Quadratic covariations for the solution to a stochastic heat equation with space-time white noise

Let u(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u(t,x)$\end{document} be the solution to a stochastic heat equation ∂∂tu=12∂2∂x2u+∂2∂t∂xX(t,x),t≥0,x∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\partial }{\partial t}u=\frac{1}{2} \frac{\partial ^{2}}{\partial x^{2}}u+ \frac{\partial ^{2}}{\partial t\,\partial x}X(t,x),\quad t\geq 0, x\in { \mathbb{R}} $$\end{document} with initial condition u(0,x)≡0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u(0,x)\equiv 0$\end{document}, where Ẋ is a space-time white noise. This paper is an attempt to study stochastic analysis questions of the solution u(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u(t,x)$\end{document}. In fact, it is well known that the solution is a Gaussian process such that the process t↦u(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\mapsto u(t,x)$\end{document} is a bi-fractional Brownian motion with Hurst indices H=K=12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H=K=\frac{1}{2}$\end{document} for every real number x. However, the many properties of the process x↦u(⋅,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\mapsto u(\cdot ,x)$\end{document} are unknown. In this paper we consider the generalized quadratic covariations of the two processes x↦u(⋅,x),t↦u(t,⋅)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\mapsto u(\cdot ,x),t\mapsto u(t,\cdot )$\end{document}. We show that x↦u(⋅,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\mapsto u(\cdot ,x)$\end{document} admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with “Itô’s integral”, but it is not a semimartingale. Moreover, some generalized Itô formulas and Bouleau–Yor identities are introduced.


