THE STOCHASTIC HEAT EQUATION DRIVEN BY A GAUSSIAN NOISE: GERM MARKOV PROPERTY

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1 Counications on Stochastic Analysis Vol. 2, No. 2 (28) Serials Publications THE STOCHASTIC HEAT EQUATION DRIVEN BY A GAUSSIAN NOISE: GERM MARKOV PROPERTY RALUCA BALAN* AND DOYOON KIM Abstract. Let u = {u(t, x); t [, T ], x R d } be the process solution of the stochastic heat equation u t = u + F, u(, ) = driven by a Gaussian noise F, which is white in tie and has spatial covariance induced by the kernel f. In this paper we prove that the process u is locally ger Markov, if f is the Bessel kernel of order α = 2k, k N +, or f is the Riesz kernel of order α = 4k, k N Introduction This article is based on the theory of stochastic partial differential equations (s.p.d.e. s) initiated by John Walsh in 1986 [24]. This theory relies on the construction of a stochastic integral with respect to the so-called worthy artingale easures and focuses ainly on equations driven by a space-tie white noise. Recently, there has been a considerable aount of interest in s.p.d.e. s driven by a noise ter which is white in tie, but is colored in space, in the sense that it has a spatial covariance structure induced by a kernel f, which is the Fourier transfor of a tepered distribution µ. This line of research was initiated in [15], [4] for the stochastic wave equation in the case d = 2, and then generalized in [3], [6] to a larger class of s.p.d.e. s in arbitrary spatial diensions. The new theory requires an extension of Walsh s stochastic integral, to include the case when the integrand is a (Schwartz) distribution; this type of extension is needed for instance for the stochastic wave equation with d 3. Under a certain integrability condition iposed on the easure µ, one can prove that any second-order s.p.d.e. with constant coefficients has a process solution, i.e. a solution which can be identified with a ultiparaeter stochastic process u. As far as we know, the literature to date does not contain any study of the ger Markov property of this process solution. Such a study is of great iportance because it allows us to conclude that the behavior of the process in any tie-space region A is independent of its behavior outside the region, given the values of the process in a thin area around the boundary A of the region. 2 Matheatics Subject Classification. Priary 6H15; Secondary 6G6. Key words and phrases. Gaussian noise, stochastic integral, stochastic heat equation, Reproducing Kernel Hilbert Space, ger Markov property. The first author was supported by a grant fro the Natural Sciences and Engineering Research Council of Canada. The second author was supported by a post-doctoral fellowship at University of Ottawa. 229

2 23 RALUCA BALAN AND DOYOON KIM There is an iense aount of literature dedicated to the study of different types of Markov properties for ultiparaeter processes (see [2] and the references therein). In particular, the ger Markov property of ultiparaeter Gaussian processes is an area which received a lot of attention in the 197 s, which allows for the use of various tools iported fro Hilbert space analysis, like the ethod of the Reproducing Kernel Hilbert Space (RKHS). An excellent all-tie reference is the onograph [2]; earlier references include [18], [14], [17]. In the 199 s, the proble of the ger Markov property for ulti-diensional Gaussian processes appears again in the literature, this tie for processes which arise as solutions of s.p.d.e. s driven by a white noise: the case of elliptic equations was thoroughly treated in [7], while the systeatic study of this proble for the quasi-linear parabolic equations (in the case d = 1) is found in [16]. The case of hyperbolic equations turns out to be the ost difficult one: the only reference here sees to be [5], in which the authors investigate very carefully the structure of the ger σ-fields induced by the process solution of the stochastic wave equation (in the case d = 2) driven by a Lévy noise without Gaussian coponent. (This type of noise induces a very particular for for the process solution, which is used in a fundaental way for deriving its ger Markov property.) Finally, the ore recent work [19] contains a detailed analysis of the relationship between the ger and sharp σ-fields (which are used for defining the sharp and ger Markov property, respectively), in the case of the process solution associated to the Bessel equation driven by a white noise (in the case d = 2). The goal of the present paper is to investigate the structure of the RKHS associated to the process solution u of the stochastic heat equation driven by a spatial covariance kernel f, and to identify soe exaples of functions f for which the process u possesses the ger Markov property. As far as we know, this is the first attept to tackle a proble of this type in the literature. Our two ain results (Theore 4.9 and Theore 4.15) state that the process solution u has the ger Markov property, if the function f is either the Bessel kernel of order α = 2k, k N or the Riesz kernel of order α = 4k, k N. In order to prove these results, we use soe results fro the L p -theory of parabolic equations with ixed nors, due to [12], [13]. The ger Markov property of the process solution u in the case of the Bessel or Riesz kernels with α > arbitrary (or in the case of other kernel functions) reains an open proble. The paper is organized as follows. In Section 2, we introduce the fraework for the study of s.p.d.e. s, including the construction of the extended stochastic integral due to Dalang (as in [3]). In Section 3, we include all the ingredients which are necessary to forulate the result about the existence of the process solution u, and we exaine the structure of the Gaussian space and the RKHS associated to this process. In Section 4, we give the definition of the ger Markov property and we prove our two ain results, by applying a fundaental result of [14]. The two appendices contain soe technical proofs. 2. The Fraework 2.1. Basic Notation. We denote by D(U) the space of all infinitely differentiable functions on R n whose support is copact and contained in the open set U, and by

3 STOCHASTIC HEAT EQUATION: GERM MARKOV PROPERTY 231 D (U) the space of distributions, i.e. continuous linear functionals on D(U). We denote by S(R n ) the Schwartz space of all rapidly decreasing functions on R n, and by S (R n ) the space of tepered distributions, i.e. continuous linear functionals on S(R n ). The space S (R n ) can be viewed as a subspace of D (R n ). An iportant subspace of S (R n ) is the space O C (Rn ) of all distributions with rapid decrease (see Chapter VII, section 5, [22]). For an arbitrary function φ on R n, the translation by x is denoted with φ x, i.e. φ x (y) = φ(x + y), and the reflection by is denoted with φ, i.e. φ(x) = φ( x). These notions have obvious extensions to distributions. The Fourier transfor of φ L 1 (R n ) is defined by Fφ(ξ) = (2π) n/2 exp( iξ x)φ(x)dx. The ap R n F : S(R n ) S(R n ) can be extended to L 2 (R n ) and S (R n ). For every subset U R n, we denote ϕ, ψ L2 (U) = ϕ(x)ψ(x)dx, whenever the integral is defined The Gaussian Noise. Let T > and f be a locally integrable function on R d. As in [3], for every ϕ, ψ D((, T ) R d ) we define T J f (ϕ, ψ) = ϕ, ψ = ϕ(t, x)f(x y)ψ(t, y)dydxdt. (2.1) R d R d The Gaussian noise entioned in the Introduction will be a zero-ean Gaussian process with covariance J f. The existence of this process is based on the following Bochner Schwartz type result (see Theore 2, p. 157, [11]): Lea 2.1. The bi-functional J f is nonnegative-definite if and only if there exists a tepered easure µ on R d such that f(x) = e iξ x µ(dξ), x R d. R d In this case, for every ϕ, ψ D((, T ) R d ) T ϕ, ψ = Fϕ(t, ξ)fψ(t, ξ)µ(dξ)dt, R d where Fϕ(t, ) denotes the Fourier transfor of ϕ(t, ). The basic exaple of kernel functions is the white noise kernel: f(x) = δ(x), µ(dξ) = dξ. More interesting exaples of kernel functions f arise when µ(dξ) = g( ξ 2 )dξ, for a certain function g on (, ). In this case, f becoes the inverse Fourier transfor of the tepered distribution ξ g( ξ 2 ) and we can define the operator g( ) : S(R d ) S (R d ) by F[g( )φ](ξ) = g( ξ 2 )Fφ(ξ), or equivalently U g( )φ = φ f. (see p , [1], or p , [23]). Here are soe typical exaples: Exaple 2.2. If µ(dξ) = ξ α dξ, then f is the Riesz kernel of order α: f(x) = R α (x) := γ α x d+α, < α < d,

