application of the Fourier method for parabolicequations with random initial conditions and obtain bounds for the distribution of thesupremum of solutions of these equations. Similar problems for hyperbolic equations areconsidered in [1, 2]. A survey of the corresponding results can be found in [3, 4]. In whatfollows we consider random initial conditions from the Orlicz spaces of random variables.

The paper is organized as follows. Section 1 contains necessary definitions and resultsof the theory of the Orlicz space. The setting of the problem as well as statements ofthe main results of the paper is given in Section 2. Conditions for the convergence ofstochastic processes in C(T ) and bounds for the distribution of the supremum of solutionsof the corresponding equations are presented in Section 3. The proofs of the main resultsare placed in Section 4.

1. Stochastic processes belonging to an Orlicz space

Definition 1.1 ([3]). An even, continuous, convex function U(x) such that U(x) > 0for x �= 0 is called a C-function.

Let {Ω,�,P} be a standard probability space.

Definition 1.2 ([4]). The space LU (Ω) of random variables ξ(ω) = ξ, ω ∈ Ω, is calledthe Orlicz space generated by a C-function U(x) if, for any ξ ∈ LU (Ω), there exists aconstant rξ such that EU(ξ/rξ) < ∞.

The Orlicz space LU (Ω) is a Banach space with respect to the norm

(1) ‖ξ‖LU= inf

{r > 0: EU

(ξ

r

)≤ 1

}.

Definition 1.3. Let X(t) = {X(t), t ∈ T} be a stochastic process. We say that Xbelongs to the Orlicz space LU (Ω) if, for all t ∈ T , the random variable X(t) belongs tothe space LU (Ω).

Lemma 1.1 ([3]). Let ξ ∈ LU (Ω) and let EU(ξ/r) ≤ a for some r > 0 and a > 0. Then‖ξ‖LU

≤ rmax{0; a}.

2000 Mathematics Subject Classification. Primary 60G60, 60G17.Key words and phrases. Parabolic equations, Orlicz spaces.

c©2010 American Mathematical Society

71

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72 YU. V. KOZACHENKO AND K. I. VERESH

Lemma 1.2 ([3]). If ξ ∈ LU (Ω), then, for all x > 0,

P{|ξ| > x} ≤ 1

U(

x‖ξ‖LU

) .Definition 1.4. We say that a C-function U is subordinate to a C-function V anddenote U ≺ V if there exist two numbers x0 ≥ 0 and C > 0 such that U(x) ≤ V(Cx) forall x such that |x| > x0. We say that two C-functions U(x) and V(x) are equivalent ifU(x) ≺ V(x) and V(x) ≺ U(x).

Theorem 1.1 ([3]). Let a C-function U be subordinate to a C-function V. Then

LV(Ω) ⊂ LU (Ω)

and there exists a constant K such that

‖ξ‖LU≤ K‖ξ‖LV

for all ξ ∈ LV(Ω). If two C-functions U(x) and V(x) are equivalent, then LU (Ω) = LV(Ω)and the norms ‖ · ‖LU

and ‖ · ‖LV are equivalent, too.

Definition 1.5 ([5]). Let U(x) be a C-function such that V(x) = x2 is subordinate tothe function U(x). A family Δ of centered random variables (E ξ = 0, ξ ∈ Δ) from theOrlicz space LU (Ω) is called a strictly Orlicz family if there exists a constant CΔ suchthat ∥∥∥∥

∑i∈I

λiξi

∥∥∥∥LU

≤ CΔ

(E

(∑i∈I

λiξi

)2)1/2

for all finite collections of random variables ξi ∈ Δ, i ∈ I, and for all λi ∈ R1, i ∈ I.

Definition 1.6. A stochastic process x = {x(t), t ∈ T}, x ∈ LU (Ω), is called a strictlyOrlicz process if the collection of the random variables x = {x(t), t ∈ T} is a strictly Orliczfamily. Two stochastic processes x = {x(t), t ∈ T} and y = {y(t), t ∈ T} are called jointlystrictly Orlicz processes if the collection of the random variables {x(t), y(t), t ∈ T} is astrictly Orlicz family.

Definition 1.7 ([3]). We say that the g-condition holds for a C-function U if there aresome constants z0 ≥ 0, k > 0, and A > 0 such that

U(x)U(y) ≤ AU(kxy)

for all x ≥ z0 and y > z0.