Introduction
Let u(t, x) denote the solution to the stochastic heat equation Clearly, we have where p(t, x) = 1 √ 2π t e -|x| 2 2t is the heat kernel. Then the processes (t, x) → u(t, x), t → u(t, ·) and x → u(·, x) are Gaussian. Mueller and Tribe [19] were first to find that for all x, y ∈ R d , as t tends to infinity, if the initial value u(0, x) = 0. By taking a special initial value (a two-sided R d -valued Brownian motion), the authors studied the stationary pinned string and hitting probabilities of a random string. When d = 1 Swanson [30] has showed that (see also Pospisil and Tribe [25]) (1.2) and the process t → u(t, x) has a nontrivial quartic variation. This shows that, for every x ∈ R, the process t → u(t, x) coincides with the bi-fractional Brownian motion and it is not a semimartingale, so a stochastic integral with respect to the process t → u(t, x) cannot be defined in the classical Itô sense. Some surveys and complete literature for bifractional Brownian motion can be found in Houdré and Villa [13], Lei and Nualart [16], Russo and Tudor [27], Tudor and Xiao [33] and Yan et al. [35], and the references therein. Moreover, for more general parabolic SPDEs, many authors studied the regularity results and stochastic calculus with respect to their solutions. We mention the work of Balan and Kim [2], Da Prato et al. [4], Deya and Tindel [5], Gradinaru et al. [12], Lanconelli [14,15], León and Tindel et al. [17], Nualart and Vuillermot [21], Ouahhabi and Tudor [22], Pardoux [23], Pospisil and Tribe [25], Torres et al. [31], Tudor [32], Tudor and Xiao [34], Zambotti [38], and the references therein. In this paper, as an attempt we study stochastic analysis questions of the solution process u = {u(t, x), t ∈ [0, T], x ∈ R} of (1.1) with u(0, x) = 0 associated with quadratic variation. We shall see (in Sect. 3) that the process x → u(·, x) admits a nontrivial finite quadratic variation in any finite interval, and, moreover, we shall also see (in Sect. 4) that the forward integral of some adapted processes with respect to x → u(·, x) coincides with "Itô's integral". On the other hand, as a noise, the stochastic process u = {u(t, x), t ≥ 0, x ∈ R} is very rough in time and it is not white in space. However, the process x → u(·, x) admits some characteristics similar to Brownian motion. These results, together with the work of Mueller and Tribe [19], Pospisil and Tribe [25], and Swanson [30] point out that the process u = {u(t, x)} as a noise admits the next special structures: • It is very rough in time and similar to fractional Brownian motion with Hurst index H = 1 4 , but it has not stationary increments. • It is not white in space, but its quadratic variation coincides with the classical Brownian motion and it is not self-similar.
• The process in space variable is not a semimartingale, but the forward integral of some adapted processes with respect to the process in space variable coincides with "Itô's integral". • The process u = {u(t, x)} admits a simple representation via a Wiener integral with respect to the Brownian sheet. • Though the process u = {u(t, x)} is Gaussian, as a noise, its time and space parts are farraginous. We cannot decompose its covariance as the product of two independent parts. This is very different from fractional noise and white noise. In fact, we have for all t ≥ s > 0 and x, y ∈ R. Therefore, it seems interesting to study the integrals and some related stochastic (partial) differential equations. For example, one can consider the following "iterated" stochastic partial differential equations: where u 0 is a space-time white noise. Of course, one can also consider some sample path properties and singular integrals with t ≥ 0 and I x = [0, x] for x ≥ 0 and I x = [x, 0] for x ≤ 0. We will carry out these projects in some forthcoming work. In the present paper our objects are to study the quadratic covariations of x → u(·, x) and t → u(t, ·), and introduce some generalized Itô formulas associated with {u(·, x), x ∈ R} and {u(t, ·), t ≥ 0}, respectively, and moreover we also consider their local times and Bouleau-Yor's identities. To expound our aim, let us start with a basic definition. Let u = {u(t, x), t ∈ [0, T], x ∈ R} be the solution process of (1.1) with u(0, x) = 0 and denote for all t, s > 0 and z ≥ 0. An elementary calculation can show that for all t, s > 0 and x, y ∈ R. This simple estimate inspires us to consider the following limits: for all t ≥ 0 and x ∈ R. That is, for all t ≥ 0 and x ∈ R. Thus, the next definition is natural.
for all t ≥ 0, x ∈ R, ε, δ > 0, where f is a measurable function on R. The limits lim δ→0 I 1 δ (f , t, x) and lim ε→0 I 2 ε (f , t, x) are called the partial quadratic covariations (PQC, in short) in space and in time, respectively, of f (u) and u, provided these limits exist in probability. We denote them by [f (W ), W ] (SQ) x and [f (B), B] (TQ) t , respectively.
These expressions say that the process W = {W x = u(·, x), x ∈ R} admits a nontrivial finite quadratic variation in any finite interval I x . This is also a main motivation to study the solution of (1.1). More work for stochastic calculus with respect to a continuous finite quadratic variation process can be found in Errami and Russo [7] and Russo and Vallois [29]. This paper is organized as follows. In Sect. 2, we establish some technical estimates associated with the solution of (1.1), and as some applications we introduce Wiener integrals with respect to the two processes B = {B t = u(t, ·), t ≥ 0} and W = {W x = u(·, x), x ∈ R}, respectively. In Sect. 3 we show that the quadratic variation [W , W ] (SQ) exists in L 2 (Ω) and equals |x| in every finite interval I x . For a given t > 0, by estimating in L 2 for all ε > 0, respectively, we construct a Banach space H t of measurable functions such that the PQC [f (W ), W ] (SQ) in space exists in L 2 (Ω) for all f ∈ H t , and in particular we have In Sect. 4, as an application of Sect. 3, we show that the Itô's formula holds for all t > 0, x ∈ R, where the integral I x f (W y )δW y denotes the Skorohod integral, F is an absolutely continuous function with the derivative F = f ∈ H t . In order to obtain the above Itô formula, we first introduce a standard Itô type formula for all F ∈ C 2 (R) satisfying some suitable conditions. It is important to note that the Gaussian process W = {W x = u(·, x), x ∈ R} does not satisfy the condition in Alós et al. [1] since for all t ≥ 0 and x ∈ R. We need to give the proof of Eq. (1.6). Moreover, we also show that the forward integral (see Russo-Vallois [29]) with 0 ≤ β < √ π 4 √ t , where the notation ucp lim denotes the uniform convergence in probability on each compact interval. This is very similar to Brownian motion, but the process W = {W x = u(·, x), x ∈ R} is not a semimartingale. In Sect. 5 we consider some questions associated with the local time In particular, we show that the Bouleau-Yor type identity holds for all f ∈ H t . In Sect. 6 we consider some analysis questions associated with the quadratic covariation of the process B = {B t = u(t, ·), t ≥ 0}.

