We consider a Cauchy problem for stochastic heat equation driven by a real harmonizable fractional stable process Z with Hurst parameter H>1/21/2$]]> and stability index α>11$]]>. It is shown that the approximations for its solution, which are defined by truncating the LePage series for Z, converge to the solution.
Partial differential equations with randomness are widely used to model physical, chemical, biological phenomena, financial asset prices, economical processes, etc. The popularity of such models is due to the combination of deterministic and stochastic features among their characteristics.
The majority of existing literature is devoted to the case where the random noise has some Gaussian or sub-Gaussian distribution. To mention only few papers, a heat equation with Gaussian noise was considered in [1, 2, 8, 14], and a wave equation in [3, 9, 14]. Equations with sub-Gaussian measures were intensively studied in [6, 7]. The articles [10, 11] consider equations with general stochastic measures.
The research carried out in the cited articles does not allow one to consider phenomena where the randomness has a heavy-tailed distribution. But heavy tails are ubiquitous when modeling extreme risks, so it is quite important to consider equations with heavy-tailed noise.
The main object of this article is a stochastic heat equation in which the source of randomness is a real harmonizable fractional stable process Z. The solution is understood in the mild sense, with the integral defined pathwise as a fractional integral [15].
We consider approximations for the solution of this equation, which are obtained by truncating the LePage representation series of Z. The main result of this paper is that such representations converge to the true solution.
The paper is organized as follows. Section 2 contains basic facts about stable random variables and related processes. It also establishes an auxiliary analytical lemma. In Section 3, we formulate and prove the main result of this article.
PreliminariesStable random variables and related processes
In this paper, we consider only symmetric α-stable (SαS) random variables with α∈(1,2). We further provide basic information about such variables and related objects; for a more detailed exposition, we refer the reader to [13].
A random variable ξ is symmetric α-stable (SαS) with scale parameter σα,σ≥0, if it has the characteristic function
E[eiλξ]=e−|σλ|α.
Given some linear space of SαS random variables, the scale parameter is a quasi-norm on this space, denoted ‖·‖α.
To construct families of stable random variables, in particular, stable random processes, one frequently uses some stable random measures. We will be interested in the so-called complex rotationally invariant SαS measure μ on R. By definition this is a complex-valued random measure on B(R) with the following properties:
for any Borel set A∈B(R), the random variable Reμ(A) is SαS with scale parameter equal to λ(A), the Lebesgue measure of A;
for any A∈B(R), the random variable μ(A) is rotationally invariant, that is, for any θ∈R, the distribution of eiθμ(A) coincides with that of μ(A);
for any disjoint sets A1,…,An∈B(R), the random variables μ(A1),…, μ(An) are independent.
For a function f:R→C with
‖f‖Lα(R)α=∫R|f(x)|αdx<∞,
it is possible to define the stochastic integral
I(f)=∫Rf(x)μ(dx)
such that ReI(f) is an SαS random variable with scale parameter ‖f‖Lα(R)α. In other words,
‖ReI(f)‖αα=‖f‖Lα(R)α,
that is, the real part of the stochastic integral I(·) maps Lα(R) isometrically into some family of SαS random variables.
Let now T be a parametric set. For a measurable function f:T×R→C such that f(t,·)∈Lα(R) for all t∈T, we may define the random field {Z(t),t∈T} by
Z(t)=Re∫Rf(t,x)μ(dx).
This random field Z(t) has the so-called LePage series representation constructed as follows. Let φ be an arbitrary positive probability density on R, and let the independent families Γk,k≥1,ξk,k≥1,gk,k≥1, of random variables satisfy:
Γk,k≥1, is a sequence of Poisson arrival times with unit intensity;
ξk,k≥1, are independent random variables having density φ;
gk,k≥1, are independent complex-valued rotationally invariant Gaussian random variables1
Note that the rotational invariance implies E[gk]=0.
with E[|Regk|α]=1.
Then the random field Z(t),t∈T, defined by (1) has the same finite-dimensional distributions as
Z′(t)=CαRe∑k≥1Γk−1/αφ(ξk)−1/αf(t,ξk)gk,
where
Cα=(Γ(2−α)cosπα21−α)1/α;
the series converges almost surely for all t∈T (see [5, Lemma 1] and [13, Theorem 1.4.2]).
In the rest of our paper, C denotes a generic constant whose value may change from line to line; Ca,b,… denotes a constant depending on a,b,….