2. Setting of the problem and main results

Consider a boundary value problem for a parabolic equation with two independentvariables 0 ≤ x ≤ π and t ≥ 0, namely

∂

∂x

(p∂V

∂x

)− qV − ρ

∂V

∂t= 0,(2)

V (t, 0) = 0, V (t, π) = 0,(3)

V (x, t)∣∣t=0

= ξ(x),(4)

where ξ(x) is a continuous with probability one stochastic process belonging to the Orliczspace LU (Ω).

The functions p = (p(x), x ∈ [0, π]), q = (q(x), x ∈ [0, π]), and ρ = (ρ(x), x ∈ [0, π]) inequation (2) are such that

1) p(x) > 0, q(x) ≥ 0, and ρ(x) > 0 for all x ∈ [0, π];2) ρ(x) and p(x) are twice continuously differentiable functions on [0, π];


HEAT EQUATION WITH RANDOM INITIAL CONDITIONS 73

3) q(x) is a continuously differentiable function on [0, π].

We now state the two main results of the paper.

Theorem 2.1. Let the initial condition ξ = {ξ(x), x ∈ [0, π]} on the right hand sideof (4) be a strictly Orlicz stochastic process belonging to the Orlicz space LU (Ω) of randomvariables, where U(x) is a C-function such that the function V (x) = x2 is subordinate toU(x) and condition g holds for U(x). Assume that the stochastic process ξ is separableand mean square continuous, E ξ(x) = 0, and E ξ(x)ξ(y) = B(x, y). Let

(5) V (t, x) =

∞∑k=1

ξke−λktXk(x),

where Xk(x) are eigenfunctions and λk are eigenvalues of the Sturm–Liouville problem

L(v) =d

dx

(pdv

dx

)− qv + λρv = 0,(6)

v(0) = 0, v(π) = 0,(7)

ξk =

∫ π

0

Xk(x)ξ(x)ρ(x) dx.

Let there exist a continuous increasing function ϕ = {ϕ(λ), λ > 0} such that ϕ(λ) > 0 forλ > 0 and Ψ(λ) = λ/ϕ(λ), λ > 0, increases for λ ≥ v0, where v0 is a certain constant.We also assume that

(8) sup|x−y|≤h

(E(ξ(x)− ξ(y))2

)1/2 ≤ C

(ϕ

(1

h+ v0

))−1

.

Moreover, let

(9)

∫ ε

0

U (−1)

(π

2

(ϕ(−1)

(c

v

)− v0

)+ 1

)dv < ∞

for all ε > 0 and c > 0.If the series

(10)∞∑k=1

∞∑l=1

|E ξkξl|ϕ(λk + v0)ϕ(λl + v0)

converges, then, for all σ > 0, the series∞∑k=1

ξke−λktλm

k X(s)k (x) = Sms(t, x)

converges with probability one and uniformly in 0 ≤ x ≤ π and t ≥ σ, where X(s)k (x)

denotes the derivative of order s in x. (The following combinations of the parameters sand m are possible: s ∈ {0, 1, 2} if m = 0 or s = 0 if m = 1; the corresponding series arethe derivatives in t and x of orders m and s, respectively, of the random function V (t, x)defined by the right hand side of (5).)

Assume further that the function V (t, s) satisfies with probability one equation (2) inthe domain 0 ≤ x ≤ π, t ≥ σ and that condition 2) holds. Moreover let

V (t, x) → ξ(x) as t → 0

in probability and uniformly with respect to x ∈ [0, π].If the function ϕ is such that

(11)

∫ ε

0

U (−1)

((π

2

(ϕ(−1)

(c

v

)− v0

)+ 1

)2)dv < ∞



for all ε > 0 and c > 0 and moreover

(12)∞∑k=1

∞∑l=1

|E ξkξl|ϕ(λk + v0)ϕ(λl + v0) < ∞,

then V (t, x) → ξ(x) with probability one as t → 0 and uniformly with respect to x ∈ [0, π].

Theorem 2.2. Let the assumptions of Theorem 2.1 hold. Moreover let conditions (11)and (12) be satisfied for the function ϕ. Denote VT = {0 ≤ x ≤ π, 0 ≤ t ≤ T}. Then

(13) P

{sup

t,x∈VT

|V (t, x)| > ε

}≤(U

(ε

B(θ)

))−1

,

where

B(θ) =1

θ(1− θ)

∫ ω0θ

0

χU

((π

2

[ϕ−1

(R

v

)− v0

]+ 1

)(T

2

[ϕ−1

(R

v

)− v0

]+ 1

))dv.