Some basic estimates
In this section we will establish divergence integral and some technical estimates associated with the solution 2t is the heat kernel. Recall that the temporal process B = {B t = u(t, ·), t ≥ 0} and spatial process W = {W x = u(·, x), x ∈ R}. As mentioned earlier, the temporal process B = {B t } is neither a semimartingale nor a Markov process. Moreover, from the next discussion we shall also see that the spatial process W = {W x } is neither a semimartingale nor a Markov process. However, as two Gaussian processes, one can develop the stochastic calculus of variations with respect to them. We refer to Alós et al. [1], Nualart [20] and the references therein for more details of stochastic calculus of Gaussian process.
As usual, we assume that H and H are the reproducing kernel Hilbert spaces asso- Then the Wiener integrals Denote by S and S the sets of smooth functionals of the form and respectively. The divergence integrals δ and δ are the adjoint of derivative operators D and D , respectively. We say that random variables v ∈ L 2 (Ω; H ) and w ∈ L 2 (Ω; H ) belong to the domains of the divergence operators δ and δ , respectively, denoted by Dom(δ ) and Dom(δ ) if respectively, for all F ∈ S and F ∈ S . In these cases δ (v) and δ (w) are defined by the duality relationships respectively, for any v ∈ D ,1,2 and w ∈ D ,1,2 . We have D ,1,2 ⊂ Dom(δ ) and D ,1,2 ⊂ Dom(δ ). We will use the notations to express the Skorohod integrals, and the indefinite Skorohod integrals are defined as respectively. It is important to note that we can localize the domains of the operators D , D , δ and δ via the usual manner. If L is a class of random variables (or processes) we denote by L loc the set of random variables F such that there exists a sequence {(Ω n , F n ), n ≥ 1} ⊂ F × L with the following properties: s. on Ω n . Take the operators D and δ for instance. If F ∈ D ,1,2 loc , and (Ω n , F n ) localizes F in D ,1,2 , then D F is defined without ambiguity by D F = D F n on Ω n , n ≥ 1. Then, if v ∈ D ,1,2 loc , the divergence δ (v) is defined as a random variable determined by the conditions where (Ω n , v n ) is a localizing sequence for v, but it may depend on the localizing sequence.
At the end of this section, we introduce some estimates for the spatial process W = {W x = u(·, x)} and temporal processes B = {B t = u(t, ·)}. For simplicity throughout this paper we let C stand for a positive constant depending only on the subscripts and its value may be different at different places, and this assumption also holds true for c. Moreover, the notation F G means that there are positive constants c 1 and c 2 such that in the common domain of definition for F and G. We have for all t, s > 0 and z ≥ 0.
By taking a special initial value (a two-sided R d -valued Brownian motion), Mueller and Tribe [19] showed that the above estimate is reversible. However, when the initial value u(0, x) = 0, the above estimate is not reversible unless x = y. In fact, we have with u = s t by the continuity. On the other hand, we also have for all t ≥ s > 0 and u ∈ R. However, by the facts that for all x ≥ 0, we have for all t > 0 and z ≥ 0, and for t > s ≥ 0 and z ≥ 0.
Proof For all t, s, r > 0 and x ∈ R, we have for all t, s, r > 0 and x ∈ R.

Lemma 2.3 For all t > 0 and x, y, z
Proof For all t > 0 and x, y, z ∈ R, we have Consider the function f : Then, by the mean value theorem, we have for all u, v ≥ 0 and some ξ between u and v. It follows that for all t > 0 and x, y, z ∈ R.
Consider the function Then, by the mean value theorem, we have for some s ≤ ξ ≤ t and the lemma follows.

Lemma 2.5
For all t > 0 and x > y > x > y we have Proof Define the function f : Then, similar to the proof of Lemma 2.3, we have for all t > 0 and x > y > x > y . By mean value theorem it follows that . This completes the proof.
Proof Given t > s > 0 and x ∈ R, we have for all 0 ≤ z ≤ 1. In fact, we have This completes the proof.
Then, under the conditions of Lemma 2.6, we have In particular, we have for all t > 0 and x, y, z ∈ R.
Proof Given t > 0 and x > y. We have by (2.5) and the lemma follows.