Fractional integration
We will use the pathwise fractional integration; for more detail, see [12, 15]. Let functions f,g:[a,b]→R be such that, for some β∈(0,1), the following fractional derivatives are defined:
(Da+βf)(x)=1Γ(1−β)(f(x)(x−a)β+β∫axf(x)−f(u)(x−u)1+βdu)1(a,b)(x),(Db−1−βg)(x)=1Γ(β)(g(x)(b−x)1−β+(1−β)∫xbg(x)−g(u)(x−u)2−βdu)1(a,b)(x).
Provided that Da+βf∈L1[a,b] and Db−1−βgb−∈L∞[a,b], where gb−(x)=g(x)−g(b), the fractional integral is defined as
∫abf(x)dg(x)=∫ab(Da+βf)(x)(Db−1−βgb−)(x)dx.
It is worth mentioning that for f∈Cν[a,b] and g∈Cμ[a,b] with μ+ν>11$]]>, the fractional integral ∫abf(x)dg(x) is well defined for any β∈(1−ν,μ) and equals the limit of Riemann sums.
Estimates of Fourier-type integrals
The following result specifies the rate of convergence in the Riemann–Lebesgue lemma. It may be known that, for example, for periodic functions and integer parameter, this is Zygmund’s theorem; however, we failed to find it in the literature. Moreover, a similar reasoning will be used later in the proof of our main results, so we found it suitable to present its proof.
Letf∈C[a,b]andh:[0,+∞)→[0,+∞)be a nondecreasing function such that|f(x)−f(y)|≤h(|x−y|)for allx,y∈[a,b]. Then, for any nonzeroλ∈R,|∫abf(x)eiλxdx|≤3(b−a)h(|λ|−1)+2|λ|−1supx∈[a,b]|f(x)|.
If |λ|≤(b−a)−1, then
|∫abf(x)eiλxdx|≤∫ab|f(x)eiλx|dx≤supx∈[a,b]|f(x)|(b−a)≤|λ|−1supx∈[a,b]|f(x)|.
Otherwise, set n=[|λ|(b−a)]+1, so that |λ|(b−a)≤n≤2|λ|(b−a). Consider the equipartition of [a,b] by points xk=a+(b−a)k/n, k=0,…,n. Then, for any |λ|≥(b−a)−1, the following relations hold:
|∫abf(x)eiλxdx|≤|∑k=1n∫xk−1xkf(xk)eiλxdx|+|∑k=1n∫xk−1xk(f(x)−f(xk))eiλxdx|≤|∑k=1nf(xk)eiλxk−eiλxk−1iλ|+∑k=1n∫xk−1xk|f(x)−f(xk)|dx≤|∑k=1nf(xk)eiλxkλ−∑k=0n−1f(xk+1)eiλxkλ|+h(b−an)(b−a)≤|f(b)eiλbλ−f(a)eiλaλ|+|∑k=1n−1(f(xk)−f(xk+1))eiλxkλ|+h(b−an)(b−a)≤2supx∈[a,b]|f(x)||λ|+∑k=1n−11|λ|h(b−an)+h(b−an)(b−a)≤2supx∈[a,b]|f(x)||λ|+∑k=1n−12(b−a)nh(1|λ|)+h(1|λ|)(b−a)≤2supx∈[a,b]|f(x)||λ|+3(b−a)h(1|λ|).
□
Stochastic heat equation with stable noise and its approximations
Consider a Cauchy problem for the one-dimensional heat equation
dtU(t,x)=12∂2∂x2U(t,x)dt+σ(t,x)dZ(t),t>0,x∈R,U(0,x)=U0(x).0,x\in \mathbb{R},\hspace{1em}\\{} U(0,x)=U_{0}(x).\hspace{1em}\end{array}\right.\]]]>
Here σ(t,x) is a bounded function that is jointly Hölder continuous of order γ∈(1/2,1), that is,
|σ(t1,x1)−σ(t2,x2)|≤C(|t1−t2|γ+|x1−x2|γ),
and U0 is a bounded measurable function. The random force in this equation is a real harmonizable fractional stable process
Z(t)=Re∫Reitx−1|x|1/α+HM(dx),
where M is a complex rotationally invariant SαS measure μ on R, defined in Section 2.1, and H∈(1/2,1) is the Hurst parameter of the process. In what follows, we denote
f(t,x)=eitx−1|x|1/α+H.
It is well known (see, e.g., [13]) that Z is an H-self-similar process having a continuous modification; henceforth, we assume that Z itself is continuous. Moreover, it is almost surely pathwise Hölder continuous with any exponent γ∈(0,H) (see [5]).
We consider Eq. (3) in the mild sense. We recall that a mild solution is given by the variation-of-constants formula
U(t,x)=∫Rρ(t,x−y)U0(y)dy+∫0tdZ(s)∫Rρ(t−s,x−y)σ(s,y)dy,
where ρ(t,x)=(4πt)−1/2exp{−|x|24t}.
