A stochastic heat equation on [0,T]×R driven by a general stochastic measure dμ(t) is investigated in this paper. For the integrator μ, we assume the σ-additivity in probability only. The existence, uniqueness, and Hölder regularity of the solution are proved.
In this paper, we consider a stochastic heat equation that can formally be written as
du(t,x)=a2∂2u(t,x)∂x2dt+f(t,x,u(t,x))dt+σ(t,x)dμ(t),u(0,x)=u0(x),
where (t,x)∈[0,T]×R, a∈R, a≠0, and μ is a stochastic measure (SM) defined on the Borel σ-algebra of [0,T]. We consider a solution to the formal equation (1) in the mild sense (see Eq. (5)). We prove the existence and uniqueness of the solution and obtain Hölder regularity of its paths under some general conditions for the stochastic part of equation.
A similar problem for μ dependent on the spatial variable x was considered in [5]. The stochastic heat equation on fractals was studied in [7], and a review of results on equations driven by SMs is given in [6].
For equations driven by white noise, the regularity of paths of solutions was considered in [10, Chapter 3]. Equations driven by fractional noise were studied in [9, Chapter 2]. In many papers, the regularity of solutions was considered in appropriate function spaces; see, for example, [2] and references therein.
Preliminaries
Let L0=L0(Ω,F,P) be the set of (equivalence classes of) all real-valued random variables defined on a complete probability space (Ω,F,P). The convergence in L0 is understood as the convergence in probability. Let X be an arbitrary set, and B be a σ-algebra of subsets of X.
Any σ-additive mapping μ:B→L0 is called a stochastic measure (SM).
In other words, μ is a vector measure with values in L0. In [3], such μ is called a general SM.
Examples of SMs are the following. Let X=[0,T]⊂R+, B be the σ-algebra of Borel subsets of [0,T], and N(t) be a square-integrable martingale. Then μ(A)=∫0T1A(t)dN(t) is an SM. If WH(t) is a fractional Brownian motion with Hurst index H>1/21/2$]]> and f:[0,T]→R is a bounded measurable function, then μ(A)=∫0Tf(t)1A(t)dWH(t) is also an SM, as follows from [4, Theorem 1.1]. An α-stable random measure defined on a σ-algebra is an SM [8, Chapter 3]. Theorem 8.3.1 of [3] states the conditions under which the increments of a real-valued Lévy process generate an SM.
For a deterministic measurable function g:X→R and SM μ, an integral of the form ∫Xgdμ is defined and studied in [3, Chapter 7]; see also [1]. In particular, every bounded measurable g is integrable w.r.t. any μ. An analogue of the Lebesgue dominated convergence theorem holds for this integral [3, Proposition 7.1.1].
We consider the Besov spacesB22α([c,d]). Recall that the norm in this classical space for 0<α<1 may be introduced by
‖g‖B22α([c,d])=‖g‖L2([c,d])+(∫0d−c(w2(g,r))2r−2α−1dr)1/2,
where
w2(g,r)=sup0≤h≤r(∫cd−h|g(s+h)−g(s)|2ds)1/2.
For all n≥1, 1≤k≤2n, put Δkn(t)=((k−1)2−nt,k2−nt].
The following estimate is a key tool for the proof of Hölder regularity of the stochastic integral. In our estimates, C and C(ω) will denote a constant and a random constant, respectively, which may be different from formula to formula.
(Lemma 3.2 [5]).
Let SM μ be defined on the Borel σ-algebra of[0,t],Zbe an arbitrary set, andq(z,s):Z×[0,t]→Rbe a function such that for some1/2<α<1and for eachz∈Z,q(z,·)∈B22α([0,t]). Then the random functionη(z)=∫[0,t]q(z,s)dμ(s),z∈Z,has a versionη˜(z)such that for some constant C (independent of z, ω) and eachω∈Ω,|η˜(z)|≤|q(z,0)μ([0,t])|+C‖q(z,·)‖B22α([0,t]){∑n≥12n(1−2α)∑1≤k≤2n|μ(Δkn(t))|2}1/2.
From Lemma 3.1 [5] it follows that, for ε>00$]]>,
∑n≥12−nε∑1≤k≤2n|μ(Δkn(t))|2<+∞a.s.