4 232 RALUCA BALAN AND DOYOON KIM where γ α is an appropriate constant. In this case, for every φ S(R d ) we have ( ) α/2 φ = φ R α. Exaple 2.3. If µ(dξ) = (1 + ξ 2 ) α/2 dξ, then f is the Bessel kernel of order α: f(x) = B α (x) := c α τ (α d)/2 1 e τ x 2 /(4τ) dτ, α >, where c α is an appropriate constant. In this case, for every φ S(R d ) we have (1 ) α/2 φ = φ B α. Exaple 2.4. If µ(dξ) = e 4π2 α ξ 2, then f is the heat kernel f(x) = G α (x) := (4πα) d/2 e x 2 /(4α), α >. In this case, for every φ S(R d ) we have e α φ = φ G α. In what follows, we let F = {F (ϕ); ϕ D((, T ) R d )} be a zero-ean Gaussian process with covariance J f, i.e. ϕ, ψ D((, T ) R d ) E(F (ϕ)f (ψ)) = ϕ, ψ. (2.2) The Gaussian space H F of the process F is defined as the closed linear subspace of L 2 (Ω), generated by the variables {F (ϕ), ϕ D((, T ) R d )} The Stochastic Integral. In this subsection, we suarize the construction of the generalized stochastic integral M(ϕ) = T ϕ(t, x)m(dt, dx) with R d respect to the artingale easure M induced by the noise F, due to Dalang (see [3]). For our purposes, it is enough to consider only the case of deterinistic integrands. Here is the construction procedure: Step. For each ϕ D((, T ) R d ), we set M(ϕ) = F (ϕ) H F. Step 1. Let B b (R d ) be the class of all bounded Borel subsets of R d and E (d) be the class of all linear cobinations of functions 1 [,t] A, t [, T ], A B b (R d ). We endow E (d) with the inner product, given by forula (2.1) and we denote by the corresponding nor. For each t (, T ), A B b (R d ), there exists a sequence {ϕ n } n D((, T ) R d ) such that ϕ n 1 [,t] A and suppϕ n K for all n (see [4], p. 19). By the bounded convergence theore, ϕ n 1 [,t] A +, where + is the nor defined in Step 2 below. Hence ϕ n 1 [,t] A and E(M(ϕ ) M(ϕ n )) 2 = ϕ ϕ n as, n, i.e. the sequence {M(ϕ n )} n is Cauchy in L 2 (Ω). A standard arguent shows that its liit does not depend on {ϕ n } n. We set M t (A) = M(1 [,t] A ) = L2(Ω) li n M(ϕ n ) H F. (Note that the process M = {M t (A); t [, T ], A B(R d )} is a worthy artingale easure, in the sense of [24].) We extend M by linearity to E (d). A liiting arguent and relation (2.2) shows that, for every ϕ, ψ E (d) E(M(ϕ)M(ψ)) = ϕ, ψ. (2.3) Step 2. Let P (d) + = {ϕ : [, T ] R d R easurable; ϕ + < }, where ϕ 2 + := T ϕ(t, x) f(x y) ϕ(t, y) dydxdt. We endow P R R (d) d d + with the inner product, given by forula (2.1) and we denote by the corresponding

5 STOCHASTIC HEAT EQUATION: GERM MARKOV PROPERTY 233 nor; note that +. An arguent siilar to that used in the proof of Proposition 2.3, [24], shows that E (d) is dense in P (d) + with respect to +, and hence for each ϕ P (d) +, there exists a sequence {ϕ n } n E (d) such that ϕ n ϕ +. As in Step 1, we set M(ϕ) = L2 (Ω) li n M(ϕ n ) H F. Note that relation (2.3) holds for every ϕ, ψ P (d) +. Step 3. Let P (d) be the copletion of E (d) with respect to <, >. The space P (d) is the largest space of integrands ϕ for which we can define the stochastic integral M(ϕ). According to [3],p. 9, the space P (d) has the following alternative definition. Let P (d) = {ϕ : [, T ] S (R d ); Fϕ(t, ) function t [, T ], (t, ξ) Fϕ(t, ξ) easurable, ϕ < }, where ϕ 2 = T Fϕ(t, ξ) R 2 µ(dξ)dt. Let E (d) d = {ϕ P (d) + ; ϕ(t, ) S(R d ), t [, T ]} and P (d) be the closure of E (d) in P (d) with respect to. Note that ϕ, ψ = T R d Fϕ(t, ξ)fψ(t, ξ)µ(dξ)dt, ϕ, ψ P (d). For each ϕ P (d), there exists a sequence {ϕ n} n E (d) such that ϕ n ϕ. As in Step 1, we set M(ϕ) = L2(Ω) li n M(ϕ n ) H F. Note that relation (2.3) holds for every ϕ, ψ P (d), and hence ϕ M(ϕ) is an isoetry between P (d) and H F. (2.4) In suary, the previous construction is based on the diagra D((, T ) R d ) E (d) P (d) + P (d) P (d) and the following 3 approxiation techniques: indicator functions can be approxiated by functions in D((, T ) R d ); functions in P (d) + can be approxiated by indicator functions; distributions in P (d) can be approxiated by sooth functions in P (d) +. In particular, we conclude that D((, T ) R d ) is dense in P (d) with respect to Alternative Characterization of the Space P (d). As in [9], for every ϕ, ψ D(R d ), define ϕ, ψ,x := ϕ(x)f(x y)ψ(y)dydx = Fϕ(ξ)Fψ(ξ) µ(dξ), R d R d R d and let P (d),x be the copletion of D(Rd ) with respect to,,x. Note that ϕ, ψ = T E (d) ϕ(t, ), ψ(t, ),x dt, ϕ, ψ P (d). (2.5)