The Cauchy problem (3) has a solution given by (5), where the integral with respect to Z is understood as a fractional integral.
Take some β∈(1−H,γ). Fix t>00$]]> and x∈Rd and denote f(s)=∫Rρ(t−s,x−y)σ(s,y)dy, s∈[0,t]. In view of the Hölder continuity of Z, the fractional derivative Dt−1−βZt− is almost surely bounded on [0,t]. So in order to prove the claim, we only need to show that D0+βf is integrable on [0,t]. To this end, write
∫0t|D0+βf(s)|ds≤C∫0t(|f(s)|sβ+∫0s|f(s)−f(u)|(s−u)1+βdu)ds.
Since σ is bounded, so is f, supplying the finiteness of the first integral. To establish that of the second one, use the change of variable y=x+zt−s and note that ρ(t,x)=ρ(1,x/t)/t to represent f as
f(s)=∫Rρ(1,z)σ(s,x+zt−s)dz.
Therefore, for any u<s<t,
|f(s)−f(u)|≤∫Rρ(1,z)|σ(s,x+zt−s)−σ(s,x+zt−u)|dz+∫Rρ(1,z)|σ(s,x+zt−u)−σ(u,x+zt−u)|dz=:I1+I2.
Thanks to (4), I2≤C(s−u)γ and
I1≤∫Rρ(1,z)|zt−s−zt−u|γdz≤C|t−s−t−u|γ=C(s−ut−s+t−u)γ≤C(t−u)−γ/2(s−u)γ.
Consequently,
∫0t∫0s|f(s)−f(u)|(s−u)1+βduds≤C∫0t∫0s(t−u)−γ/2(s−u)γ−1−βduds≤C∫0t(t−s)−γ/2sγ−βds<∞,
which concludes the proof. □
The process Z is a particular example of a random field given by (1). In view of this, we assume that Z is given by its LePage series representation (2) corresponding to the density
φ(x)=Kη|x|−1|log|x|+1|−1−η,
where η is some positive number, and Kη=(∫R|x|−1|ln|x|+1|−1−ηdx)−1 is a normalizing constant.
To simplify the following reasoning, we assume that
(Ω,F,P)=(ΩΓ⊗Ωξ⊗Ωg,FΓ⊗Fξ⊗Fg,PΓ⊗Pξ⊗Pg)
and
Γk(ω)=Γk(ωΓ),ξk(ω)=ξk(ωξ),gk(ω)=gk(ωg)
for all ω=(ωΓ,ωξ,ωg)∈Ω,k≥1.
Let us consider the approximation of the process Z by partial sums of its LePage series (2). Specifically, define
ZN(t)=CαRe∑k=1NΓk−1/αφ(ξk)−1/αf(t,ξk)gk.
The following result establishes a uniform, in N, Hölder continuity of the family {ZN,N≥1}.
For anyθ∈(0,H)andT>00$]]>, there is an almost surely finite random variableC=Cθ,T(ω)such that, for allN≥1andt,s∈[0,T], we have the following inequality:|ZN(t)−ZN(s)|≤Cθ(ω)|t−s|θ.
For fixed ωΓ∈ΩΓ, ωξ∈Ωξ, and t>00$]]>, the sequence
{ZN(t)=ZN(t,(ωΓ,ωξ,ωg)),N≥1}
is a martingale on (Ωg,Fg,Pg). Then, for any N0≥1 and t,s>00$]]>, the Doob inequality yields
Eg[sup1≤N≤N0(ZN(t)−ZN(s))2]≤CEg[(ZN0(t)−ZN0(s))2].
Letting N0→∞ and applying the Fatou lemma, we get
Eg[supN≥1(ZN(t)−ZN(s))2]≤CEg[(Z(t)−Z(s))2].
Using further a reasoning similar to that used in [5, Theorem 1], we get, for any t,s∈[0,T],
Eg[supN≥1(ZN(t)−ZN(s))2]≤CT(ω)|t−s|2H|log|t−s|+1|a
with some a>00$]]>. Consequently, for any θ∈(0,H),
supN≥1|ZN(t)−ZN(s)|≤Cθ,T(ω)|t−s|θ,
as required. □
Taking into account that the processes ZN approximate Z, it is natural to consider corresponding approximations of a mild solution to (3):
UN(t,x)=∫Rρ(t,x−y)U0(y)dy+∫0tdZN(s)∫Rρ(t−s,x−y)σ(s,y)dy.
The following theorem is the main result of this paper.
For anyt≥0andx∈R, we have the convergenceUN(t,x)→U(t,x),N→∞,almost surely.