Here χU is defined by relation (15) below, R = CΔ ·√W√2max(2Cx, L), the meaning

of the constant L is explained in Lemma 4.5,

W =∞∑k=1

∞∑l=1

|E ξkξl|ϕ(λk + v0)ϕ(λl + v0),

CΔ is a constant involved in Definition 1.5, and CX is a constant such that |Xk(x)| < CX .

3. Conditions for the convergence in probability of stochastic processes

in C(T ) and bounds for the distribution of the supremum

Theorem 3.1 (Theorem 3.2 in [6]). Let (T, d) be a compact metric space and let

Xn = {Xn(t), t ∈ T}be a sequence of stochastic processes in LU (Ω). Assume that the g-condition is satisfiedfor the function U and that all the processes Xn(t) are separable in (T, d) and thatρn(t, s) = ‖Xn(t)−Xn(s)‖Lu

. Let

1) ρ(t, s) ≤ z(d(t, s)), where ρ(t, s) = supn≥1 ρn(t, s) and where z = {z(x), x > 0}is a function such that z(x) → 0 as x → 0;

2) Nρ(u) denotes the metric capacity of the space (T, ρ);

3)∫ ε

0U (−1)(Nρ(u)) du < ∞ for all ε > 0.

Then, for all σ > 0,

limε→0

supn≥1

P

⎧⎪⎨⎪⎩ sup

t,s∈Td(t,s)<ε

|Xn(t)−Xn(s)| > σ

⎫⎪⎬⎪⎭ = 0.

Theorem 3.2 ([3]). Let (T, d) be a compact metric space and let C(T ) be the Banachspace of continuous functions equipped with the uniform norm. Let Xn = {Xn(t), t ∈ T},n ≥ 1, be a sequence of separable stochastic processes. If

1) the sequence (Xn(t), n ≥ 1) converges in probability for all t ∈ TS, where TS isan arbitrary set that is dense in T ;

2) for all σ > 0,

limε→0

supn≥1

P

⎧⎪⎨⎪⎩ sup

t,s∈Td(t,s)<ε

|Xn(t)−Xn(s)| > σ

⎫⎪⎬⎪⎭ = 0,



then all processes Xn(t) are continuous with probability one and the sequence Xn(t)converges in probability in C(T ) to X(t) ∈ C(T ).

Theorem 3.3. Let Rk be the k-dimensional Euclidean space,

d(t, s) = max1≤i≤k

|ti − si|,

T = {(t1, . . . , tk) : 0 ≤ ti ≤ Ti, i = 1, . . . , k}, and let Xn = {Xn(t), t ∈ T}, n = 1, 2, . . . ,be a sequence of separable stochastic processes belonging to the Orlicz space LU (Ω), wherethe function U satisfies the g-condition. Assume that

1) for all t ∈ T ,

Xn(t) → X(t) as n → ∞in probability;

2)

supd(t,s)≤h

supn≥1

‖Xn(t)−Xn(s)‖ ≤ σ(h),

where σ = {σ(h), h > 0} is a continuous increasing function such that σ(h) → 0as h → 0;

3) for some ε > 0,

∫ ε

0

U (−1)

(k∏

i=1

(Ti

2σ(−1)(u)+ 1

))du < ∞,

where σ(−1)(u) is the inverse function to σ(u).

Then the processes Xn(t) are continuous with probability one and converge in proba-bility in the space C(T ).

Proof. The proof of this theorem follows from Theorems 3.1 and 3.2, since

Nρ(u) ≤k∏

i=1

(Ti

2σ(−1)(u)+ 1

). �

Theorem 3.4. Let (T, ρ) be a compact metric space and let N(u) denote the metriccapacity of the space (T, ρ), that is, N(u) denotes the minimum number of closed ballsof radius u that cover (T, ρ). Assume that X = {X(t), t ∈ T} is a separable stochasticprocess belonging to the space LU (Ω), where the function U is such that the g-conditionholds. Let there exist an increasing continuous function

σ = σ(h), 0 ≤ h ≤ supt,s∈T

ρ(t, s),

such that

supρ(t,s)≤h

‖X(t)−X(s)‖U ≤ σ(h).