6 234 RALUCA BALAN AND DOYOON KIM Reark 2.5. Note that for every ϕ P (d), ϕ(t, ) P(d),x for a.e. t [, T ]. Since E (d) is dense in P (d), one can prove that the ap t ϕ(t, ) is strongly easurable fro [, T ] to P (d),x (in the sense of Definition on p.649, [8]). Using (2.5), we conclude that P (d) L 2 ((, T ), P (d),x ) and ϕ = ϕ L2 ((,T ),P (d) ), ϕ P(d). 3. The Process Solution 3.1. The Equation and its Solution. We consider the stochastic heat equation with vanishing initial conditions, written forally as: u t u = F, in (, T ) R d, u(, ) =. (3.1) The solution of this equation is defined forally as follows. Suppose for the oent that F is a rando variable with values in S (R d+1 ). For every fixed ω Ω, let {u(ϕ); ϕ D((, T ) R d )} be the distribution solution of (3.1). It is known that u(ϕ) = (G F )(ϕ) = F (ϕ G) (3.2) where G is the fundaental solution of the heat equation: { ) (4πt) G(t, x) = d/2 exp ( x 2 4t if t >, x R d if t, x R d (Since G O C (Rd+1 ), we have ϕ G S(R d+1 ) for every ϕ D(R d+1 ) and the convolution G F is well defined; see Theore XI, Chapter VII, [22].) Going back to our fraework, we have the following lea. Lea 3.1. If 1 µ(dξ) <, (3.3) R 1 + ξ 2 d then: (a) (G tx ) ~ P (d) + for every (t, x) [, T ] R d ; (b) ϕ G P (d) for every ϕ D((, T ) R d ). Proof. (a) Set g t,x = (G tx ) ~. Note that g tx is easurable and g tx + = g tx, since g tx. We have Fg t,x (s, ξ) = c d 1 t>s exp( iξx (t s) ξ 2 ), where c d is an appropriate constant. Hence g tx 2 = T Fg t,x (s, ξ) 2 µ(dξ) ds N(T ) R d,x 1 µ(dξ) <, R 1 + ξ 2 d where N(T ) is a constant depending on T. (b) We apply Reark 4, [3] to the function ψ = ϕ G. For this we need to check that: (i) t Fψ(t, ξ) is continuous ξ R d ; (ii) there exists a nonnegative function k(t, ξ) which is square-integrable with respect to dt µ(dξ) such that Fψ(t, ξ) k(t, ξ), t [, T ], ξ R d. (i) is clearly satisfied, and (ii) will follow fro (3.3) once we prove that Fψ(t, ξ) N 1 + ξ 2 := k(t, ξ), t [, T ], ξ Rd, (3.4)

7 STOCHASTIC HEAT EQUATION: GERM MARKOV PROPERTY 235 where N is a constant. Since (1 + ξ 2 ) Fψ(t, x) = Fφ(t, ξ) φ(t, ) L1(R d ) where φ = (1 )ψ, relation (3.4) follows if we prove that φ(t, ) L1(R d ) N. Note that φ t φ = (1 )ϕ in (, T ) R d, φ(t, x) =, since ψ is the unique solution of: ψ t ψ = ϕ in (, T ) R d, ψ(t, x) =. Thus φ(s, y) = T G(t s, x y)(1 )ϕ(t, x) dx dt. Fro this we calculate s R d φ(s, ) L1(R ), which turns out to be bounded (note that (1 )ϕ has a copact d support in (, T ) R d ). A consequence of Lea 3.1.(b) is the fact that M(ϕ G) is well-defined for every ϕ D((, T ) R d ). By analogy with (3.2) (and using a slight abuse of terinology), we introduce the following definition: Definition 3.2. The process {u(ϕ); ϕ D((, T ) R d )} defined by T ( ) u(ϕ) := M(ϕ G) = ϕ(t + s, x + y)g(s, y)dyds M(dt, dx) R d R d R + is called the distribution-valued solution of the stochastic heat equation (3.1), with vanishing initial conditions. Theore 3.3 ([3], [6]). Let {u(ϕ); ϕ D((, T ) R d )} be the distribution-valued solution of the stochastic heat equation (3.1). In order that there exists a jointly easurable process X = {X(t, x); t [, T ], x R d } such that T u(ϕ) = X(t, x)ϕ(t, x)dxdt ϕ D((, T ) R d ) a.s. R d it is necessary and sufficient that (3.3) holds. In this case, X is a odification of the process u = {u(t, x); t [, T ], x R d } defined by u(t, x) := M((G tx ) ~ T ) = G(t s, x y)m(ds, dy). (3.5) R d Definition 3.4. The process u = {u(t, x); t [, T ], x R d } defined by (3.5) is called the process solution of the stochastic heat equation (3.1), with vanishing initial conditions. Note that the process u is a zero-ean Gaussian process. In the present paper, we are interested in exaining the ger Markov property of the process u The Gaussian space. The Gaussian space H u of the process u is defined as the closed linear subspace of L 2 (Ω), generated by the variables {u(t, x), t [, T ], x R d }. The next result shows that this space coincides with the space H F. Lea 3.5. We have H u = H F. Proof. a) First, we prove that H u H F. Recall that u(t, x) = M((G tx ) ~ ) and (G tx ) ~ P (d) + P (d), by Lea 3.1.(a). Since D((, T ) Rd ) is dense in P (d) with respect to, there exists a sequence {ϕ n } n 1 D((, T ) R d ) such that

8 236 RALUCA BALAN AND DOYOON KIM ϕ n (G tx ) ~. Hence E(M(ϕ n ) u(t, x)) 2. But M(ϕ n ) H F for all n and therefore u(t, x) H F. b) Let H u be the closed linear subspace of L 2 (Ω), generated by the variables {u(η), η D((, T ) R d )}. To prove that H F H u, let ϕ D((, T ) R d ) be arbitrary and η = ϕ t ϕ D((, T ) R d ). Then ϕ = η G and F (ϕ) = M(ϕ) = M(η G) = u(η) H u. c) Finally, we prove that H u H u. Let η D((, T ) R d ) be arbitrary. Note that u(η) = M(ϕ) where ϕ = η G. Let K be a copact set such that supp η [, T ] K. For each n 1, let {Q (n), Z} be a partition of (, T ) R d such that each Q (n) = R (n) S (n), where R (n) is an interval in [, T ] and S (n) is a cube, we choose t for all t R (n). We consider the Rieann in R d. Suppose that ax Z Q (n) as n. For each Q (n) (t (n), x (n) ) Q (n) su: such that t (n) where I n = { Z; t (n) (s, y). We clai that ϕ n (s, y) = Q (n) G(t (n) s, x (n) y)η(t (n), x (n) ), I n > s, x (n) K}. Clearly ϕ n (s, y) ϕ(s, y) for every ϕ n ϕ. (3.6) Fro here, it follows that E(M(ϕ n ) M(ϕ)) 2. This concludes that proof, since ϕ n = I n a (n) (G (n) t ) ~ and hence M(ϕ n ) = I n a (n) u(t (n), x (n) ) H u, where a (n),x (n) = Q (n) η(t (n), x (n) ). The proof of (3.6) is given in Appendix A The RKHS. Let H u = {h Y ; h Y (t, x) = E(Y u(t, x)), Y H u } be the RKHS of the process u, endowed with the inner product h Y, h Z H u := E(Y Z), Y, Z H u. Note that any function h H u is continuous on [, T ] R d and satisfies h(, ) =. Moreover, H u is the closure of {R((t, x), ); (t, x) [, T ] R d } with respect to H u, where R((t, x), (s, y)) = E(u(t, x)u(s, y)). By Lea 3.5 and (2.4), it follows that H u = {h(t, x) = E(M(ϕ)u(t, x)); ϕ P (d) }. Moreover, if h(t, x) = E(M(ϕ)u(t, x)), g(t, x) = E(M(η)u(t, x)) with ϕ, η P (d), then h, g H u = ϕ, η. Let g t,x = (G tx ) ~. Using the fact that u(t, x) = M(g tx ) and (2.5), we get: t h(t, x) = E(M(ϕ)u(t, x)) = ϕ, g tx = Fϕ(s, ξ)fg t,x (s, ξ) µ(dξ) ds. (3.7) R d Lea 3.6. Let h(t, x) = E(M(ϕ)u(t, x)), ϕ P (d). For η D((, T ) Rd ), let φ be a solution to the equation φ t φ = η in (, T ) R d, φ(t, ) =. Then h, η L2 ((,T ) R d ) = ϕ, φ. In particular, for any φ D((, T ) R d ), we have h, φ t φ L2((,T ) R d ) = ϕ, φ.