Fix arbitrary numbers t>00$]]> and x∈R. Let us define the functions vk(t,x) corresponding to the terms of LePage series:
vk(t,x)=CαΓk−1/αφ(ξk)−1/αRe∫0tdsf(s,ξk)∫Rρ(t−s,x−y)σ(s,y)dy.
Then
UN(t,x)=∫Rρ(t,x−y)U0(y)dy+∑k=1Nvk(t,x).
As the first step of our proof, we establish the almost sure convergence of the series ∑k=1∞vk(t,x) for all t∈[0,T] and x∈R.
Let us transform the differential
dsf(s,ξk)=ds(eisξk−1|ξk|1/α+H)=eisξk·iξk|ξk|1/α+Hdt=ieisξksignξk|ξk|1/α+H−1dt.
Then we have
vk(t,x)=CαΓk−1/αRe[iφ(ξk)−1/αsignξk|ξk|1/α+H−1gk×∫0t∫Rρ(t−s,x−y)σ(s,y)dyeisξkds].
Hence,
Eg[|vk(t,x)|2]≥Cα2Γk−2/αφ(ξk)−2/α|ξk|2/α+2H−2×|∫0t∫Rρ(t−s,x−y)σ(s,y)dyeisξkds|2.
Let us estimate the last integral:
|∫0t∫Rρ(t−s,x−y)σ(s,y)dyeisξkds|≤|∫0t∫Rρ(t−s,x−y)σ(s,x)dyeisξkds|+|∫0t∫Rρ(t−s,x−y)(σ(s,y)−σ(s,x))dyeisξkds|=:I1+I2.
Since ∫Rρ(t−s,x−y)dy=1, we have
I1=|∫0t∫Rρ(t−s,x−y)σ(s,x)dyeisξkds|=|∫0tσ(s,x)eisξkds|.
First, assume that |ξk|≥t−1. Using Lemma 1, the last expression admits the following estimate:
|∫0tσ(s,x)eisξkds|≤3tC|ξk|−γ+2|ξk|−1sups∈[0,t]|σ(s,x)|≤C|ξk|−γ.
Let us now estimate I2, taking into account that ρ(t,x)=ρ(1,x/t)/t:
|∫0t∫Rρ(t−s,x−y)(σ(s,y)−σ(s,x))dyeisξkds|=|∫0t∫Rρ(1,x−yt−s)(σ(s,y)−σ(s,x))dyt−seisξkds|=|∫0t∫Rρ(1,z)(σ(s,x+zt−s)−σ(s,x))dzeisξkds|=|∫0tτ(s)eisξk|=:I3,
where
τ(s)=∫Rρ(1,z)(σ(s,x+zt−s)−σ(s,x))dz.
We further estimate
|τ(s1)−τ(s2)|=|∫Rρ(1,z)(σ(s1,x+zt−s1)−σ(s1,x))−(σ(s2,x+zt−s2)−σ(s2,x))dz|≤|∫Rρ(1,z)(σ(s1,x+zt−s1)−σ(s1,x+zt−s2))dz|+|∫Rρ(1,z)(σ(s1,x+zt−s2)−σ(s2,x+zt−s2))dz|+∫Rρ(1,z)|σ(s1,x)−σ(s2,x)|dz=:J1+J2+J3.
Thanks to the Hölder continuity (4), J3≤C(s1−s2)γ for s2<s1. Further, similarly to the proof of Theorem 2, J1≤C(s1−s2)γ(t−ss)−γ/2 and J2≤C(s1−s2)γ. Consequently, for s2<s1,
|τ(s1)−τ(s2)|≤C(s1−s2)γ(t−s2)−γ/2.
Similarly to J1, |τ(s)|≤C(t−s)γ/2.
Further, recall that |ξk|≥t−1. As in the proof of Lemma 1, set n=[|ξk|t]+1, so that |ξk|t≤n≤2|ξk|t, and define the equidistant partition of [0,t]: tj=tj/n,j=0,…,n. Then
I3=|∑j=1n∫tj−1tjτ(s)eisξkds|≤|∑j=1n∫tj−1tjτ(tj−1)eisξkds|+∑j=1n∫tj−1tj|τ(s)−τ(tj−1)|ds≤|∑j=1nτ(tj−1)(eitjξkξk−eitj−1ξkξk)|+C∑j=1n∫tj−1tj(t−tj−1)−γ/2(s−tj−1)γds≤|τ(0)ξk|+|τ(t)ξk|+|∑j=1neitjξkξk(τ(tj)−τ(tj−1))|+C∑j=1n(t−tj−1)−γ/2(tj−tj−1)γtn≤C(|ξk|−1+|ξk|−1∑j=1n(tn)γ+(tn)γ)≤C|ξk|−1(1+n1−γ)≤C|ξk|−1(1+|ξk|1−γ)≤C|ξk|−γ.
