VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA14 10.15559/14-VMSTA14 Research Article Heat equation with general stochastic measure colored in time RadchenkoVadymvradchenko@univ.kiev.ua∗ Department of Mathematical Analysis, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine Corresponding author. 2014 512201412129138 2102014 10112014 13112014 © 2014 The Author(s). Published by VTeX2014 Open access article under the CC BY license.

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.

 Stochastic measure stochastic heat equation mild solution Hölder regularity Besov space 60H15 60G17 60G57 Introduction

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>1/2 1/2$]]> 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 spaces B22α([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 [<xref ref-type="bibr" rid="j_vmsta14_ref_005">5</xref>]).

Let SM μ be defined on the Borel σ-algebra of [0,t] , Z be an arbitrary set, and q(z,s):Z×[0,t]→R be a function such that for some 1/2<α<1 and for each z∈Z , q(z,·)∈B22α([0,t]) . Then the random function η(z)=∫[0,t]q(z,s)dμ(s),z∈Z, has a version η˜(z) such that for some constant C (independent of z, ω) and each ω∈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>0 0$]]>, ∑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 <italic>x</italic>

Consider the regularity of paths of the stochastic integral from (5).

Let Assumptions A5 and A6 hold. Then, for any fixed t∈[0,T] and γ1<β(σ)−1/2 , the stochastic function ϑ(x)=∫(0,t]dμ(s)∫Rp(t−s,x−y)σ(s,y)dy,x∈R, has a Hölder continuous version with exponent γ1 .

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)>1/2 1/2$]]>, then the integral from (2) is finite for some α>1/2 1/2$]]>. 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)>1/2 1/2$]]>, we can get any γ1<β(σ)−1/2 .  □

 Hölder continuity in <italic>t</italic>

Assume that Assumptions A5, A6, and A7 hold. Then, if γ2≤β(μ) and γ2<β(σ)−1/2 , then for any fixed x∈R , the stochastic process ϑ¯(t)=∫(0,t]dμ(s)∫Rp(t−s,x−y)σ(s,y)dy,t∈[0,T], has a Hölder continuous version with exponent γ2 .

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 of F1 . 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 of F2 . 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)−1/2 1/2\Leftrightarrow (1-\lambda )\beta (\sigma )<\beta (\sigma )-1/2$]]>, then the integral from (2) is finite for some α>1/2 1/2$]]>. 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 solution u(t,x) . If v(t,x) is another solution to (5), then for all t and x, u(t,x)=v(t,x)  a.s.

For any fixed t∈[0,T] and γ1<β(σ)−1/2 , the stochastic function u(t,x) , x∈R , has a Hölder continuous version with exponent γ1 .

In addition, let Assumption A7 hold. Then for any fixed δ>0 0$]]> and γ1 ,  γ2 such that γ1<β(σ)−1/2 , γ2≤β(μ) , and γ2<β(σ)−1/2 , the stochastic function u(t,x) has a version u˜(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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