9 STOCHASTIC HEAT EQUATION: GERM MARKOV PROPERTY 237 Proof. Using (3.7) and applying Fubini s theore (since ϕ P (d), g tx P (d) +, and η has copact support), we get h, η L2 ((,T ) R d ) = = T T h(s, ), η(s, ) L2 (R d ) ds R d Fϕ(r, ξ) T Note that for every r (, T ) and y R d φ(r, y) = T r r R d Fg s,x (r, ξ) η(s, x) dx ds µ(dξ) dr. (3.8) R d G(s r, x y)η(s, x) dx ds. Hence T Fφ(r, ξ) = Fg s,x (r, ξ)η(s, x) dx ds (3.9) r R d Fro (3.8) and (3.9), we conclude that h, η L2((,T ) R d ) = ϕ, φ (recall that φ P (d) by Lea 3.1). This finishes the proof. 4. The Ger Markov Property 4.1. The Definition. Let S [, T ] R d be an arbitrary set. Let FS u be the σ-field generated by the variables {u(t, x); (t, x) S}, KS u be the closed linear subspace of L 2 (Ω) generated by the variables {u(t, x); (t, x) S}, and KS u be the closed linear subspace of H u generated by the functions {R((t, x), ); (t, x) S}. Let GS u = FO, u HS u = KO, u HS u = KO. u O open; O S O open; O S O open; O S Definition 4.1. The process u = {u(t, x); (t, x) [, T ] R d } is locally ger Markov if for every relatively copact open set A [, T ] R d, G u and A Gu are A c conditionally independent given G A u, where A = A Ac. Based on the fact that GS u = σ(hu S ), one can prove that u is locally ger Markov if and only if for every open set A, H u = H u A (Hu H u A c A ) (see Lea 1.3, [14]), or equivalently, using the isoetry between H u and H u H u = H u A (Hu A c H u A). (4.1) (Here, denote the usual operations on Hilbert subspaces: if H is a Hilbert space, S is a closed subspace and S is its orthogonal copleent, then we write H = S S and S = H S.) On the other hand, h(t, x) = h, R((t, x), ) H u for every h H u. Hence supp h B c if and only if h (H u B). (4.2) Based on (4.1) and (4.2), we have the following fundaental result.

10 238 RALUCA BALAN AND DOYOON KIM Theore 4.2 (Theore 5.1, [14]). The Gaussian process u is locally ger Markov if and only if the following two conditions hold: (i) If h, g H u are such that (supp h) (supp g) = and supp h is copact, then < h, g > H u=. (ii) If ζ = h + g H u, where h, g are such that (supp h) (supp g) = and supp h is copact, then h, g H u. In the next two subsections, we will suppose that f is the Bessel kernel of order α > (Exaple 2.3), respectively the Riesz kernel of order < α < d (Exaple 2.2). In both cases we ust have α > d 2, in order to have condition (3.3) satisfied. Our goal is to prove that conditions (i) and (ii) of Theore 4.2 hold. For this, we will assue that α = 2k, k N + in the case of the the Bessel kernel, respectively α = 4k, k N + in the case of the Riesz kernel (and d = 4k + 1, since d 2 < α < d) The Case of the Bessel Kernel. In this subsection we will assue that f is the Bessel kernel, i.e. f = B α with α > ax{, d 2}. In this case, P (d),x = H α/2 2 (R d ), where H γ 2 (Rd ) = {ϕ S (R d ); Fϕ is a function, ϕ 2 γ,2 = Fϕ(ξ) 2 (1+ ξ 2 ) γ dξ < } R d denotes the fractional Sobolev space of index γ R (see e.g. [1], p ). By Reark 2.5, P (d) L 2 ((, T ), H α/2 2 (R d )). On the other hand, we have the denseness of E (d) in L 2 ((, T ), H α/2 2 (R d )) (for a proof, one ay see the proof of Theore 3.1 in [12]). Thus P (d) = L 2 ((, T ), H α/2 2 (R d )) and ϕ = ϕ L2, ϕ ((,T ),H α/2 2 (R d P(d) )). For each t [, T ], let ϕ 1 (t, ) := (1 ) α/2 ϕ(t, ) and note that the ap ϕ ϕ 1 is an isoetry between L 2 ((, T ), H α/2 2 (R d )) and L 2 ((, T ), H α/2 2 (R d )). Since Fϕ 1 (t, ξ) = (1 + ξ 2 ) α/2 Fϕ(t, ξ), it is not difficult to see that: ϕ L 2 ((, T ), H α/2 2 (R d )) ϕ 1, φ L2 ((,T ) R d ) = ϕ, φ, φ D((, T ) R d ). (4.3) To investigate the relationship between h and ϕ 1, we need soe general results fro the L p -theory of parabolic equations. Recall that H2 (R d ) = L 2 (R d ), and H γ 2 (Rd ) H γ 2 (Rd ) if γ > γ. For every γ, β R, (1 ) γ/2 is a unitary isoorphis between H β 2 (Rd ) and H γ β 2 (R d ). In particular, (1 ) γ/2 is a unitary isoorphis between H γ 2 (Rd ) and L 2 (R d ). For every v H γ 2 (Rd ), φ D(R d ), we denote (v, φ) = [(1 ) γ/2 v](x)[(1 ) γ/2 φ](x)dx. R d Note that, if v H γ 2 (Rd ) and γ, then (v, φ) = v, φ L2 (R d ) for all φ D(R d ).