The quadratic covariation of the spatial process W = {W x , |x| ≤ M}
In this section, we study the existence of the PQC [f (W ), W ] (SQ) . We fix a time parameter t > 0. Recall that for ε > 0 and x ∈ R, and provided the limit exists in probability. In [25], Pospisil and Tribe have showed that the following convergence holds.
Lemma 3.1 Let x ∈ R and let x n j = jx n ; j = 0, 1, . . . , n. Then we have in L 2 (Ω), as n tends to infinity.
In this section we are not only to show that in L 2 , but also to find a Banach space of Borel functions such that exists in L 2 for all Borel functions f belonging to the Banach space.
Proof of Proposition 3.1 From Russo and Vallois [29] we only need to show that with some α > 0, as ε → 0, by the above lemma, where We have Recall that Noting that for all ε > 0 and y, z ∈ I x , we get Now, let us estimate the above function ε → φ t,y (ε). We have Combining this with Lemma 2.5, we complete the proof. Now, we discuss the existence of the PQC [f (W ), W ] (SQ) . Consider the decomposition for ε > 0, and define the set and μ t (R) = C|x| < ∞, which implies that the set E of elementary functions of the form is dense in H t , where f i ∈ R and {x i , 0 ≤ i ≤ l} is a finite sequence of real numbers such that x i < x i+1 . Moreover, H t includes all Borel functions f satisfying the condition for all x ∈ [-M, M].
In order to prove the theorem it is enough to prove the following two statements with f ∈ H t : (1) For any ε > 0 and x ∈ [-M, M], I 1,± ε (f , x, ·) ∈ L 2 (Ω). That is, (2) I 1,ε (f , x, t) and I 1,+ ε (f , x, t) are two Cauchy sequences in L 2 (Ω) for all t > 0 and x ∈ [-M, M]. That is, for all x ∈ R, as ε 1 , ε 2 ↓ 0. We split the proof of the two statements into two parts.
for all ε > 0. Now, let us estimate the expression for all ε 1 , ε 2 > 0 and y, y ∈ I x . To estimate the above expression, it is enough to assume that f ∈ E by denseness, and moreover, by approximating we can assume that f is an infinitely differentiable function with compact support. It follows from the duality relationship (2.1) that for all y, y ∈ I x and ε 1 , ε 2 > 0. In order to end the proof we claim the for all ε > 0. This shows that Next, let us estimate 4 j=1 Λ j . We have where ρ 2 = σ 2 t,y σ 2 t,yμ 2 t,y,y and ϕ(x, y) is the density function of (W y , W y ), that is, Combining this with the identity we get A straightforward calculation shows that for all m ≥ 1 and It follows that Thus, we get the estimate Combining this with (3.9), (3.10), Lemma 2.3 and Lemma 2.5, we get for all ε > 0 and x ∈ R. This shows that Similarly, one can show the estimate and the first statement follows.
Step I. The following convergence holds: as ε 1 , ε 2 → 0. We have Consider the next function on R + (see Sect. 2): Then we have Notice that, by Taylor's expansion, ). It follows that for all 0 < ε 2 < ε 1 < 1 and y, y ∈ I x , which shows that the convergence (3.11) holds since f ∈ H t .
Step II. The following convergence holds: as ε 1 , ε 2 → 0. Keeping the notations in Step I, we have for all ε and It follows from (2.8) that for all 0 < ε 2 < ε 1 < 1 and y, y ∈ I x , which implies that Similarly, we can show the next convergence: as ε 1 , ε 2 → 0.
Then the convergence holds in L 2 (Ω) for all x ∈ R.

The Itô formula for the spatial process
In this section, as an application of the previous section we discuss the Itô calculus for the process W = {W x , |x| ≤ M} and fix a time parameter t > 0. For a continuous process X admitting a finite quadratic variation [X, X], Russo and Vallois [29] have introduced the following Itô formula: is called the forward integral, where the notation ucp lim denotes the uniform convergence in probability on each compact interval. We refer to Russo and Vallois [29] and the references therein for more details of stochastic calculus of continuous processes with finite quadratic variations. It follows from the previous section (the quadratic variation of for all F ∈ C 2 (R). Thus, by smooth approximating, we have the next Itô type formula.