The problem
Consider equation (1) in the following mild sense:
u(t,x)=∫Rp(t,x−y)u0(y)dy+∫0tds∫Rp(t−s,x−y)f(s,y,u(s,y))dy+∫(0,t]dμ(s)∫Rp(t−s,x−y)σ(s,y)dy.
Here
p(t,x)=12aπte−x24a2t
is the Gaussian heat kernel, u(t,x)=u(t,x,ω):[0,T]×R×Ω→R is an unknown measurable random function, and μ is an SM defined on the Borel σ-algebra of [0,T]. The integrals of random functions w.r.t. dy and ds are taken for each fixed ω∈Ω.
Throughout this paper, we will use the following assumptions.
u0(y)=u0(y,ω):R×Ω→R is measurable and ω-wise bounded, |u0(y,ω)|≤C(ω).
u0(y) is Hölder continuous:
|u0(y1)−u0(y2)|≤C(ω)|y1−y2|β(u0),β(u0)≥1/2.
f(s,y,v):[0,T]×R×R→R is measurable and bounded: |f(s,y,v)|≤C.
f(s,y,v) is uniformly Lipschitz in y,v∈R:
|f(s,y1,v1)−f(s,y2,v2)|≤C(|y1−y2|+|v1−v2|).
σ(s,y):[0,T]×R→R is measurable and bounded: |σ(s,y)|≤C.
σ(s,y) is Hölder continuous:
|σ(s1,y1)−σ(s2,y2)|≤C(|s1−s2|β(σ)+|y1−y2|β(σ)),β(σ)>1/2.1/2.\]]]>
μ is Hölder continuous:
|μ((s1,s2])|≤C(ω)|s1−s2|β(μ),s1,s2∈[0,T],β(μ)>0.0.\]]]>
Recall that ∫Rp(t,x)dx=1.
Hölder continuity in x
Consider the regularity of paths of the stochastic integral from (5).
Let Assumptions A5 and A6 hold. Then, for any fixedt∈[0,T]andγ1<β(σ)−1/2, the stochastic functionϑ(x)=∫(0,t]dμ(s)∫Rp(t−s,x−y)σ(s,y)dy,x∈R,has a Hölder continuous version with exponentγ1.
Denote
q(z,s)=∫R(p(t−s,x1−y)−p(t−s,x2−y))σ(s,y)dy,z=(x1,x2,t),
and apply (3) to η(z)=ϑ(x1)−ϑ(x2). We will estimate the Besov space norm in (3). Consider the difference
q(z,s+h)−q(z,s)=(∫Rp(t−s−h,x1−y)σ(s+h,y)dy−∫Rp(t−s,x1−y)σ(s,y)dy)−(∫Rp(t−s−h,x2−y)σ(s+h,y)dy−∫Rp(t−s,x2−y)σ(s,y)dy):=D1−D2.
Using (6) and the change of variables
v=x1−y2at−s−h,v=x1−y2at−s,
we get
|D1|=C|∫Re−v2σ(s+h,x1−2avt−s−h)dv−∫Re−v2σ(s,x1−2avt−s)dv|≤A6C∫Re−v2(|h|β(σ)+|v(t−s−h−t−s)|β(σ))dv=C∫Re−v2(|h|β(σ)+|v|β(σ)|h|β(σ)|t−s−h+t−s|β(σ))dv≤C|h|β(σ)∫Re−v2(1+|v|β(σ)t−sβ(σ))dv=C|h|β(σ)(t−s)−β(σ)/2.
By a similar way, we can estimate |D2| and obtain
|q(z,s+h)−q(z,s)|≤C|h|β(σ)(t−s)−β(σ)/2.
Further, consider
q(z,s+h)−q(z,s)=(∫Rp(t−s−h,x1−y)σ(s+h,y)dy−∫Rp(t−s−h,x2−y)σ(s+h,y)dy)−(∫Rp(t−s,x1−y)σ(s,y)dy−∫Rp(t−s,x2−y)σ(s,y)dy):=E1−E2.
Using (6) and the substitutions
v=x1−y2at−s−h,v=x2−y2at−s−h,
we get
|E1|=C|∫Re−v2σ(s+h,x1−2avt−s−h)dv−∫Re−v2σ(s+h,x2−2avt−s−h)dv|≤A6C∫Re−v2|x1−x2|β(σ)dv=C|x1−x2|β(σ).
Similarly, we can estimate |E2| (we consider |E1| for h=0) and obtain
|q(z,s+h)−q(z,s)|≤C|x1−x2|β(σ).