11 STOCHASTIC HEAT EQUATION: GERM MARKOV PROPERTY 239 Definition 4.3. If t v(t, ) is a function fro [, T ] to H γ 2 (Rd ), with γ R, we say that v is a solution of dv = ( v + g)dt in (, T ) R d, v(, ) = if for any t (, T ) and for any ψ D(R d ), we have (v(t, ), ψ) = t (v(s, ), ψ)ds + t (g(s, ), ψ)ds. (4.4) We write v H γ 2, (T ) if v xx L 2 ((, T ), H γ 2 2 (R d )), v(, ) =, and there exists g L 2 ((, T ), H γ 2 2 (R d )) satisfying (4.4). By v H γ 2, (T ) we ean v H γ 2, (T ) = v xx L2((,T ),H γ 2 2 (R d )) + g L 2((,T ),H γ 2 2 (R d )). Reark 4.4. It is known that (4.4) iplies that: t (, T ), φ D((, T ) R d ) (v(t, ), φ(t, )) = t (v(s, ), ( / t + )φ(s, ))ds + t (g(s, ), φ(s, ))ds (see e.g. Proposition 1, [9] for a stochastic version of this result). In particular, (4.4) iplies that: φ D((, T ) R d ) T (v(s, ), ( / t )φ(s, ))ds = T (by taking t = T and using the fact that φ(t, ) = ). (g(s, ), φ(s, ))ds (4.5) Theore 4.5 ([12], [13]). Given g L 2 ((, T ), H γ 2 (Rd )), γ R, there exists a unique solution v H γ+2 2, (T ) to the equation dv = ( v + g)dt in (, T ) R d, v(, ) =. Moreover, there exists a constant N (independent of v) such that We now return to our fraework. v H γ+2 2, (T ) N g L 2 ((,T ),L 2 (R d )). Theore 4.6. Let h(t, x) = E(M(ϕ)u(t, x)), where u is defined in (3.5) and ϕ P (d). Set ϕ 1 = (1 ) α/2 ϕ. Then h is the unique solution in H α/2+2 2, (T ) to the equation dh = ( h + ϕ 1 ) dt in (, T ) R d, h(, ) =. (4.6) Proof. Recall that ϕ 1 L 2 ((, T ), H α/2 2 (R d )). Thus by Theore 4.5 there exists a unique solution v H α/2+2 2, (T ) to the equation (4.6). We are now proving that h = v. This will follow once we prove that h, η L2((,T ) R d ) = v, η L2((,T ) R d ), η D((, T ) R d ). Let η D((, T ) R d ) be arbitrary and φ be the unique solution of By Lea 3.6 and (4.3) φ t φ = η in (, T ) R d, φ(t, ) =. h, η L2 ((,T ) R d ) = ϕ, φ = ϕ 1, φ L2 ((,T ) R d ).

12 24 RALUCA BALAN AND DOYOON KIM On the other hand, by (4.5) we have This concludes the proof. v, η L2 ((,T ) R d ) = ϕ 1, φ L2 ((,T ) R d ) Corollary 4.7. If f is the Bessel kernel of order α, then H u = H α/2+2 2, (T ) and the nors in the two spaces are equivalent. Proof. Fro the arguent at the beginning of subsection 3.3, for every h H u, we have h(t, x) = EM(ϕ)u(t, x), where u is defined in (3.5) and ϕ P (d). Then by Theore 4.6, h H α/2+2 2, (T ). For v H α/2+2 2, (T ), there is a ϕ 1 L 2 ((, T ), H α/2 2 (R d )) satisfying (4.4) with ϕ 1 in place of g. Then ϕ := (1 ) α/2 ϕ 1 L 2 ((, T ), H α/2 2 (R d )) = P (d). Set h(t, x) = EM(ϕ)u(t, x) H u, then by Theore 4.6 it follows that v = h, so v H u. Now notice that h H u = ϕ = ϕ L2 ((,T ),H α/2 2 (R d )) = (1 ) α/2 ϕ L2 ((,T ),H α/2 2 (R d )) = ϕ 1 L2 ((,T ),H α/2 2 (R d )) h H α/2+2 2, (T ), where indicates the equivalence of the nors, which follows fro the definition of the nor of H α/2+2 2, (T ) and the estiate in Theore 4.5. This finishes the proof. Reark 4.8. Under the conditions of Theore 4.6, we can also say that h is the unique solution in W 1,2 2 ((, T ) R d ) of: h t = h + ϕ 1 in (, T ) R d, h(, ) =. Here W 1,2 2 ((, T ) R d ) is the space of all easurable functions v : (, T ) R d R, such that the weak derivatives v t, v xi, v xi x j exist and are in L 2 ((, T ) R d ). We are now ready to prove the ain result of this subsection. Theore 4.9. Suppose that f is the Bessel kernel of order α = 2k, k N + such that α > d 2. Then the process solution u of the stochastic heat equation (3.1) with vanishing initial conditions is locally ger Markov. Proof. We need to verify conditions (i) and (ii) of Theore 4.2. We first verify condition (i). Let h, g H u are such that (supp h) (supp g) = and supp h is copact. We have to prove that h, g H u =. We know that there exist ϕ, η P (d) such that h(t, x) = E(M(ϕ)u(t, x)) and g(t, x) = E(M(η)u(t, x)). Then by Theore 4.6 and Corollary 4.7 (also recall the definition of the nor of H γ 2, (T )), h, g H u = h, g H k+2 2, (T ) = h xx, g xx L2 ((,T ),H k 2 (Rd )) + ϕ 1, η 1 L2 ((,T ),H k 2 (Rd )),

13 STOCHASTIC HEAT EQUATION: GERM MARKOV PROPERTY 241 where ϕ 1 = (1 ) k ϕ and η 1 = (1 ) k η. Note that (see also Reark 4.8) supp h xx supp h, supp ϕ 1 supp h. Siilar inclusions hold for g and η 1. Thus it is clear that h xx, g xx L2((,T ),H k 2 (Rd )) = ϕ 1, η 1 L2((,T ),H k 2 (Rd )) =, fro which we arrive at h, g H u =. This finishes the proof of the condition (i). To prove the condition (ii), let ζ = h + g H u, where h and g are such that (supp h) (supp g) = and supp h is copact. We have to prove that h H u. Let χ be an infinitely differentiable function such that χ = 1 on supp h and χ = on an open set containing supp g. By Corollary 4.7, it follows that ζ H2, k+2 (T ), and hence h = χζ Hk+2 2, (T ) = Hu. The theore is proved The Case of the Riesz Kernel. In this subsection, we will assue that f is the Riesz kernel, i.e. f = R α with ax{d 2, } < α = 4k < d, k N +. According to [21], for any < β < d/2 we can define the Riesz potential I β ϕ(y) := ϕ R β (y) = 1 γ n (β) R d ϕ(y) x y d β dy for all ϕ L 2(R d ). Let q = q d,β > 2 be such that 1/q = 1/2 β/d. The space I β (L 2 (R d )) of all Riesz potentials has the following properties (see [21]): (1) I β (L 2 (R d )) L q (R d ) and there exists a constant N > such that I β ϕ Lq(R d ) N ϕ L2(R d ), for all ϕ L 2 (R d ). (2) For every f I β (L 2 (R d )), we define the Riesz derivative D β f as li ε D β ε f in L 2 (R d ), where (D β ε f)(x) = 1 ( l yf)(x) dy (is independent of l), c d,l (β) y d+β y >ε l yf = (I τ y ) l f and (τ y f)(x) = f(x y). Then F(D β f)(ξ) = Ff(ξ) ξ β, f S(R d ). (3) I β (L 2 (R d )) = {f L q (R d ); D β f exists and is in L 2 (R d )}. (4) D β is the left inverse of I β in L 2 (R d ), i.e. D β (I β ϕ) = ϕ for all ϕ L 2 (R d ). (5) D(R d ) is dense in I β (L 2 (R d )) with respect to the nor f Lq (R d ) + D β f L2(R d ). In what follows, we will use these properties with β = α/2 = 2k, where d 2 < 4k < d. We let q = q d,4k > 2 be such that 1/q = 1/2 2k/d. Proposition 4.1. Let 2 < d/(2k) and 1/q = 1/2 2k/d. If f I 2k (L 2 (R d )), where k N +, then where Ff is an eleent in S (R d ). F ( D 2k f ) = ξ 2k Ff,