Assume further that

(14)

∫ ε

0

χU

(N(σ(−1)(u)

))du < ∞

for some ε, where

(15) χU (n) =

{n, n < U(z0),

CUU(−1)(n), n ≥ U(z0),



and where CU = K(1 + U(z0))max(1, A), z0, k, and A are the constants involved inDefinition 1.7, and σ(−1)(h) is the inverse function to σ(h). Then the random variablesupt∈T |X(t)| belongs to the space LU (Ω) with probability one and

(16)

∥∥∥∥ supt∈T

|X(t)|∥∥∥∥U

≤ ‖X(t0)‖U +1

θ(1− θ)

∫ w0θ

0

χU

(N(σ(−1)(u)

))du = B(t0, θ),

where t0 is an arbitrary point of the set T and w0 = σ(supt∈T ρ(t0, t)

), 0 < θ < 1.

Moreover,

(17) P

{supt∈T

|X(t)| > ε

}≤(U

(ε

B(t0, θ)

))−1

for all ε > 0.

Proof. This theorem is a particular case of Theorem 2.2 and Lemma 2.3 of [7]. �

Remark 3.1 ([7]). Theorem 3.4 remains valid if the number ω0 on the right hand sideof (16) is replaced by 2 supt∈T ‖X(t)‖U .

Corollary 3.1. Let

T = {(x, t) : 0 ≤ x ≤ b, c ≤ t ≤ d}and ρ((x, t), (x1, t1)) = max(|x− x1|, |t− t1|) in Theorem 3.4. Further let

(18)

∫ ε

0

U (−1)

((b

2σ(−1)(u) + 1+ 1

)(d− c

2σ(−1)(u)+ 1

))du < ∞

for some ε > 0 and

(19)

B(t0, θ) ≤ B(t0, θ)

=∥∥X(t0)

∥∥U+

1

θ(1− θ)

∫ w0θ

0

χU

((b

2σ(−1)(u)+ 1

)(d− c

2σ(−1)(u)+ 1

))du,

where t0 = (x0, t0). Then condition (14) holds. Moreover

(20) P

{supt∈T

|X(t)| > ε

}≤(U

(ε

B(t0, θ)

))−1

for all ε > 0.

Proof. The corollary follows from Theorem 3.4, since

N(u) ≤(

b

2u+ 1

)(d− c

2u+ 1

). �

Remark 3.2. Corollary 3.1 remains valid if the number ω0 on the right hand side of (19)is replaced by 2 supt∈T ‖X(t)‖U .

4. Auxiliary results and proofs of the main theorems

Consider the series∞∑k=1

ξke−λktλm

k X(s)k = Sms(t, x)

for two sets of parameters s and m: either for s ∈ {0, 1, 2} and m = 0 or for s = 0 andm = 1.

Let ξ(x), 0 ≤ x ≤ π, be a strictly Orlicz process such that the series Sms(t, x) convergein the mean square sense for all 0 ≤ x ≤ π and t ≥ 0. Then it follows from [5] that all



Sms(t, x) are strictly Orlicz stochastic processes in the domain 0 ≤ x ≤ π, 0 ≤ t ≤ T ,where T > 0 is an arbitrary number. Moreover, the sums

SmsN (t, x) =

N∑k=1

ξke−λktλm

k X(s)k , N = 1, 2, . . . ,

are also strictly Orlicz processes.

Lemma 4.1. Let ε be an arbitrary positive number. The series Sms(t, x) converge withprobability one uniformly in the domain Dε = [0 ≤ x ≤ π]× [ε,∞).

Proof. According to [8] we have supk=1,∞ sup0≤x≤π

∣∣Xk(x)∣∣ ≤ Cx, where Cx is a constant

and where

Xn(x) = λn

∫ π

0

G(x, s)Xn(s)ρ(s) ds,

G(x, s) =

{u(x)v(s), x ≤ s,

u(s)v(x), x > s.

Here u(x) and v(x), x ∈ [0, π], are some twice continuously differentiable functions forx ∈ [0, π]. Hence

X ′n(x) = λn

∫ π

0

G∗(x, s)Xn(s)ρ(s) ds,(21)

X ′′n(x) = λn

(∫ π

0

G∗∗(x, s)Xn(s)ρ(s) ds+ (v′(x)u(x)− v(x)u′(x))Xn(x)

),(22)

where

G∗(x, s) =

{u′(x)v(s) if x ≤ s,

u(s)v′(x) if x > s,

G∗∗(x, s) =

{u′′(x)v(s) if x ≤ s,

u(s)v′′(x) if x > s.