Theorem 4.1 Let f ∈ H t be left continuous. If F is an absolutely continuous function with
the derivative F = f , then the following Itô type formula holds: x .
Clearly, this is an analogue of Föllmer-Protter-Shiryayev's formula. It is an improvement in terms of the hypothesis on f and it is also quite interesting itself. Some details and more work could be found in Eisenbaum [6], Feng-Zhao [8], Föllmer et al. [9], Moret-Nualart [18], Peskir [24], Rogers-Walsh [26], Russo-Vallois [28,29], Yan et al. [36,37], and the references therein. It is well known that when W is a semimartingale, the forward integral coincides with the Itô integral. However, the following theorem points out that the two integrals are coincident for the process Proof of Theorem 4.1 If f ∈ C 1 (R), then this is Itô's formula since For f / ∈ C 1 (R), by a localization argument we may assume that the function f is uniformly bounded. In fact, for any k ≥ 0 we may consider the set and such that f [k] vanishes outside. Then f [k] is uniformly bounded and If the theorem is true for all uniformly bounded functions on H t , then we get the desired formula on the set Ω k . Letting k tend to infinity we deduce the Itô formula (4.1).
Let now F = f ∈ H t be uniformly bounded and left continuous. Consider the function ζ on R by where c is a normalizing constant such that R ζ (x) dx = 1. Define the mollifiers ζ n (x) := nζ (nx), n = 1, 2, . . . , (4.4) and the sequence of smooth functions Then F n ∈ C ∞ (R) for all n ≥ 1 and the Itô formula holds for all n ≥ 1, where f n = F n . Moreover, by using Lebesgue's dominated convergence theorem, one can prove that, as n → ∞, for each x, in L 2 (Ω) by Corollary 3.1, as n tends to infinity. It follows that in L 2 (Ω), as n tends to infinity. This completes the proof since the integral is closed in L 2 (Ω). Now, we consider the Itô formula including the Skorohod integral of the spatial process W = {W x }. with 0 ≤ β < √ π 4 √ t , then the following Itô type formula holds: for all x ∈ [-M, M].
According to the two theorems above we get the next relationship: if f satisfies the growth condition (4.6). Similar to the proof of Theorem 4.1 one can introduce Theorem 4.2. But we need to give the following standard Itô type formula: for all F ∈ C 2 (R) satisfying the condition F(y) , F (y) , F (y) ≤ Ce βy 2 , y ∈ R (4.10) It is important to note that one have given a standard Itô formula for a large class of Gaussian processes in Alós et al. [1]. However, the process x → W x does not satisfy the condition in Alós et al. [1] since for all t ≥ 0 and x ∈ R. So, we need to give the proof of the formula (4.9).
Proof of (4.9) Let us fix x ∈ [-M, M] and let π ≡ {x n j = jx n ; j = 0, 1, . . . , n} be a partition of [0, x]. Clearly, the growth condition (4.6) implies that for some constant c > 0 and all p < In particular, the estimate (4.11) holds for p = 2. Using the Taylor expansion, we have where W j (θ j ) = W x n j-1 + θ j (W x n j -W x n j-1 ) with θ j ∈ (0, 1) being a random variable. By the duality relationship (2.1) we have Now, in order to end the proof we claim that the following convergences in L 2 hold: 14) (4.15) as n tends to infinity.
To prove the first convergence, it is enough to establish that as n tends to infinity. By the -Minkowski inequality we have Now, we prove the third convergence. We have Suppose that n ≥ m, and for any j = 1, . . . , n we denote by x m(n) j the point of the mth partition that is closer to x n j from the left hand. Then we obtain Clearly, we have Λ n (2, 2) → 0 (n, m → ∞) by Lemma 3.1 and the estimate (4.11), by (2.4), the estimate (4.11) and Minkowski's inequality. Thus, we obtain the third convergence, i.e., J n → t 0 F (W y ) dy in L 1 . Finally, to end the proof we address the second convergence: We need to show that in L 2 (Ω; H ), as n tends to infinity. We have by (2.6) and (2.8). Combining this with (4.16) and the estimate (4.11), we get in L 2 (Ω; H ), as n tends to infinity. It follows that in L 2 (Ω), as n tends to infinity. This completes the proof since the integral · 0 u s δW s is closed in L 2 (Ω).

The Bouleau-Yor identity for the spatial process
In this section, we consider the local time of the process W = {W x , x ∈ [0, M]}. Our main object is to prove that the integral R g(a)L (x, da) is well-defined and that the identity  More work on this subject can be found in Bouleau-Yor [3], Eisenbaum [6], Föllmer et al. [9], Feng-Zhao [8], Peskir [24], Rogers-Walsh [26], Yan et al. [36,37], and the references therein.
Recall that, for any closed interval I ⊂ R + and for any a ∈ R, the local time L(a, I) of u is defined as the density of the occupation measure μ I defined by It can be shown (see Geman and Horowitz [10], Theorem 6.4) that the following occupation density formula holds: for every Borel function g(a, x) ≥ 0 on I × R. Thus, some estimates in Sect. 2 and Theorem 21.9 in Geman-Horowitz [10] together imply that the following result holds. for all x ≥ 0, which gives the first identity. In the same way one can obtain the second identity, and by subtracting the last identity from the previous one, we get the third identity.
According to Theorem 5.1, we get an analogue of the Itô formula (Bouleau-Yor type formula).

Corollary 5.3 Let f ∈ H t be a left continuous function with right limits. If F is an absolutely continuous function with F = f , then the Itô type formula
holds for all x ≥ 0.
Recall that, if F is the difference of two convex functions, then F is an absolutely continuous function with derivative of bounded variation. Thus, the Itô-Tanaka formula F(W x ) = F(0) + holds.