14 Proof. Note that (F ( D 2k f ), φ ) = ( ξ 2k Ff, φ ) = ( Ff, ξ 2k φ ). 242 RALUCA BALAN AND DOYOON KIM Proof. First of all, ξ 2k Ff is well-defined as an eleent of S since ξ 2k = (ξ ξ 2 d )k is infinitely differentiable. Due to the fact that D(R d ) is dense in I 2k (L 2 (R d )), there exists a sequence {f n } D(R d ) such that f n f Lq (R d ) + D 2k f n D 2k f L2 (R d ). Note that F ( D 2k f n ) = ξ 2k Ff n S(R d ). Also note that ( F ( D 2k f n ), φ ) = ( ξ 2k Ff n, φ ) = ( Ff n, ξ 2k φ ) = ( f n, F 1 ( ξ 2k φ )) ( f, F 1 ( ξ 2k φ )) = ( ξ 2k Ff, φ ), where the convergence is possible because f n f in L q (R d ). This along with the fact that D 2k f n D 2k f in L 2 (R d ) iplies that F ( D 2k f ) = ξ 2k Ff. Corollary Let 2 < d/(2k) and 1/q = 1/2 2k/d. If f I 2k (L 2 (R d )), where k N +, then D 2k f = ( ) k f. On the other hand, ( F ( ( ) k f ), φ ) = ( ( ) k f, F 1 φ ) = ( f, ( ) k F 1 φ ) = ( f, F 1 ( ξ 2k φ )) = ( Ff, ξ 2k φ ). Lea Let 2 < d/2k. There exists a linear operator J : P (d) L 2 ((, T ) R d ) such that J is one-to-one and onto, and satisfies: for any ϕ D((, T ) R d ). Jϕ(t, x) = I 2k (ϕ(t, )) (x) Proof. For all ϕ D((, T ) R d ), set J(ϕ) = I 2k ϕ(t, y) (ϕ(t, )) (x) = c d,k dy, R x y d 2k d where c d,k is an appropriate constant. Denote ϕ := J(ϕ). We see that ϕ (t, x) is easurable in (t, x) (, T ) R d. Since ϕ(t, ) S(R d ) for each fixed t (, T ), we have Fϕ (t, )(ξ) = Fϕ(t, ξ) ξ 2k in the sense that ϕ (t, x)φ(x) dx = Fϕ(t, ξ) ξ 2k Fφ(ξ) dξ, R d R d where φ S(R d ). Also notice that, for a.e. t (, T ), Fϕ(t, ξ) ξ 2k L 2 (R d )

15 STOCHASTIC HEAT EQUATION: GERM MARKOV PROPERTY 243 because ϕ P (d) (recall that α = 4k), i.e., T R d Fϕ(t, ξ) ξ 2k 2 dξ dt <. Thus, for a.e. t (, T ), as a function of x R d, ϕ (t, x) L 2 (R d ) and Thus Fϕ (t, )(ξ) = Fϕ(t, ξ) ξ 2k. T J(ϕ) 2 L 2((,T ) R d ) = = T R d ϕ (t, x) 2 dx dt R d Fϕ(t, ξ) 2 ξ 4k dξ dt = ϕ 2. Fro this and the fact that D((, T ) R d ) is dense in P (d), we extend J to all eleents in P (d). It is easy to see that J is linear as well as one-to-one. To prove the fact that J is onto, take ϕ L 2 ((, T ) R d ). Also take a sequence {ϕ n } D((, T ) R d ) such that ϕ n ϕ in L 2 ((, T ) R d ). Let ϕ n = ( ) k ϕ n. Then and T Fϕ n (t, ξ) = ξ 2k Fϕ n (t, ξ) Fϕ n (t, ξ) 2 ξ 4k dξ dt = Fϕ n 2 dξ dt. R d R d Thus {ϕ n } is a Cauchy sequence in P (d), so there is a ϕ P(d) such that ϕ n ϕ in P (d). To prove J(ϕ) = ϕ, we only need to prove that which follows fro Fro the above result it follows that: I 2k (ϕ n ) = ϕ n, T F ( I 2k (ϕ n ) ) = ξ 2k Fϕ n. Corollary For ϕ, η P (d), let ϕ := J(ϕ) and η := J(η). Then Now we set ϕ, η = ϕ, η L2 ((,T ) R d ). ϕ 1 (t, x) = I 2k (ϕ (t, )) (x) = c d,k ϕ (t, y) dy, R x y d 2k d where ϕ L 2 ((, T ) R d ). Notice that ϕ 1 is a easurable function of (t, x) (, T ) R d. Since ϕ (t, x) L 2 (R d ) for a.e. t (, T ), ϕ 1 (t, ) L q (R d ), ϕ 1 (t, ) Lq (R d ) N ϕ (t, ) L2 (R d ), for a.e. t (, T ), where 1/q = 1/2 2k/d. This iplies that ϕ 1 L 2,q ((, T ) R d ),

16 244 RALUCA BALAN AND DOYOON KIM ( ) 1/2 T ϕ 1 L2,q ((,T ) R d ) = ϕ 1 (t, ) 2 L q(r d ) dt N ϕ L2 ((,T ) R ). d Fro the L p -theory of parabolic equations with ixed nors, there exists a unique function w W 1,2 2,q ((, T ) Rd ) satisfying w t w = ϕ 1, w(, ) =. (4.7) Lea Let h(t, x) = EM(ϕ)u(t, x), where u is defined in (3.5). Let ϕ = J(ϕ), ϕ 1 be a function defined as above, and w be the solution to (4.7). Then w = h. Proof. For η D((, T ) R d ), find a function φ, a unique solution to Then by Lea 3.6 and φ t φ = η, φ(t, ) =. h, η L2 ((,T ) R d ) = ϕ, φ w, η L2 ((,T ) R d ) = w t w, φ L2 ((,T ) R d ) = ϕ 1, φ L2 ((,T ) R d ). Find a sequence {ϕ n } D((, T ) R d ) such that ϕ n ϕ in L 2 ((, T ) R d ). Let ψ n = I 2k (ϕ n ) and w n be the solution to ( / t )w n = ψ n and w n (, ) =. Then w, η L2((,T ) R d ) = li w n, η L2((,T ) R n d ) = li ψ n, φ L2((,T ) R n d ) T = li n I2k (ϕ n ), φ L2 ((,T ) R d ) = li Fϕ n (t, ξ)fφ(t, ξ) ξ 2k dξ dt. n R d On the other hand, T h, η L2((,T ) R d ) = ϕ, φ = li Fϕ n (t, ξ)fφ(t, ξ) ξ 4k dξ dt n R d T = li Fϕ n (t, ξ)fφ(t, ξ) ξ 2k dξ dt, n R d where ϕ n = ( ) 2k ϕ n. This finishes the proof. Now we prove the ain result of this subsection. Theore Suppose that f is the Riesz kernel of order α = 4k, k N + and d = 4k + 1. Then the process solution u of the stochastic heat equation (3.1) with vanishing initial conditions is locally ger Markov. Proof. Again we need to verify conditions (i) and (ii) in Theore 4.2. We first verify condition (i). Let h(t, x) = EM(ϕ)u(t, x) and g(t, x) = EM(η)u(t, x), where ϕ, η P (d), such that supp h supp g =. We need to show that ϕ, η =. Let ϕ = J(ϕ) and η = J(η), where J is the operator defined in Lea Also let ϕ 1 = I 2k (ϕ (t, )) and η 1 = I 2k (η (t, )). For a.e. t (, T ), we have ϕ (t, x) = D 2k ϕ 1 (t, x) = ( ) k ϕ 1,