Otherwise, if |ξk|≤t−1, then
I3≤|∫0tτ(s)eisξkds|≤∫0t|τ(s)|ds≤C.
Then, for |ξk|≥t−1,
Eg[|vk(t,x)|2]≤Cα2Γk−2/αφ(ξk)−2/α|ξk|2/α+2H−2|∫0t∫Rρ(t−s,x−y)σ(s,y)dyeisξkds|2≤CΓk−2/α|ξk|2−2H−2γ|ln|ξk|+1|2(1+η)/α,
whereas, for |ξk|<t−1,
Eg[|vk(t,x)|2]≤CΓk−2/α|ξk|2−2H|ln|ξk|+1|2(1+η)/α.
Hence,
Eξ,g[|vk(t,x)|2]≤CΓk−2/α[∫|y|≥t−1|y|2−2H−2γ(|ln|y||+1)2(1+η)/αφ(y)dy+∫|y|<t−1|y|2−2H(|ln|y||+1)2(1+η)/αφ(y)dy]=CΓk−2/α[∫|y|≥t−1|y|1−2H−2γ(|ln|y||+1)(−1+2/α)(1+η)dy+∫|y|<t−1|y|1−2H(|ln|y||+1)(−1+2/α)(1+η)dy].
The first integral converges since 1−2H−2γ<−1, whereas the second one converges since 1−2H>−1-1$]]>. Therefore,
∑k=1∞Eξ,g[|vk(t,x)|2]≤∑k=1∞CkΓk−2/α.
By the strong law of large numbers, Γk∼1k,k→+∞, PΓ-almost surely. Therefore, ∑k=1∞Eξ,g[|vk(t,x)|2]<∞PΓ-almost surely.
In particular, ∑k=1∞Eg[|vk(t,x)|2] converges Pξ⊗PΓ-almost surely. For fixed ωξ∈Ωξ and ωΓ∈ΩΓ, the random variables {vk(t,x),k≥1} are independent centered Gaussian random variables; moreover,
∑k=1∞E[|vk(t,x)|2]<∞.
Then, by the Kolmogorov theorem, ∑k=1∞vk(t,x) converges Pξ⊗PΓ⊗Pg-almost surely, as claimed.
It remains to prove that the sum U0(t,x)+∑k=1∞vk(t,x) is equal to U(t,x). We first show that ZN(t)→Z(t),N→∞, almost surely in Cθ[0,T] for any T>00$]]>, θ∈(0,H). Taking into account that, for any t∈[0,T], ZN(t)→Z(t),N→∞, almost surely, we get that there is a set Ω0⊂Ω such that P(Ω0)=1 and ZN(t)→Z(t),N→∞, for any t∈[0,T]∩Q, ω∈Ω0. Thanks to Proposition 3, the sequence {ZN,N≥1} is almost surely bounded in Cθ[0,T] for any θ∈(0,H) and T>00$]]>. Therefore, it is precompact in each of these spaces. Take arbitrary θ∈(0,H). Without loss of generality, the sequence {ZN,N≥1} is precompact in Cθ[0,T] for any ω∈Ω0. Fix ω∈Ω0 and let {ZNk,k≥1} be any subsequence of {ZN,N≥1}. In view of precompactness, it must contain a subsequence convergent in Cθ; to avoid cumbersome notation, we assume that ZNk→Y, k→∞. In particular, ZNk(t)→Y(t), k→∞, t∈[0,T]∩Q. In view of continuity, Z(t)=Y(t) for any t∈[0,T]. Since any subsequence of {ZN,N≥1} contains a subsequence convergent to Z in Cθ[0,T], the sequence itself converges to Z.
Now, thanks to the integrability of the fractional derivative of f(s)=∫Rρ(t−s,x−y)σ(s,y)dy, which was shown in the proof of Theorem 2, the convergence established in the previous paragraph yields
∫0tdZN(s)∫Rρ(t−s,x−y)σ(s,y)dy→∫0tdZ(s)∫Rρ(t−s,x−y)σ(s,y)dy
as n→∞, concluding the proof. □
It is possible to consider (3) with Z being a real harmonizable multifractional stable motion considered in [4]. Making some minor changes, one can show that Theorem 4 is valid in this case as well.
Acknowledgements
The authors thank an anonymous referee for his careful reading of the manuscript and valuable remarks, which led to a significant improvement of the article.
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