The product of (9) raised to the power λ and (10) raised to the power 1−λ, 0<λ<1, now satisfies
|q(z,s+h)−q(z,s)|≤C|h|λβ(σ)(t−s)−β(σ)λ/2|x1−x2|(1−λ)β(σ),w2(q(z,·),r)≤Crλβ(σ)|x1−x2|(1−λ)β(σ).
If λβ(σ)>1/21/2$]]>, then the integral from (2) is finite for some α>1/21/2$]]>. In this case, the integral does not exceed C|x1−x2|(1−λ)β(σ).
From the estimate of E1 for h=0 we obtain
|q(z,0)|≤C|x1−x2|β(σ),‖q(z,·)‖L2([0,t])≤C|x1−x2|β(σ).
Therefore, we have
|ϑ(x1)−ϑ(x2)|≤C(ω)|x1−x2|γ1,γ1=(1−λ)β(σ).
Under the restriction λβ(σ)>1/21/2$]]>, we can get any γ1<β(σ)−1/2. □
Hölder continuity in t
Assume that Assumptions A5, A6, and A7 hold. Then, ifγ2≤β(μ)andγ2<β(σ)−1/2, then for any fixedx∈R, the stochastic processϑ¯(t)=∫(0,t]dμ(s)∫Rp(t−s,x−y)σ(s,y)dy,t∈[0,T],has a Hölder continuous version with exponentγ2.
For t1<t2, we have
ϑ¯(t2)−ϑ¯(t1)=∫(t1,t2]dμ(s)∫Rp(t2−s,x−y)σ(s,y)dy+∫(0,t1]dμ(s)∫R(p(t2−s,x−y)−p(t1−s,x−y))σ(s,y)dy:=F1+F2.Step 1. Estimation ofF1. Consider segments [0,T], Δkn(T)=((k−1)2−nT,k2−nT], and the function
q¯(z,s)=∫Rp(t2−s,x−y)σ(s,y)dy,s∈[t1,t2],z=(x,t2).
From the estimates of D1 in (7) and (8) it follows that
|q¯(z,s+h)−q¯(z,s)|≤C|h|β(σ)(t2−s)−β(σ)/2,s∈[t1−h,t2−h).
Let kn1 and kn2 be such that t1∈Δkn1n(T), t2∈Δkn2n(T). For the functions
q¯n(z,s)=∑k=12nq¯(z,(k−1)2−nT∨t1∧t2)1Δkn(T)(s),
the analogue of the Lebesgue theorem [3, Proposition 7.1.1] implies that
|∫(t1,t2]q¯(z,s)dμ(s)|=|plimn→∞∫(t1,t2]q¯n(z,s)dμ(s)|=|∫(t1,t2]q¯0(z,s)dμ(s)+∑n=1∞(∫(t1,t2]q¯n(z,s)dμ(s)−∫(t1,t2]q¯n−1(z,s)dμ(s))|≤|q¯(z,t1)μ((t1,t2])|+∑n=n0∞|(q¯(z,kn12−nT)−q¯(z,t1))μ(Δ(kn1+1)n(T))|+∑n=n0∞∑k:kn1≤2k−2<kn2−2|(q¯(z,(2k−1)2−nT)−q¯(z,(2k−2)2−nT))μ(Δ(2k)n(T))|+∑n=n0∞|(q¯(z,(kn2−1)2−nT)−q¯(z,(kn2−2)2−nT))μ(((kn2−1)2−nT,t2])|.
Here n0 is such that
2−n0T<t2−t1≤2−n0+1T.
We have
kn2−kn1≤(t2−t1)2n+2/T,n≥n0.