Thus

|X ′n(x)| ≤ λn

∫ π

0

|G∗(x, s)| · |Xn(s)|ρ(s) ds ≤ λnCx

∫ π

0

|G∗(x, s)| ds ≤ C∗λn,

where C∗ is a constant and where |X ′′n(x)| ≤ λnC

∗∗ for some constant C∗∗ > 0. Hence

supt>ε

0≤x≤π

Sms(t, x) ≤∞∑k=1

|ξk| supt>ε

e−λktλmk

∣∣∣X(s)k (x)

∣∣∣

≤ C

∞∑k=1

|ξk|e−λkελm+1k ,

where we set C = max(C∗, C∗∗, 1). The latter series converges with probability one ifthe series

∑∞k=1 E |ξk|e−λkελm+1

k converges. Since

E |ξk| ≤(EC2

k

)1/2=

(E

(∫ π

0

Xk(x)ξ(x)ρ(x) dx

)2)1/2

=

(∫ π

0

∫ π

0

Xk(x)Xk(y)E ξ(x)ξ(y)ρ(x)ρ(y) dx dy

)1/2

=

(∫ π

0

∫ π

0

Xk(x)Xk(y)B(x, y)ρ(x)ρ(y) dx dy

) 12

≤ b,



where b is a constant, we obtain

(23)∞∑k=1

E |ξk|e−λkελm+1k ≤ b

∞∑k=1

e−λkελm+1k .

According to [8], √λn = dn+ O

(1

n

),

where d is a positive constant. Thus√λn ≤ d1n,

√λn ≥ d2n

for sufficiently large n, where 0 < d2 < d1 are some constants. Thus series (23) converges,since so does the series

∞∑k=1

e−d22εk

2

k2(m+1). �

Lemma 4.2. Let T = [0,∞) and let a function Xλ(u), λ > 0, u ∈ T , be such that

1) supu∈T |Xλ(u)| ≤ B;2)∣∣Xλ(u)−Xλ(v)

∣∣ ≤ Cλ|u− v| for all u, v ∈ T .

Let ϕ(λ), λ > 0, be a continuous increasing function such that ϕ(λ) > 0 for all λ > 0and the function λ/ϕ(λ) is increasing for λ > v0 for some constant v0 ≥ 0. Then

(24) |Xλ(u)−Xλ(v)| ≤ max(C; 2B)ϕ(λ+ v0)

ϕ (|u− v|−1 + v0)

for all λ ≥ 0 and v > 0.

Proof. Let λ ≥ |u− v|−1. Then

|Xλ(u)−Xλ(v)|2B

≤ 1 ≤ ϕ(λ+ v0)

ϕ(|u− v|−1 + v0);

that is, inequality (24) holds.Now let λ ≤ |u− v|−1. Since λ/ϕ(λ) is an increasing function for λ > v0, we get

|Xλ(u)−Xλ(v)| ≤ Cλ|u− v| ≤ Cλ+ v0

|u− v|−1 + v0

≤ Cϕ(λ+ v0)

ϕ(|u− v|−1 + v0);

that is, inequality (24) holds in this case, too. �Lemma 4.3. Let ξ(x) be a continuous with probability one process on [0, T ]. Assumethat

E |V (t, x)− ξ(x)|2 → 0 as t → 0

for all 0 ≤ x ≤ π and that there exists a continuous increasing function σ = (σ(h), 0 <h ≤ π) such that σ(h) → 0 as h → 0 and

(25) sup0<t<t0

sup|x−y|≤hx,y∈[0,π]

(E |V (t, x)− V (t, y)|2

)1/2 ≤ σ(h),

where 0 < t0 < T . If

(26)

∫ ε

0

U (−1)

(π

2σ(−1)(h)+ 1

)dh < ∞

for all ε > 0, then the series V (t, x) converges in probability to ξ(x) as t → 0 uniformlyin the interval [0, π].



Proof. Since V (t, x) is a strictly Orlicz process, we obtain

‖V (t, x)− V (t, y)‖ ≤ CΔ

(E(V (t, x)− V (t, y)

)2)1/2.

Thus Lemma 4.3 follows from Theorem 3.3, since V (t, x) → ξ(x) in probability. �

Remark 4.1. Relation (26) holds if σ(h) = Ch in condition (25).

Proof of Remark 4.1. If U(x) x2, then

U(x) ≥ Cx2

for sufficiently large x > 0, namely for x such that

x ≥ U (−1)(Cx2

).