17 STOCHASTIC HEAT EQUATION: GERM MARKOV PROPERTY 245 where the second equality follows fro Corollary Let φ(t, x) D((, T ) R d ) such that φ(t, x) = on supp ϕ 1. Then for a.e. t (, T ), ϕ (t, x)φ(t, x) dx = ϕ 1 (t, x)( ) k φ(t, x) dx. R d R d Thus T T ϕ (t, x)φ(t, x) dx dt = ϕ 1 (t, x)( ) k φ(t, x) dx dt =. R d R d This shows that supp ϕ supp ϕ 1. Siilarly, we have supp η supp η 1. Hence we have ϕ, η L2((,T ) R d ) =. This and Corollary 4.13 prove that ϕ, η =. We now verify condition (ii). Assue that ζ = h + g, where ζ(t, x) = EM(ν)u(t, x), supp h supp g =, and supp h is copact. We have to prove that h H u. Let χ be an infinitely differentiable function defined on [, T ] R d such that χ 1, χ = 1 on supp µ, and χ = in an (relative) open set containing supp ν. Then χζ = h and h W 1,2 2,q ((, T ) Rd ). Let ( / t )ζ = ν 1, where ν 1 = I 2k (ν ) and ν = J(ν). Now we set ϕ 1 := χν 1, η 1 := (1 χ)ν 1, ϕ := χν. Since ϕ L 2 ((, T ) R d ) there exists a sequence {ϕ n } D((, T ) R d ) such that ϕ n ϕ in L 2 ((, T ) R d ). Let ϕ n = ( ) k ϕ n. Then ϕ n D((, T ) R d ) and {ϕ n } is a Cauchy sequence in P (d) due to the fact that Fϕ n (t, ξ) = ξ 2k Fϕ n (t, ξ). Let ϕ be the liit in P (d) of ϕ n. Then J(ϕ) = ϕ. Now let ˆϕ 1 = I 2k (ϕ ) and ĥ(t, x) = EM(ϕ)u(t, x). We now prove that h = ĥ, which will iply that h Hu. For this, it suffices to prove that ϕ 1 = ˆϕ 1. By Lea 4.14, we have ( / t )ĥ = ˆϕ 1. Notice that ν 1 = ϕ 1 + η 1 and supp ϕ 1 supp η 1 =. Thus for a.e. t (, T ), we have supp x ϕ 1 (t, ) supp x η 1 (t, ) =. Thus by Lea B.1 it follows that, for a.e. t (, T ), We also have D 2k ϕ 1 (t, ) = ϕ (t, ). D 2k ˆϕ 1 (t, ) = ϕ (t, ) for a.e. t (, T ). Since ϕ 1 (t, ) ˆϕ 1 (t, ) I 2k (L 2 (R d )) for a.e. t (, T ), there exists a function f t (x) such that ϕ 1 (t, ) ˆϕ 1 (t, ) = I 2k (f t ) for a.e. t (, T ). Fro the fact that D 2k is the left inverse of the I 2k, we know that = D 2k (ϕ 1 (t, ) ˆϕ 1 (t, )) = D 2k I 2k (f t ) = f t. We also know that I 2k () =. Hence ϕ 1 (t, ) = ˆϕ 1 (t, ) for a.e. t (, T ). Therefore, ϕ 1 = ˆϕ 1 as eleents in L 2,q ((, T ) R d ). The theore is proved.

18 246 RALUCA BALAN AND DOYOON KIM We need to prove that T Appendix A. Proof of Relation (3.6) R d Fϕ n (s, ξ) Fϕ(s, ξ) 2 µ(dξ)ds, as n. Note that Fϕ n (s, ξ) = I n c d Q (n) exp{ iξ x (n) (t (n) s) ξ 2 }η(t (n), x (n) ) converges pointwise to T Fϕ(s, ξ) = c d exp{ iξ x (t s) ξ 2 }η(t, x) dx dt. s R d In order to apply the doinated convergence theore, we need to prove that there exists a function Ψ, which is square-integrable with respect to ds µ(dξ) such that Fϕ n (s, ξ) Ψ(s, ξ). Let N be a constant such that η N. Then Fϕ n (s, ξ) I n c d Q n η(t (n), x (n) ) exp{ (t (n) s) ξ 2 } T c d N Q (n) e (t(n) s) ξ 2 c d Ne (t s) ξ 2 dx dt I s K n c d N K (T s)1 ξ <1 + c d N K ) ξ 2 1 e(s T ξ 2 1 ξ 1 := Ψ(s, ξ), where K is a copact set containing K, and the third inequality above is due to the fact that 1 (n) Q (t, x)e (t(n) s) ξ 2 1 (n) Q (t, x)e (t s) ξ 2. Clearly Ψ is squareintegrable with respect to ds µ(dξ). This concludes the proof of (3.6). Appendix B. An Auxiliary Lea The following technical result was used in the proof of Theore 4.15 for the verification of condition (ii). Lea B.1. Let 2 < d/2k and 1/q = 1/2 2k/d. Assue that κ I 2k (L 2 (R d )) and κ = µ + ν, where µ, ν L q (R d ), supp µ supp ν =, and supp µ is copact. Then µ I 2k (L 2 (R d )), D 2k µ = χd 2k κ, where χ is an infinitely differentiable function such that χ 1, χ = 1 on supp µ, and χ = in an open set containing supp ν. Proof. Since κ I 2k (L 2 (R d )), we have D 2k ε κ := 1 c(d, l, 2k) z >ε ( l z κ ) (x) z d+2k dz D 2k κ in L 2 (R d ). Note that D 2k ε κ L 2 (R d ) (see Proposition 2.4 in [1]). Let Λ 2k ε κ := 1 c(d, l, 2k) ε< z ε ( l z κ ) (x) z d+2k dz, ε < ε.