Applying Assumptions A5, A7, (11), and the Cauchy inequality from (12) for 0<ε<2β(σ)−1, we obtain
|∫(t1,t2]q¯(z,s)dμ(s)|≤C(ω)(t2−t1)β(μ)+C(ω)∑n=n0∞2−nβ(μ)+C∑n=n0∞∑k:kn1≤2k−2<kn2−22−nβ(σ)(t2−(2k−2)2−nT)−β(σ)/2|μ(Δ(2k)n(T))|+C(ω)∑n=n0∞2−nβ(μ)≤C(ω)(t2−t1)β(μ)+C(ω)2−n0β(μ)+C(∑n=n0∞2εn2−2nβ(σ)∑k:kn1≤2k−2<kn2−2(t2−(2k−2)2−nT)−β(σ))1/2×(∑n=0∞2−εn∑k=12nμ2(Δkn(T)))1/2≤(4),(13)C(ω)(t2−t1)β(μ)+C(ω)(∑n=n0∞2εn2−2nβ(σ)∑1≤i<(kn2−kn1)/2(i2−nT)−β(σ))1/2≤C(ω)(t2−t1)β(μ)+C(ω)(∑n=n0∞2n(ε−β(σ))(kn2−kn1)1−β(σ))1/2≤(14)C(ω)(t2−t1)β(μ)+C(ω)(t2−t1)(1−β(σ))/22−n0(2β(σ)−ε−1)/2≤(13)C(ω)(t2−t1)β(μ)+C(ω)(t2−t1)(β(σ)−ε)/2≤C(ω)(t2−t1)γ2.Step 2. Estimation ofF2. Now denote
q˜(z,s)=∫R(p(t2−s,x−y)−p(t1−s,x−y))σ(s,y)dy,s∈[0,t1],z=(x,t1,t2).
Using the change of variables
v=x−y2at2−s,v=x−y2at1−s,
we get
|q˜(z,s)|=C|∫Re−v2σ(s,x−2avt2−s)dv−∫Re−v2σ(s,x−2avt1−s)dv|≤A6C∫Re−v2|v(t2−s−t1−s)|β(σ)dv≤(8)C(t2−t1)β(σ)(t2−s)−β(σ)/2.
Also, analogously to (7) and (8), we have
|q˜(z,s+h)−q˜(z,s)|≤C|h|β(σ)(t1−s)−β(σ)/2.
From (15) and (16) for 0<λ<1 and 0≤s≤t1−h, we obtain
|q˜(z,s+h)−q˜(z,s)|≤C|h|λβ(σ)(t1−s)−β(σ)λ/2(t2−t1)(1−λ)β(σ)(t2−s−h)−(1−λ)β(σ)/2,w2(q˜(z,·),r)≤Crλβ(σ)(t2−t1)(1−λ)β(σ).
If λβ(σ)>1/2⇔(1−λ)β(σ)<β(σ)−1/21/2\Leftrightarrow (1-\lambda )\beta (\sigma )<\beta (\sigma )-1/2$]]>, then the integral from (2) is finite for some α>1/21/2$]]>. In this case, the integral does not exceed C(t2−t1)(1−λ)β(σ).
From (15) we get
|q˜(z,0)|≤C(t2−t1)β(σ)/2,‖q˜(z,·)‖L2([0,t1])≤C(t2−t1)β(σ)/2.
Therefore, from (3) we have |F2|≤C(ω)(t2−t1)γ2, which finishes the proof. □
Solution to the equation
Suppose that Assumptions A1–A6 hold.
Equation (5) has a solutionu(t,x). Ifv(t,x)is another solution to (5), then for all t and x,u(t,x)=v(t,x) a.s.
For any fixedt∈[0,T]andγ1<β(σ)−1/2, the stochastic functionu(t,x),x∈R, has a Hölder continuous version with exponentγ1.
In addition, let Assumption A7 hold. Then for any fixedδ>00$]]>andγ1,γ2such thatγ1<β(σ)−1/2,γ2≤β(μ), andγ2<β(σ)−1/2, the stochastic functionu(t,x)has a versionu˜(t,x)such that|u˜(t1,x1)−u˜(t2,x2)|≤C(ω)(|t1−t2|γ2+|x1−x2|γ1),ti∈[δ,T],xi∈R.
Consider the standard iteration process. Take u(0)(t,x)=0 and set
u(n+1)(t,x)=∫Rp(t,x−y)u0(y)dy+∫0tds∫Rp(t−s,x−y)f(s,y,u(n)(s,y))dy+∫(0,t]dμ(s)∫Rp(t−s,x−y)σ(s,y)dy.
Further, we can repeat the proof of Theorem [5]. Instead of reference to Lemmas 5.1 and 6.1 of [5], we can refer to Lemmas 2 and 3 of this paper. □
For u, we obtained less regularity than for elements of equation (5). However, a solution to a heat equation usually has the same regularity or even more regular than the coefficients. One may expect that using other methods gives (17) with exponents γ2≤β(μ)∧γ2<β(σ) and γ1<β(σ).
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