This means that

U (−1)(t) ≤√

t

C

for sufficiently large t. Thus

∫ ε

0

U (−1)

(π

2σ(−1)(h)+ 1

)dh =

∫ ε

0

U (−1)

(CπC

2h+ 1

)dh

≤∫ ε

0

√π

2h+

1

Cdh ≤

∫ ε

0

√π

2hdh+

1

C

∫ ε

0

dh =√2πε+

ε

C< ∞

for sufficiently small ε, since

σ(−1)(h) =h

C. �

Lemma 4.4. Let ξ(x), 0 ≤ x ≤ π, be a strictly Orlicz stochastic process. Assume thatthe assumptions of Theorem 2.1 hold. Let ϕ(x) be a function such that the assumptionsof Lemma 4.2 hold and let

(27)

∫ ε

0

U (−1)

(π

2

(ϕ(−1)

( ch

)− v0

)+ 1

)dh < ∞,

where c > 0 is an arbitrary constant. If

(28)

∞∑k=1

∞∑l=1

|E ξkξl|ϕ(λk + v0)ϕ(λl + v0) < ∞,

then

V (t, x) → ξ(x) as t → 0

in probability and uniformly in the interval [0, π].

Proof. We derive Lemma 4.4 from Lemma 4.3. First we check the assumptions ofLemma 4.3.

Note that the assumptions of Lemma 4.4 imply that

E |V (t, x)− ξ(x)|2 → 0 as t → 0.



Indeed,

(29)

E |V (t, x)− ξ(x)|2 = E

( ∞∑k=1

ξke−λktXk(x)− ξkXk(x)

)2

= E

( ∞∑k=1

ξkXk(x)(e−λkt − 1

))2

=∞∑k=1

∞∑l=1

E ξkξlXk(x)Xl(x)(e−λkt − 1

) (e−λlt − 1

),

since ξ(x) =∑∞

k=1 ξkXk(x). The function Xλ(t) = e−λt satisfies the assumptions ofLemma 4.2 with B = 1 and C = 1. Since |Xk(x)| ≤ Cx, we get

(30)

E |V (t, x)− ξ(x)|2

≤∞∑k=1

∞∑l=1

|E ξkξlXk(x)Xl(x)|ϕ(λk + v0)ϕ(λl + v0)ϕ−2

(1

t+ v0

)

≤ C2X

∑k=1

∑l=1

|E ξkξl|ϕ(λk + v0)ϕ(λl + v0)ϕ−2

(1

t+ v0

)→ 0 as t → 0.

Now we find a function σ(h) such that

(31) sup0<t<t0

sup|x−x1|≤hx1,x2∈[0,π]

(E |V (t, x)− V (t, x1)|2

)1/2 ≤ σ(h).

It is easy to see that

E(V (t, x)− V (t, x1))2

= E∞∑k=1

(ξkXk(x)e

−λkt − ξkXk(x1)e−λkt

)2

= E

( ∞∑k=1

e−λkt(ξk(Xk(x)−Xk(x1))

))2

=

∞∑k=1

∞∑l=1

e−λkte−λlt(E ξkξl)(Xk(x)−Xk(x1))(Xl(x)−Xl(x1))

≤∞∑k=1

∞∑l=1

e−λkte−λlt |E ξkξl| · |Xk(x)−Xk(x1)| · |Xl(x)−Xl(x1)|

≤∞∑k=1

∞∑l=1

|E ξkξl| · |Xk(x)−Xk(x1)| · |Xl(x)−Xl(x1)|,

Xk(x) = λk

∫ π

0

G(x, y)Xk(y)ρ(y) dy,

G(x, s) =

{u(x)v(y), if x ≤ y,

u(y)v(x), if x > y,



and

|Xk(x)−Xk(x1)| =∣∣∣∣λk

∫ π

0

(G(x, y)−G(x1, y))Xk(y)ρ(y) dy

∣∣∣∣≤ λk

∫ π

0

|G(x, y)−G(x1, y)| · |Xk(y)|ρ(y) dy

≤ λkCX

∫ π

0

|G(x, y)−G(x1, y)|ρ(y) dy,

where

CX = CX sup ρ(x).

LetC ′

v = sup0≤y≤π

|v′(y)|, C ′u = sup

0≤x≤π|u′(x)|,

Cv = sup0≤y≤π

|v(y)|, Cu = sup0≤x≤π

|u(x)|.

Consider the integral

I =

∫ π

0

|G(x, y)−G(x1, y)| dy

=

∫ x1

0

|G(x, y)−G(x1, y)| dy +∫ x

x1

|G(x, y)−G(x1, y)| dy

+

∫ π

x

|G(x, y)−G(x1, y)| dy

= I1 + I2 + I3,

where

I1 =

∫ x1

0

|u(y)(v(x)− v(x1))| dy ≤∫ x1

0

CuC′v|x− x1| dy ≤ πCvC

′u|x− x1|,

I2 ≤∫ x

x1

(sup

0≤y≤π|u(y)| sup

0≤x≤π|v(x)|+ sup

0≤x≤π|u(x)| sup

0≤y≤π|v(y)|

)dy ≤ 2CuCv|x− x1|,

I3 =

∫ π

x

|v(y)(u(x)− u(x1))| dy ≤ πCvC′u|x− x1|,

that is,

I ≤ |x− x1|(CvC′u + 2CuCv + CuCv) ≤ |x− x1|K,

where K is a constant.We have proved that

(32) |Xk(x)−Xk(x1)| ≤ D|x− x1|λk,

where D = (CvC′u + 2CuCv + CuC

′v)CX .

The estimate |Xk(x)| ≤ CX and Lemma 4.2 imply that

(33)∣∣Xk(x)−Xk(x1)

∣∣ ≤ max(2CX , D) · ϕ(λk + v0)

ϕ(|x− x1|−1 + v0),

whence

E(V (t, x)− V (t, y))2 =

∞∑k=1

∞∑l=1

(E ξkξl)(Xk(x)−Xk(y))(Xl(x)−Xl(y))

≤∞∑k=1

∞∑l=1

|E ξkξl| 4D2ϕ(λk + v0)ϕ(λl + v0)ϕ−2

(1

|x− y| + v0

).



Thus

σ(h) = C

(ϕ

(1

h+ v0

))−1

and ∫ ε

0

U (−1)

(π

2σ(−1)(h)+ 1

)dh < ∞

in view of

σ(−1)(h) =1

ϕ(−1)(C/h)− v0.

According to Theorem 3.3, condition (8) implies that the process ξ(x) is continuouswith probability one. Thus all the assumptions of Lemma 4.3 are satisfied. �

Lemma 4.5. Let Xk(x), 0 ≤ x ≤ π, be defined as above. Assume that

(34)∣∣Xk(x)−Xk(x1)

∣∣ ≤ Lλk|x− x1|for 0 ≤ x ≤ π. (We have shown in the proof of Lemma 4.4 that L = D, where D isdefined by (32), although the values of the constant D can be estimated more accuratelyin several particular cases.) Assume that the series

(35)

∞∑k=1

∞∑l=1

|E ξkξl| · ϕ(λk + v0)ϕ(λl + v0) = W

converges, where the function ϕ is defined in Lemma 4.2. Then

supmax(|t−t1|,|x−x1|)≤h

(E(z(t, x)− z(t1, x1))

2)1/2 ≤

√2max(2CX , L)

ϕ(1/h+ v0)

√W

for t1 ≥ t ≥ 0 and 0 ≤ x, x1 ≤ π, where CX is a constant such that Xk(x) < CX andwhere z(t, x) = V (t, x)− ξ(x).

Proof. It is easy to see that

z(t, x)− z(t1, x1) =∞∑k=1

ξk(Xk(x)

(e−λkt − 1

)−Xk(x1)

(e−λkt1 − 1

)),

E(z(t, x)− z(t1, x1))2 =

∞∑k=1

∞∑l=1

E ξkξl · akal ≤∞∑k=1

∞∑l=1

|E ξkξl| · |ak| · |al|,(36)

where

ak =(Xk(x)

(e−λkt − 1

)−Xk(x1)

(e−λkt1 − 1

)).

Note that

|ak| ≤ |Xk(x)−Xk(x1)| ·∣∣e−λkt − 1

∣∣+ |Xk(x1)| ·∣∣e−λkt − e−λkt1

∣∣≤ |Xk(x)−Xk(x1)|+ Cx

∣∣e−λkt − e−λkt1∣∣ .

Lemma 4.2 implies that

|Xk(x)−Xk(x1)| ≤ max(2Cx, L) ·ϕ(λk + v0)

ϕ(|x− x1|−1 + v0).

Since the function e−λt satisfies the assumptions of Lemma 4.2 with C = 1 and B = 1,

|ak| ≤ 2max(2Cx, L)ϕ(λk + v0)

ϕ(1/h+ v0),

where h = max(|t− t1|, |x− x1|) and Lemma 4.5 follows from (36). �



Proof of Theorem 2.1. The almost sure uniform convergence of the series Sms(t, x) withrespect to 0 ≤ x ≤ π and 0 < t < ε is proved in Lemma 4.1. It follows from Lemma 4.4that if condition (10) holds, then

V (t, x) → ξ(x) as t → 0

in probability and uniformly in the interval [0, π]. Now we prove that conditions (11)and (12) imply that

V (t, x) → ξ(x) as t → 0

with probability one and uniformly in the interval 0 ≤ x ≤ π. Consider the stochasticprocess z(t, x) = V (t, x)− ξ(x) in the domain 0 ≤ t, t1 ≤ t ≤ π. Lemma 4.5 implies that

supmax(|t−t1|,|x−x1|)≤h

‖z(t, x)− z(t1, x1)‖LU

≤ CΔ supmax(|t−t1|,|x−x1|)≤h

(E(z(t, x)− z(t1, x1))

2)1/2 ≤ Rϕ

(1

h+ v0

),

where R = CΔ

√W√2max(2CX , L). Thus the assumptions of Corollary 3.1 hold for the

process z(t, x), since σ(h) = Rϕ(1/h+ v0), whence∫ ε

0

U (−1)

((b

2σ(−1)+ 1

)(d− c

2σ(−1)+ 1

))du

=

∫ ε

0

U (−1)

((π

2

(ϕ(−1)

(R

u

)− v0

)+ 1

)(t

2

(ϕ(−1)

(R

u

)− v0

)+ 1

))du

≤∫ ε

0

U (−1)

((π

2

(ϕ(−1)

(R

u

)− v0

)+ 1

)2)

du < ∞.

Therefore

P

{sup

0≤x≤π, 0≤t≤t

|z(t, x)| > y

}≤(U

(y

B(θ)

))−1

for a sufficiently small θ, where

B(θ) = CΔ

(E(z(0, 0))2

)1/2+

1

θ(1− θ)

∫ sθ

0

U (−1)

((π

2

(ϕ(−1)

(Ru

)− v0

)+ 1

)2)du,

s = 2 supt∈Vt‖z(t, x)‖LU

, and Vt = {0 ≤ x ≤ π, 0 ≤ t ≤ t}. Since

‖z(t, x)‖LU≤ CΔ

√E(z(t, x))2,

relation (30) implies that

‖z(t, x)‖LU≤ g

1

ϕ(1/t+ v0),

where

g = CΔCX

( ∞∑k=1

∞∑l=1

|E ξkξl|)

· ϕ(λk + v0)ϕ(λl + v0)1/2

and E(z(0, 0))2 = 0.

Thus B(θ) → 0 as t → 0, whence

P

{sup

0≤x≤π, 0≤t≤t

|Z(t, x)| > ε

}−−−→t→0

0,

that is,

sup0≤x≤π, 0≤t≤t

|Z(t, x)| −−−→t→0

0



in probability. Since the function sup0≤x≤π, 0≤t≤t |Z(t, x)| decreases with t, we haveproved that

sup0≤x≤π, 0≤t≤t

|Z(t, x)| −−−→t→0

0

with probability one, that is, V (t, x) → ξ(x) with probability one and uniformly in0 ≤ x ≤ π. �Proof of Theorem 2.2. Theorem 2.2 follows from Corollary 3.1 and from the proof ofTheorem 2.1 by putting t0 = (0, 0) in Corollary 3.1. �

5. Concluding remarks

Conditions justifying an application of the Fourier method for parabolic equationswith initial conditions that are stochastic processes belonging to the Orlicz spaces ofrandom variables are obtained in the paper. Some bounds for the distribution of thesupremum of solutions of such equations are found.

Bibliography

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7. Yu. V. Kozachenko and M. M. Perestyuk, On the uniform convergence of wavelet expansionsof random processes belonging to the Orlicz spaces of random variables. I, Ukr. Matem. Zh. 59(2007), no. 12, 1647–1659; English transl. in Ukrain. Math. J. 59 (2007), no. 12, 1850–1869.MR2411593 (2009b:60058)

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Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics

and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 2,

Kiev 03127, Ukraine

E-mail address: [email protected]

Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics

and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 2,

Kiev 03127, Ukraine

E-mail address: [email protected]

Received 2/FEB/2009

Translated by OLEG KLESOV


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THE HEAT EQUATION WITH RANDOM INITIAL CONDITIONS · Teor Imovr. ta Matem. Statist. Theor.ProbabilityandMath.Statist. Vip. 80, 2009 No.80,2010,Pages71–84 S0094-9000(2010)00795-2