19 STOCHASTIC HEAT EQUATION: GERM MARKOV PROPERTY 247 Since Λ 2k ε κ = D 2k ε κ D 2k ε κ, we have Λ 2k ε κ L 2 (R d ). Moreover, Λ 2k ε 1 κ Λ 2k ε 2 κ = ε 1 < z ε 2 ( l z κ ) (x) z d+2k dz = D 2k ε 1 κ D 2k ε 2 κ. For this and the fact that D 2k ε κ is Cauchy in L 2 (R d ), it follows that {Λ 2k ε κ} is a Cauchy sequence in L 2 (R d ). Consider D 2k ε (χκ), which is well-defined because χκ L q (R d ). Note that ( D 2k ε (χκ) = Λ 2k l ε (χκ) + z χκ ) (x) dz z ε z d+2k and Λ 2k ε (χκ) = χλ 2k ε κ (B.1) for a sufficiently sall ε >. The proof of (B.1) is stated below. Also note that ( Λ 2k ε (χκ) L 2 (R d l ), z χκ ) (x) z d+2k dz L 2 (R d ), z ε where the latter follows fro the fact that χκ L 2 (R d ) (χ has a copact support and 2 q). In addition, Λ 2k ε κ approaches a function in L 2 (R d ) as ε. Thus D 2k ε (χκ) converges a function in L 2 (R d ). This shows that µ = χκ I 2k (L 2 (R d )). To prove D 2k µ = χd 2k κ, we can just use Corollary Let us prove (B.1) as follows. For each x supp µ, find δ x > such that B(x, δ x ) supp ν =. Consider {B(x, δ x /5) : x supp µ}. Since supp µ is copact, we have a finite subset {x i : i I} supp µ such that B(x i, δ xi /5) supp µ. Let Then and O 1 := i I B(x i, δ xi /4), i I O 2 := i I B(x i, δ xi /3), O 3 := i I B(x i, δ xi /2). O i supp µ, i = 1, 2, 3 O i supp ν =, i = 1, 2, 3. We ay assue that χ = 1 on O 2 and χ = on O3. c We also assue that ε < δ/(2l), where δ = in i I δ xi. By definition we have Λ 2k ε (χκ) (x) = 1 ( l z (χκ) ) (x) c ε< z ε z d+2k dz = 1 l k= ( 1)k c k l χ(x kz)κ(x kz) c z d+2k dz ε< z ε Let x O 1, i.e. x B(x i, δ xi /4). Then x i (x kz) x i x + kz δ xi /4 + lδ/(2l)

20 248 RALUCA BALAN AND DOYOON KIM δ xi /4 + δ xi /12 = δ xi /3. Thus x kz O 2. This shows that χ(x kz) = 1 and Λ 2k ε (χκ) (x) = χ(x)λ 2k ε κ(x), x O 1. Now let x O c 1. Then for every x i, i I, x i (x kz) x i x l z δ xi /4 lδ/(2l) δ xi /4 δ xi /2 = δ xi /5. This iplies that x kz / supp µ. In that case we have χ(x kz)κ(x kz) = because κ(x kz) = in case χ(x kz) >. Thus Λ 2k ε (χκ) (x) =. In addition, χ(x)λ 2k ε κ(x) = (recall that x O1 c and x kz / supp µ) because κ(x kz) = in case χ(x) >. Indeed, if x O 3 \ O 1, then there exists x j, j I, such that x B(x j, δ xj /2). Then x j (x kz) x j x + l z < δ xj. Thus by the choice of δ xj it follows that x kz / supp ν, that is, κ(x kz) =. Acknowledgent. The authors are grateful to an anonyous referee who carefully read the article and ade soe suggestions for future research. References 1. Aleida, A. and Sako, S. G.: Characterization of Riesz and Bessel potentials on variable Lebesgue spaces. J. Funct. Spaces Appl. 4 (26) Balan, R. M. and Ivanoff, B. G.: A Markov property for set-indexed processes, J. Theoret. Prob. 15 (22) Dalang, R. C.: Extending artingale easure stochastic integral with applications to spatially hoogenous s.p.d.e. s. Electr. J. Probab. 4 (1999) no. 6, Erratu in Electr. J. Probab. 4 (1999) no. 6, Dalang, R. C. and Frangos, N. E.: The stochastic wave equation in two spatial diensions. Ann. Probab. 26 (1998) Dalang, R. C. and Hou, Q.: On Markov properties of Lévy waves in two diensions. Stoch. Proc. Appl. 72 (1997) Dalang, R. C. and Mueller, C.: Soe non-linear s.p.d.e. s that are second order in tie. Electr. J. Probab. 8 (23) no. 1, Donati-Martin, C. and Nualart, D.: Markov property for elliptic stochastic partial differential equations. Stoch. Stoch. Rep. 46 (1994) Evans, L. C. (1998) Partial Differential Equations. Graduate Studis in Matheatics. 19, Aer. Math. Soc. Providence, RI. 9. Ferrante, M. and Sanz-Sole, M.: SPDE s with colored noise: analytic and stochastic approaches. ESAIM: Prob. Stat. 1 (26) Folland, G. B.: Introduction to Partial Differential Equations, Second Edition, Princeton University Press, Princeton, Gel fand, I. M. and Vilenkin, N. Y.: Generalized Functions, Vol. 4, Acadeic Press, New York, Krylov, N. V.: An analytic approach to SPDEs, in: Stochastic partial differential equations: six perspectives, 64 (1999) Math. Surveys Monogr., , Aer. Math. Soc., Providence, RI. 13. Krylov, N. V.: The heat equation in L q ((, T ), L p )-spaces with weights. SIAM J. Math. Anal., 32 (21) (electronic).

21 STOCHASTIC HEAT EQUATION: GERM MARKOV PROPERTY Kunsch, H.: Gaussian Markov rando fields. J. Fac. Sci. Univ. Tokyo. Sec. IA. 26 (1979) Mueller, C.: Long tie existence for the wave equation with a noise ter. Ann. Probab. 25 (1997) Nualart, D. and Pardoux, E.: Markov field properties of solutions of white noise driven quasi-linear parabolic p.d.e. s. Stoch. Stoch, Rep. 48 (1994) Nualart, D., and Sanz, M.: A Markov property for two-paraeter Gaussian processes. Stochastica 3 (1979) Pitt, L. D.: A Markov property for Gaussian processes with ultidiansional paraeters. Arch. Rational Mech. Anal. 43 (1971) Pitt, L. D. and Robeva, R. S.: On the sharp Markov property for Gaussian rando fields and spectral synthesis in spaces of Bessel potentials. Ann. Probab. 31 (23) Rozanov, Yu. A.: Markov Rando Fields, Springer, Berlin, Sako, S. G.: On spaces of Riesz potentials. Izv. Acad. Nauk SSSR, 4 (1976) Schwartz, L.: Théorie des distributions, Herann, Paris, Stein, E.M.: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, Walsh, J. B.: An introduction to stochastic partial differential equations. Ecole d Eté de Probabilités de Saint-Flour XIV. Lecture Notes in Math. 118 (1986) Springer- Verlag, Berlin. Raluca Balan: Departent of Matheatics and Statistics, University of Ottawa, 585, King Edward Avenue, Ottawa, Ontario K1N 6N5 Canada E-ail address: rbalan@uottawa.ca URL: rbalan Doyoon Ki: Departent of Matheatics, University of Southern California, Los Angeles, 362 Veront Avenue, KAP18, Los Angeles, CA 989, USA E-ail address: doyoonki@usc.edu URL: