A Local time correspondence for stochastic partial differential equations
aa r X i v : . [ m a t h . P R ] N ov A LOCAL-TIME CORRESPONDENCE FORSTOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
MOHAMMUD FOONDUN, DAVAR KHOSHNEVISAN, AND EULALIA NUALART
Abstract.
It is frequently the case that a white-noise-driven parabolic and/or hyperbolicstochastic partial differential equation (SPDE) can have random-field solutions only in spa-tial dimension one. Here we show that in many cases, where the “spatial operator” is the L -generator of a L´evy process X , a linear SPDE has a random-field solution if and only ifthe symmetrization of X possesses local times. This result gives a probabilistic reason forthe lack of existence of random-field solutions in dimensions strictly bigger than one.In addition, we prove that the solution to the SPDE is [H¨older] continuous in its spatialvariable if and only if the said local time is [H¨older] continuous in its spatial variable. We alsoproduce examples where the random-field solution exists, but is almost surely unbounded inevery open subset of space-time. Our results are based on first establishing a quasi-isometrybetween the linear L -space of the weak solutions of a family of linear SPDEs, on one hand,and the Dirichlet space generated by the symmetrization of X , on the other hand.We study mainly linear equations in order to present the local-time correspondence at amodest technical level. However, some of our work has consequences for nonlinear SPDEsas well. We demonstrate this assertion by studying a family of parabolic SPDEs that haveadditive nonlinearities. For those equations we prove that if the linearized problem hasa random-field solution, then so does the nonlinear SPDE. Moreover, the solution to thelinearized equation is [H¨older] continuous if and only if the solution to the nonlinear equationis. And the solutions are bounded and unbounded together as well. Finally, we prove thatin the cases that the solutions are unbounded, they almost surely blow up at exactly thesame points. Date : November 11, 2007.2000
Mathematics Subject Classification.
Primary. 60H15, 60J55; Secondary. 35R60, 35D05.
Key words and phrases.
Stochastic heat equation, stochastic wave equation, Gaussian noise, existence ofprocess solutions, local times, isomorphism theorems.The research of D.K. is supported in part by the National Science Foundation grant DMS-0404729. Contents Introduction
More general equations
Existence of functions-valued solutions: Proof of Theorem 2.1
Spatial continuity: Proof of Theorem 2.4
An aside on temporal continuity
Spatial and joint continuity: Proofs of Theorems 2.5 and 2.6
Heat equation via generators of Markov processes − ǫ A semilinear parabolic problem
Introduction
We consider the stochastic heat equation inspired by the fundamental works of Pardoux(1975a, 1975b, 1972), Krylov and Rozovskii (1979a, 1979b, 1977), and Funaki (1983). Let˙ w denote space-time white noise, t is nonnegative, x is in R d , and the Laplacian acts on the x variable. Then, we have:(1.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ t H ( t , x ) = (∆ H )( t , x ) + ˙ w ( t , x ) ,H (0 , x ) = 0 . Let us consider also the stochastic wave equation of Caba˜na (1970):(1.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ tt W ( t , x ) = (∆ W )( t , x ) + ˙ w ( t , x ) ,W (0 , x ) = ∂ t W (0 , x ) = 0 . One of the common features of (1.1) and (1.2) is that they suffer from a curse of dimen-sionality . Namely, these equations can have random-field solutions only in dimension one.Moreover, this curse of dimensionality appears to extend beyond the linear parabolic setting
OCAL TIMES AND SPDES 3 of (1.1), or the linear hyperbolic setting of (1.2). For instance, see Perkins (2002, CorollaryIII.4.3) for an example from superprocesses, and Walsh (1986, Chapter 9) for an examplefrom statistical mechanics.One can informally ascribe this curse of dimensionality to the “fact” that while the Lapla-cian smooths, white noise roughens. In one dimension, the roughening effect of white noiseturns out to be small relative to the smoothing properties of the Laplacian, and we thushave a random-field solution. However, in dimensions greater than one white noise is muchtoo rough, and the Laplacian cannot smooth the solution enough to yield a random field.Dalang and Frangos (1998) were able to construct a first fully-rigorous explanation of thecurse of dimensionality. They do so by first replacing white noise by a Gaussian noise that iswhite in time and colored in space. And then they describe precisely the roughening effectof the noise on the solution, viewed as a random generalized function. See also Brze´zniakand van Neerven (2003), Dalang and Mueller (2003), Millet and Sanz-Sol´e (1999), Peszat(2002), and Peszat and Zabczyk (2000). More recently, Dalang and Sanz-Sol´e (2005) studyfully nonlinear stochastic wave equations driven by noises that are white in time and coloredin space, and operators that are arbitrary powers of the Laplacian.In this article we present a different explanation of this phenomenon. Our approachis to describe accurately the smoothing effect of the Laplacian in the presence of whitenoise. Whereas the answer of Dalang and Frangos (1998) is analytic, ours is probabilistic.For instance, we will see soon that (1.1) and (1.2) have solutions only in dimension onebecause d -dimensional Brownian motion has local times only in dimension one [Theorem2.1]. Similarly, when d = 1, the solution to (1.1) [and/or (1.2)] is continuous in x becausethe local time of one-dimensional Brownian motion is continuous in its spatial variable.The methods that we employ also give us a local-time paradigm that makes precise theclaim that the stochastic PDEs (1.1) and (1.2) “have random-field solutions in dimension d = 2 − ǫ for all ǫ ∈ (0 , § § §
3. Section 4 containsthe proof of the necessary and sufficient local-time condition for continuity of our SPDEs intheir space variables. Although we are not aware of any interesting connections between localtimes and temporal regularity of the solutions of SPDEs, we have included § §
2. We also produceexamples of SPDEs that have random-field solutions which are almost surely unbounded inevery open space-time set [Example 5.5].
FOONDUN, KHOSHNEVISAN, AND NUALART
Section 6 discusses issues of H¨older continuity in either space, or time, variable. In § − ǫ for all ǫ ∈ (0 , § § Y := { Y t } t ≥ on R d ,consider the occupation measure[s],(1.3) Z ( t , ϕ ) := Z t ϕ ( Y s ) ds for all t ≥ ϕ : R d → R + .We can identify each Z ( t , • ) with a measure in the usual way. Then, we say that Y has localtimes when Z ( t , dx ) ≪ dx for all t . The local times of Y are themselves defined by Z ( t , x ) := Z ( t , dx ) /dx . It follows that if Y has local times, then Z ( t , ϕ ) = R R d Z ( t , x ) ϕ ( x ) dx a.s. forevery t ≥ ϕ : R d → R + . And the converse holds also.2. More general equations
In order to describe when (1.1) and (1.2) have random-field solutions, and why, we studymore general equations.Let L denote the generator of a d -dimensional L´evy process X := { X t } t ≥ with charac-teristic exponent Ψ. We can normalize things so that E exp( iξ · X t ) = exp( − t Ψ( ξ )), and OCAL TIMES AND SPDES 5 consider L as an L -generator with domain(2.1) Dom L := (cid:26) f ∈ L ( R d ) : Z R d | ˆ f ( ξ ) | | Ψ( ξ ) | dξ < ∞ (cid:27) . As usual, ˆ f denotes the Fourier transform of f ; we opt for the normalization(2.2) ˆ f ( ξ ) := Z R d e iξ · x f ( x ) dx for all f ∈ L ( R d ).[The general L -theory of Markov processes is described in great depth in Fukushima,¯Oshima, and Takeda (1994) in the symmetric case, and Ma and R¨ockner (1992) for thegeneral case.]In this way, we can—and will—view L as a generalized convolution operator with Fouriermultiplier ˆ L ( ξ ) := − Ψ( ξ ).We consider two families of stochastic partial differential equations. The first is the sto-chastic heat equation for L :(2.3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ t H ( t , x ) = ( L H )( t , x ) + ˙ w ( t , x ) ,H (0 , x ) = 0 , where x ranges over R d and t over R + := [0 , ∞ ), and the operator L acts on the x variable.We also consider hyperbolic SPDEs of the following wave type:(2.4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ tt W ( t , x ) = ( L W )( t , x ) + ˙ w ( t , x ) ,W (0 , x ) = ∂ t W (0 , x ) = 0 . Recall that X := { X t } t ≥ is a L´evy process whose generator is L . Let X ′ denote anindependent copy of X and define its symmetrization, a la Paul L´evy, by(2.5) ¯ X t := X t − X ′ t for all t ≥ . It is a standard fact that ¯ X is a symmetric L´evy process with characteristic exponent 2Re Ψ.The following is our first main result, though the precise meaning of its terminology is yetto be explained. Theorem 2.1.
The stochastic heat equation (2.3) has random-field solutions if and only ifthe symmetric L´evy process ¯ X has local times. The same is true for the stochastic waveequation (2.4) , provided that the process X is itself symmetric.Remark . Brze´zniak and van Neerven (2003) consider parabolic equations of the type(2.3), where L is a pseudo-differential operator with a symbol that is bounded below inour language [bounded above in theirs]. If we apply their theory with our constant-symboloperator L , then their bounded-below condition is equivalent to the symmetry of the L´evy FOONDUN, KHOSHNEVISAN, AND NUALART process X , and their main result is equivalent to the parabolic portion of Theorem 2.1, underthe added condition that X is symmetric. (cid:3) It is well-known that when d ≥
2, L´evy processes in R d do not have local times (Hawkes, 1986).It follows from this that neither (2.3) nor (2.4) [under a symmetry assumption on X ] canever have random-field solutions in dimension greater than one. But it is possible that thereare no random-field solutions even in dimension one. Here is one such example; many othersexist. Example 2.3.
Suppose X is a strictly stable process in R with stability index α ∈ (0 , X has local times if and only if α >
1; see, for example, Hawkes (1986). If X is itself symmetric, then L = − ( − ∆) α/ is the α -dimensional fractional Laplacian (Stein, 1970, Chapter V, § α > (cid:3) The local-time correspondence of Theorem 2.1 is not a mere accident. In fact, the nexttwo theorems suggest far deeper connections between the solutions to the linear SPDEs ofthis paper and the theory of local times of Markov processes. We emphasize that the nexttwo theorems assume the existence of a random-field solution to one of the stochastic PDEs(2.3) and/or (2.4). Therefore, they are inherently one dimensional statements.
Theorem 2.4.
Assume that d = 1 and the stochastic heat equation (2.3) has a random-fieldsolution { H ( t , x ) } t ≥ ,x ∈ R . Then, the following are equivalent: (1) There exists t > such that x H ( t , x ) is a.s. continuous. (2) For all t > , x H ( t , x ) is a.s. continuous. (3) The local times of ¯ X are a.s. continuous in their spatial variable.The same equivalence is true for the solution W to the stochastic wave equation (2.4) , pro-vided that the process X is itself symmetric. Theorem 2.5.
Assume that d = 1 and the stochastic heat equation (2.3) has a random-fieldsolution { H ( t , x ) } t ≥ ,x ∈ R . Then, the following are equivalent: (1) There exists t > such that x H ( t , x ) is a.s. H¨older continuous. (2) For all t > , x H ( t , x ) is a.s. H¨older continuous. (3) The local times of ¯ X are a.s. H¨older continuous in their spatial variable.If the process X is itself symmetric, then the preceding conditions are also equivalent tothe H¨older continuity of the solution W to the stochastic wave equation (2.4) in the spatialvariable. Finally, the critical H¨older indices of x H ( t , x ) , x W ( t , x ) and that of thelocal times of ¯ X are the same. OCAL TIMES AND SPDES 7
Blumenthal and Getoor (1961) have introduced several “indices” that describe variousproperties of a L´evy process. We recall below their lower index β ′′ :(2.6) β ′′ := lim inf | ξ |→∞ log Re Ψ( ξ )log | ξ | = sup (cid:26) α ≥ | ξ |→∞ Re Ψ( ξ ) | ξ | α = ∞ (cid:27) . Theorem 2.6. If β ′′ > d , then the stochastic heat equation (2.3) has a random-field solutionthat is jointly H¨older continuous. The critical H¨older index is ≤ ( β ′′ − d ) / for the spacevariable and ≤ ( β ′′ − d ) / β ′′ for the time variable. Furthermore, if X is symmetric, then thesame assertions hold for the solution to the stochastic wave equation (2.4) .Remark . If X is in the domain of attraction of Brownian motion on R , then β ′′ = 2,and the critical temporal and spatial index bounds of Theorem 2.6 are respectively 1 / /
4. These numbers are well known to be the optimal H¨older indices. In fact, the relativelysimple H¨older-index bounds of Theorem 2.6 are frequently sharp; see Example 5.4. (cid:3)
It is well known that 0 ≤ β ′′ ≤ β ′′ ≤
2. [But of course the original proof of Blumenthal and Getoor is simpler.]Several things need to be made clear here; the first being the meaning of a “solution.”With this aim in mind, we treat the two equations separately.We make precise sense of the stochastic heat equation (2.3) in much the same man-ner as Walsh (1986); see also Brze´zniak and van Neerven (2003), Dalang (1999, 2001),and Da Prato (2007). It is well known that it is much harder to give precise meaningto stochastic hyperbolic equations of the wave type (2.4), even when L is the Laplacian(Dalang and L´evˆeque, 2004a; , 2004b; Dalang and Mueller, 2003; Quer-Sardanyons and Sanz-Sol´e, 2003;Mueller, 1997; Dalang, 1999; Mueller, 1993). Thus, as part of the present work, we introducea simple and direct method that makes rigorous sense of (2.4) and other linear SPDEs ofthis type. We believe this method to be of some independent interest.2.1. The parabolic case.
In order to describe the meaning of (2.3) we need to first intro-duce some notation.Let { P t } t ≥ denote the semigroup of the driving L´evy process X ; that is, ( P t f )( x ) :=E f ( x + X t ) for all bounded Borel-measurable functions f : R d → R [say], all x ∈ R d , andall t ≥
0. [As usual, X := 0.] Formally speaking, P t = exp( t L ).The dual semigroup is denoted by P ∗ , so that ( P ∗ t f )( x ) = E f ( x − X t ). It is easy to seethat P and P ∗ are adjoint in L ( R d ) in the sense that(2.7) ( P t f , g ) = ( f , P ∗ t g ) for all f, g ∈ L ( R d ). FOONDUN, KHOSHNEVISAN, AND NUALART
Needless to say, ( · , · ) denotes the usual Hilbertian inner product on L ( R d ).Let S ( R d ) denote the class of all rapidly decreasing test functions on R d , and recall that:(i) S ( R d ) ⊂ Dom( L ) ∩ C ∞ ( R d ) ⊂ L ( R d ) ∩ C ∞ ( R d ); and (ii) for all ϕ, ψ ∈ S ( R d ),(2.8) lim h ↓ (cid:18) P t + h ϕ − P t ϕh , ψ (cid:19) = ( L P t ϕ , ψ ) . Thus, v ( t , x ) := ( P t ϕ )( x ) solves the Kolmogorov equation ∂ t v ( t , x ) = ( L v )( t , x ), subject tothe initial condition that v (0 , x ) = ϕ ( x ). This identifies the Green’s function for ∂ t − L =0. Hence, we can adapt the Green-function method of Walsh (1986, Chapter 3), withoutany great difficulties, to deduce that a weak solution to (2.3) is the Gaussian random field { H ( t , ϕ ); t ≥ , ϕ ∈ S ( R d ) } , where(2.9) H ( t , ϕ ) := Z t Z R d ( P ∗ t − s ϕ )( y ) w ( dy ds ) . This is defined simply as a Wiener integral.
Proposition 2.8.
The Gaussian random field { H ( t , ϕ ); t ≥ , ϕ ∈ S ( R d ) } is well defined.Moreover, the process ϕ H ( t , ϕ ) is a.s. linear for each t ≥ .Proof. On one hand, the Wiener isometry tells us that(2.10) E (cid:0) | H ( t , ϕ ) | (cid:1) = Z t k P ∗ t − s ϕ k L ( R d ) ds. On the other hand, it is known that each P ∗ s is a contraction on L ( R d ). Indeed, it is nothard to check directly that if ℓ d denotes the Lebesgue measure on R d , then the dual L´evyprocess − X is ℓ d -symmetric (Fukushima et al., 1994, pp. 27–28). Therefore, the assertedcontraction property of P ∗ s follows from equation (1.4.13) of Fukushima, ¯Oshima, and Takeda(1994, p. 28). It follows then that E( | H ( t , ϕ ) | ) ≤ t k ϕ k L ( R d ) , and this is finite for all t ≥ ϕ ∈ S ( R d ). This proves that u is a well-defined Gaussian random field indexedby R + × S ( R d ). The proof of the remaining property follows the argument of Dalang(1999, Section 4) quite closely, and is omitted. (cid:3) The nonrandom hyperbolic case.
It has been known for some time that hyperbolicSPDEs tend to be harder to study, and even to define precisely, than their parabolic coun-terparts. See, for instance, Dalang and Sanz-Sol´e (2007) for the most recent work on thestochastic wave equation in dimension 3.In order to define what the stochastic wave equation (2.4) means precisely, we can tryto mimic the original Green-function method of Walsh (1986). But we quickly run intothe technical problem of not being able to identify a suitable Green function (or even ameasure) for the corresponding integral equation. In order to overcome this obstacle, one
OCAL TIMES AND SPDES 9 could proceed as in Dalang and Sanz-Sol´e (2007), but generalize the role of their fractionalLaplacian. Instead, we opt for a more direct route that is particularly well suited for studyingthe SPDEs of the present type.In order to understand (2.4) better, we first consider the deterministic integro-differentialequation,(2.11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ tt u ( t , x ) = ( L u )( t , x ) + f ( t , x ) ,u (0 , x ) = ∂ t u (0 , x ) = 0 , where f : R + × R d → R is a “nice” function, and the variables t and x range respectivelyover R + and R d . We can study this equation only under the following symmetry condition:(2.12) The process X is symmetric . Equivalently, we assume that Ψ is real and nonnegative .Recall that “ˆ” denotes the Fourier transform in the x variable, and apply it informallyto (2.11) to deduce that it is equivalent to the following: For all t ≥ ξ ∈ R d ,(2.13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ tt ˆ u ( t , ξ ) = − Ψ( ξ )ˆ u ( t , ξ ) + ˆ f ( t , ξ ) , ˆ u (0 , ξ ) = ∂ t ˆ u (0 , ξ ) = 0 . This is an inhomogeneous second-order ordinary differential equation [in t ] which can besolved explicitly, via Duhamel’s principle, to produce the following “formula”:(2.14) ˆ u ( t , ξ ) = 1 p Ψ( ξ ) Z t sin (cid:16)p Ψ( ξ ) ( t − s ) (cid:17) ˆ f ( s , ξ ) ds. We invert the preceding—informally still—to obtain(2.15) u ( t , x ) = 1(2 π ) d Z R d Z t sin (cid:16)p Ψ( ξ ) ( t − s ) (cid:17)p Ψ( ξ ) e − iξ · x ˆ f ( s , ξ ) ds dξ. We can multiply this by a nice function ϕ , then integrate [ dx ] to arrive at(2.16) u ( t , ϕ ) := 1(2 π ) d Z R d Z t sin (cid:16)p Ψ( ξ ) ( t − s ) (cid:17)p Ψ( ξ ) ˆ ϕ ( ξ ) ˆ f ( s , ξ ) ds dξ. We may think of as this as the “weak/distributional solution” to (2.11).2.3.
The random hyperbolic case.
We follow standard terminology and identify thewhite noise ˙ w with the iso-Gaussian process { w ( h ) } h ∈ L ( R + × R d ) as follows:(2.17) w ( h ) := Z R d Z ∞ h ( s , x ) w ( ds dx ) . Next, we define the
Fourier transform ˆ w of white noise:(2.18) ˆ w ( h ) := w (ˆ h )(2 π ) d/ = 1(2 π ) d/ Z R d Z ∞ ˆ h ( s , ξ ) w ( ds dξ ) , all the time remembering that “ˆ h ” refers to the Fourier transform of h in its spatial variable.Suppose h ( s , x ) = ϕ ( s ) ϕ ( x ) for t ≥ x ∈ R d , where ϕ ∈ L ( R + ) and ϕ ∈ L ( R d ).Then, it follows from the Wiener isometry and Plancherel’s theorem that(2.19) k ˆ w ( h ) k L ( R + × R d ) = k h k L ( R + × R d ) . Because L ( R + ) ⊗ L ( R d ) is dense in L ( R + × R d ), this proves that ˆ w is defined continuouslyon all of L ( R + × R d ). Moreover, ˆ w corresponds to a white noise which is correlated with˙ w , as described by the following formula:(2.20) E h w ( h ) · ˆ w ( h ) i = 1(2 π ) d/ Z ∞ Z R d ˆ h ( s , ξ ) h ( s , ξ ) dξ ds, valid for all h , h ∈ L ( R + × R d ).In light of (2.16), we define the weak solution to the stochastic wave equation (2.4) as theWiener integral(2.21) W ( t , ϕ ) = 1(2 π ) d/ Z R d Z t sin (cid:16)p Ψ( ξ ) ( t − s ) (cid:17)p Ψ( ξ ) ˆ ϕ ( ξ ) ˆ w ( ds dξ ) . It is possible to verify that this method also works for the stochastic heat equation (2.3), andthat it produces a equivalent formulation of the Walsh solution (2.9). However, in the presentsetting, this method saves us from having to describe the existence [and some regularity] ofthe Green function for the integral equation (2.11).
Proposition 2.9.
If the symmetry condition (2.12) holds, then the stochastic wave equation (2.4) has a weak solution W for all ϕ ∈ S ( R d ) . Moreover, { W ( t , ϕ ); t ≥ , ϕ ∈ S ( R d ) } is a well-defined Gaussian random field, and ϕ W ( t , ϕ ) is a.s. linear for all t ≥ .Proof. We apply the Wiener isometry to obtain(2.22) E (cid:0) | W ( t , ϕ ) | (cid:1) = 1(2 π ) d Z R d Z t sin (cid:16)p Ψ( ξ ) ( t − s ) (cid:17) Ψ( ξ ) | ˆ ϕ ( ξ ) | dsdξ. Because | sin θ/θ | ≤ (cid:0) | W ( t , ϕ ) | (cid:1) ≤ π ) d Z R d Z t ( t − s ) | ˆ ϕ ( ξ ) | ds dξ = t k ϕ k L ( R d ) , (2.23) OCAL TIMES AND SPDES 11 thanks to Plancherel’s theorem. It follows immediately from this that { W ( t , ϕ ); t ≥ , ϕ ∈ S ( R d ) } is a well-defined Gaussian random field. The remainder of the proposition is stan-dard. (cid:3) Existence of functions-valued solutions: Proof of Theorem 2.1
Let u := { u ( t , ϕ ); t ≥ , ϕ ∈ S ( R d ) } denote the weak solution to either one of (2.3) or(2.4). Our present goal is to extend uniquely the Gaussian random field u to a Gaussianrandom field indexed by R + × M , where M is a maximal subset of D ( R d )—the space of allSchwartz distributions on R d . Such an M exists thanks solely to functional-analytic facts:Define, temporarily,(3.1) d t ( ϕ ) := q E (cid:0) | u ( t , ϕ ) | (cid:1) for all ϕ ∈ S ( R d ) and t ≥ u in ϕ shows that ( ϕ , ψ ) d t ( ϕ − ψ ) defines a metric for each t ≥ M t denote the completion of S ( R d ) in D ( R d ) with respect to the metric induced by d t ;and define M := ∩ t ≥ M t . The space M can be identified with the largest possible family ofcandidate test functions for weak solutions to either the stochastic heat equation (2.3) or thestochastic wave equation (2.4). Standard heuristics from PDEs then tell us that (2.3) and/or(2.4) has random-field solutions if and only if δ x ∈ M for all x ∈ R d ; this can be interpretedas an equivalent definition of random-field solutions. When δ x ∈ M we may write u ( t , x ) inplace of u ( t , δ x ). In order to prove Theorem 2.1 we will need some a priori estimates on theweak solutions of both the stochastic equations (2.3) and (2.4). We proceed by identifying M with generalized Sobolev spaces that arise in the potential theory of symmetric L´evyprocesses. Now let us begin by studying the parabolic case.3.1. The parabolic case.Proposition 3.1.
Let H denote the weak solution (2.9) to the stochastic heat equation (2.3) .Then, for all ϕ ∈ S ( R d ) , λ > , and t ≥ , (3.2) 1 − e − t/λ E ( λ ; ϕ ) ≤ E (cid:0) | H ( t , ϕ ) | (cid:1) ≤ e t/λ E ( λ ; ϕ ) , where (3.3) E ( λ ; ϕ ) := 1(2 π ) d Z R d | ˆ ϕ ( ξ ) | (1 /λ ) + ReΨ( ξ ) dξ. Next we record the following immediate but useful corollary; it follows from Proposition3.1 by simply setting λ := t . Corollary 3.2. If H denotes the weak solution to the stochastic heat equation (2.3) , thenfor all ϕ ∈ S ( R d ) and t ≥ , (3.4) E ( t ; ϕ ) ≤ E (cid:0) | H ( t , ϕ ) | (cid:1) ≤ E ( t ; ϕ ) , The preceding upper bound for E( | H ( t , ϕ ) | ) is closely tied to an energy inequality for theweakly asymmetric exclusion process. See Lemma 3.1 of Bertini and Giacomin (1999); theyascribe that lemma to H.-T. Yau.The key step of the proof of Proposition 3.1 is an elementary real-variable result which weprove next. Lemma 3.3. If g : R + → R + is Borel measurable and nonincreasing, then for all t, λ > , (3.5) (cid:0) − e − t/λ (cid:1) Z ∞ e − s/λ g ( s ) ds ≤ Z t g ( s ) ds ≤ e t/λ Z ∞ e − s/λ g ( s ) ds. Monotonicity is not needed for the upper bound on R t g ( s ) ds .Proof of Lemma 3.3. The upper bound on R t g ( s ) ds follows simply because e t − s ) /λ ≥ t ≥ s . In order to derive the lower bound we write Z ∞ e − s/λ g ( s ) ds = ∞ X n =0 Z ( n +1) tnt e − s/λ g ( s ) ds ≤ ∞ X n =0 e − nt/λ Z t g ( s + nt ) ds. (3.6)Because g is nonincreasing we can write g ( s + nt ) ≤ g ( s ) to conclude the proof. (cid:3) Proof of Proposition 3.1.
We know from (2.9) and the Wiener isometry that(3.7) E (cid:0) | H ( t , ϕ ) | (cid:1) = Z t k P ∗ s ϕ k L ( R d ) ds. Since the Fourier multiplier of P ∗ s is exp( − s Ψ( − ξ )) at ξ ∈ R d , we can apply the Planchereltheorem and deduce the following formula:(3.8) k P ∗ s ϕ k L ( R d ) = 1(2 π ) d Z R d e − s ReΨ( ξ ) | ˆ ϕ ( ξ ) | dξ. Because ReΨ( ξ ) ≥
0, Lemma 3.3 readily proves the proposition. (cid:3)
Equation (3.3) can be used to define E ( λ ; ϕ ) for all Schwartz distributions ϕ , and notonly those in S ( R d ). Moreover, it is possible to verify directly that ϕ E ( λ ; ϕ ) / definesa norm on S ( R d ). But in all but uninteresting cases, S ( R d ) is not complete in this norm.Let L L ( R d ) denote the completion of S ( R d ) in the norm E ( λ ; • ) / . Thus the Hilbert space L L ( R d ) can be identified with M . The following is a result about the potential theory of OCAL TIMES AND SPDES 13 symmetric L´evy processes, but we present a self-contained proof that does not depend onthat deep theory.
Lemma 3.4.
The space L L ( R d ) does not depend on the value of λ . Moreover, L L ( R d ) isa Hilbert space in norm E ( λ ; • ) / for each fixed λ > . Finally, the quasi-isometry (3.2) isvalid for all t ≥ , λ > , and ϕ ∈ L L ( R d ) .Proof. We write, temporarily, L L ,λ ( R d ) for L L ( R d ), and seek to prove that it is independentof the choice of λ .Define for all distributions ϕ and ψ ,(3.9) E ( λ ; ϕ , ψ ) := 12(2 π ) d "Z R d ˆ ϕ ( ξ ) ˆ ψ ( ξ )(1 /λ ) + ReΨ( ξ ) dξ + Z R d ˆ ψ ( ξ ) ˆ ϕ ( ξ )(1 /λ ) + ReΨ( ξ ) dξ . For each λ > ϕ , ψ ) E ( λ ; ϕ , ψ ) is a pre-Hilbertian inner product on S ( R d ), and E ( λ ; ϕ ) = E ( λ ; ϕ , ϕ ).Thanks to Proposition 3.1, for all α > c = c α,λ such that c − E ( α ; ϕ ) ≤ E ( λ ; ϕ ) ≤ c E ( α ; ϕ ) for all ϕ ∈ S ( R d ). This proves that L L ,λ ( R d ) = L L ,α ( R d ), whence follows the independence of L L ( R d ) from the value of λ .The remainder of the lemma is elementary. (cid:3) The space L L ( R d ) is a generalized Sobolev space, and contains many classical spaces ofBessel potentials, as the following example shows. Example 3.5.
Suppose L = − ( − ∆) s/ for s ∈ (0 , L is the generator of anisotropic stable- s L´evy process, and L L ( R d ) is the space H − s/ ( R d ) of Bessel potentials. Fora nice pedagogic treatment see the book of Folland (1976, Chapter 6). (cid:3) The hyperbolic case.
The main result of this section is the following quasi-isometry;it is the wave-equation analogue of Proposition 3.1.
Proposition 3.6.
Suppose the symmetry condition (2.12) holds, and let W := { W ( t , ϕ ); t ≥ , ϕ ∈ S ( R d ) } denote the weak solution to the stochastic wave equation (2.4) . Then, (3.10) t E (cid:0) t ; ϕ (cid:1) ≤ E (cid:0) | W ( t , ϕ ) | (cid:1) ≤ t E (cid:0) t ; ϕ (cid:1) . for all t ≥ and ϕ ∈ S ( R d ) . Moreover, we can extend W by density so that the precedingdisplay continues to remain valid when t ≥ and ϕ ∈ L L ( R d ) .Proof. Although Lemma 3.3 is not applicable, we can proceed in a similar manner as we didwhen we proved the earlier quasi-isometry result for the heat equation (Proposition 3.1).
Namely, we begin by observing that(3.11) E (cid:0) | W ( t , ϕ ) | (cid:1) = 1(2 π ) d Z t Z R d sin (cid:16)p Ψ( ξ ) s (cid:17) Ψ( ξ ) | ˆ ϕ ( ξ ) | dξ ds. See (2.21). If θ > θ is at most the minimum of one and θ . This leads to thebounds E (cid:0) | W ( t , ϕ ) | (cid:1) ≤ π ) d Z t Z R d (cid:18) s ∧ ξ ) (cid:19) | ˆ ϕ ( ξ ) | dξ ds ≤ t (2 π ) d Z R d (cid:18) t ∧ ξ ) (cid:19) | ˆ ϕ ( ξ ) | dξ. (3.12)The upper bound follows from this and the elementary inequality t ∧ z − ≤ / ( t − + z ),valid for all z ≥ (cid:0) | W ( t , ϕ ) | (cid:1) = t π ) d Z R d − sin (cid:16) p Ψ( ξ ) t (cid:17) p Ψ( ξ ) t | ˆ ϕ ( ξ ) | Ψ( ξ ) dξ. We shall analyze the integral by splitting it according to whether or not Ψ ≤ /t .Taylor’s expansion [with remainder] reveals that if θ is nonnegative, then sin θ is at most θ − ( θ /
6) + ( θ / − sin θθ ≥ θ
15 if 0 ≤ θ ≤ . Consequently, Z { Ψ ≤ /t } − sin (cid:16) p Ψ( ξ ) t (cid:17) p Ψ( ξ ) t | ˆ ϕ ( ξ ) | Ψ( ξ ) dξ ≥ t Z { Ψ ≤ /t } | ˆ ϕ ( ξ ) | dξ ≥ Z { Ψ ≤ /t } (cid:18) t ∧ ξ ) (cid:19) | ˆ ϕ ( ξ ) | dξ. (3.15)For the remaining integral we use the elementary bound 1 − (sin θ/θ ) ≥ /
2, valid for all θ >
2. This leads to the following inequalities: Z { Ψ > /t } − sin (cid:16) p Ψ( ξ ) t (cid:17) p Ψ( ξ ) t | ˆ ϕ ( ξ ) | Ψ( ξ ) dξ ≥ Z { Ψ > /t } | ˆ ϕ ( ξ ) | Ψ( ξ ) dξ = 12 Z { Ψ > /t } (cid:18) t ∧ ξ ) (cid:19) | ˆ ϕ ( ξ ) | dξ. (3.16)The proof concludes from summing up equations (3.15) and (3.16), and then plugging theend result into (3.13). (cid:3) OCAL TIMES AND SPDES 15
We now give a proof of Theorem 2.1.
Proof.
Let us begin with the proof in the case of the stochastic heat equation (2.3). Propo-sition 3.1 is a quasi-isometry of the maximal space M of test functions for weak solutionsof (2.3) into L L ( R d ). Thus, M can be identified with the Hilbert space L L ( R d ), and hence(2.3) has random-field solutions if and only if δ x ∈ L L ( R d ) for all x ∈ R d . Thanks to Lemma3.4, the stochastic heat equation (2.3) has random-field solutions if and only if(3.17) Z R d dξϑ + ReΨ( ξ ) < ∞ for some, and hence all, ϑ > X to have local times (Hawkes, 1986, Theorem 1). The remaining portion ofthe proof follows from the preceding in much the same way as the first portion was deducedfrom Proposition 3.1. (cid:3) Spatial continuity: Proof of Theorem 2.4
Proof.
We work with the stochastic heat equation (2.3) first. Without loss of generality,we may—and will—assume that (2.3) has a random-field solution H ( t , x ), and ¯ X has localtimes. Else, Theorem 2.1 finishes the proof.Let ϕ := δ x − δ y , and note that | ˆ ϕ ( ξ ) | = 2(1 − cos( ξ ( x − y ))) is a function of x − y .Because H ( t , ϕ ) = H ( t , x ) − H ( t , y ), equations (3.7) and (3.8) imply that z H ( t , z ) is acentered Gaussian process with stationary increments for each fixed t ≥ h ( r ) := 12 π Z ∞−∞ − cos( rξ )1 + Re Ψ( ξ ) dξ, defined for all r ≥ . According to Lemma 3.4, Proposition 3.1 holds for all Schwartz distributions ϕ ∈ L L ( R d ).The existence of random-field solutions is equivalent to the condition that δ x ∈ L L ( R d ) forall x ∈ R . We apply Proposition 3.1 to ϕ := δ x − δ y , with λ := 1 [say], and find that(4.2) (cid:0) − e − t (cid:1) h ( | x − y | ) ≤ E (cid:0) | H ( t , x ) − H ( t , y ) | (cid:1) ≤ e t h ( | x − y | ) . Define ¯ h to be the Hardy–Littlewood nondecreasing rearrangement of h . That is,(4.3) ¯ h ( r ) := inf { y ≥ g ( y ) > r } where g ( y ) := meas { r ≥ h ( r ) ≤ y } . Then according to the proof of Corollary 6.4.4 of Marcus and Rosen (2006, p. 274), thestationary-increments Gaussian process x H ( t , x ) has a continuous modification iff(4.4) Z + ¯ h ( r ) r | log r | / dr < ∞ . Next we claim that (4.4) is equivalent to the continuity of the local times of the symmetrizedL´evy process ¯ X . We recall that the characteristic exponent of ¯ X is 2Re Ψ, and hence by theL´evy–Khintchine formula it can be written as(4.5) 2Re Ψ( ξ ) = σ ξ + Z ∞−∞ (1 − cos( ξx )) ν ( dx ) , where ν is a σ -finite Borel measure on R with R ∞−∞ (1 ∧ x ) ν ( dx ) < ∞ . See, for exampleBertoin (1996, Theorem 1, p. 13).Suppose, first, that(4.6) either σ > Z ∞−∞ (1 ∧ | x | ) ν ( dx ) = ∞ .Then, (4.4) is also necessary and sufficient for the [joint] continuity of the local times of thesymmetrized process ¯ X ; confer with Barlow (1988, Theorems B and 1). On the other hand,if (4.6) fails to hold then ¯ X is a compound Poisson process. Because ¯ X is also a symmetricprocess, the L´evy–Khintchine formula tells us that it has zero drift. That is, ¯ X t = P Π( t ) j =1 Z i ,where { Z i } ∞ i =1 are i.i.d. and symmetric, and Π is an independent Poisson process. It followsimmediately from this that the range of ¯ X is a.s. countable in that case. This proves that theoccupation measure for ¯ X is a.s. singular with respect to Lebesgue measure, and therefore¯ X cannot possess local times. But that contradicts the original assumption that ¯ X has localtimes. Consequently, (4.4) implies (4.6), and is equivalent to the spatial continuity of [amodification of] local times of ¯ X . This proves the theorem in the parabolic case.Let us assume further the symmetry condition (2.12). Because E ( t ; ϕ ) / E (1 ; ϕ ) is boundedabove and below by positive finite constants that depend only on t > (cid:3) An aside on temporal continuity
We spend a few pages discussing matters of temporal continuity—especially temporalH¨older continuity—of weak solutions of the stochastic heat equation (2.3), as well as thestochastic wave equation (2.4).
Definition 5.1.
We call a function g ( s ) a gauge function if the following are satisfied:(1) g ( s ) is an increasing function;(2) g ( s ) is a slowly varying function at infinity;(3) g ( s ) satisfies the integrability condition,(5.1) Z + dss log(1 /s ) g (1 /s ) < ∞ . OCAL TIMES AND SPDES 17
Next, we quote a useful property of slow varying functions (Bingham et al., 1989, p. 27).
Proposition 5.2. If g is a slowly varying function and α > , then the integral R ∞ x t − α g ( t ) dt converges for every x > , and (5.2) Z ∞ x g ( t ) t α dt ∼ g ( x )( α − x α − as x → ∞ . We can now state the main theorem of this section. It gives a criteria for the temporalcontinuity of the weak solutions of our stochastic equations.
Theorem 5.3.
Let H denote the weak solution to the stochastic heat equation (2.3) . Let g be a gauge function in the sense of Definition 5.1. Choose and fix ϕ ∈ L L ( R d ) . Then, t H ( t , ϕ ) has a continuous modification if the following is satisfied (5.3) Z R d log(1 + Re Ψ( ξ )) g (1 + | Ψ( ξ ) | )1 + Re Ψ( ξ ) | ˆ ϕ ( ξ ) | dξ < ∞ . Moreover, the critical H¨older exponent of t H ( t , ϕ ) is precisely (5.4) ind E ( • ; ϕ ) := lim inf ǫ ↓ log E ( ǫ ; ϕ )log ǫ . Consequently, t H ( t , ϕ ) has a H¨older-continuous modification (a.s.) iff ind E ( • ; ϕ ) > .If the symmetry condition (2.12) holds, then (5.3) guarantees the existence of a continuousmodification of t W ( t , ϕ ) , where W denotes the weak solution to the stochastic waveequation. Furthermore, (5.4) implies the temporal H¨older continuity of t W ( t , ϕ ) of anyorder < ind E ( • ; ϕ ) . We now give two examples. The first one is about the temporal H¨older exponent while thesecond one provides a family of random-field solutions which are almost surely unboundedin every open space-time set.
Example 5.4.
Suppose d = 1 and L = − ( − ∆) α/ for some α ∈ (1 , x ∈ R . According to Theorem 2.1, δ x ∈ L L ( R ), so we can apply Theorem 5.3 with ϕ := δ x .In this case, E ( ǫ ; δ x ) = 12 π Z ∞−∞ dξ (1 /ǫ ) + | ξ | α = const · ǫ − (1 /α ) . (5.5)In particular, ind E ( • ; δ x ) = 1 − (1 /α ), whence it follows that the critical H¨older exponent of t H ( t , x ) is precisely − (2 α ) − . When L = ∆, we have α = 2 and the critical temporalexponent is 1 /
4, which agrees with a well-known folklore theorem. For example, Corollary3.4 of Walsh (1986, pp. 318–320) and its proof contain this statement for the closely-related stochastic cable equation. In particular, see the last two lines on page 319 of Walsh’s lectures( loc. cit. ). (cid:3) Example 5.5.
Choose and fix α ∈ R . According to the L´evy–Khintchine formula (Bertoin, 1996,Theorem 1, p. 13) we can find a symmetric L´evy process whose characteristic exponents sat-isfies Ψ( ξ ) ∼ ξ (log ξ ) α as ξ → ∞ . Throughout, we assume that α >
1. This ensures thatcondition (3.17) is in place; i.e., (1 + Ψ) − ∈ L ( R ). Equivalently, that both SPDEs (2.3)and (2.4) have a random-field solution.A few lines of computations show that for all x ∈ R ,(5.6) E ( ǫ ; δ x ) ≍ | log(1 /ǫ ) | − α +1 , where a ( ǫ ) ≍ b ( ǫ ) means that a ( ǫ ) /b ( ǫ ) is bounded and below by absolute constants, uni-formly for all ǫ > x ∈ R . Next, consider(5.7) d ( s , t ) := q E (cid:0) | H ( s , x ) − H ( t , x ) | (cid:1) . According to Proposition 5.6 below, (5.6) implies that(5.8) d ( s , t ) ≍ (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) | s − t | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (1 − α ) / , uniformly for all s and t in a fixed compact subset [0 , T ] of R + , say. Let N denote the metric entropy of [0 , T ] in the [pseudo-] metric d (Dudley, 1967). That is, for all ǫ >
0, wedefine N ( ǫ ) to be the minimum number of d -balls of radius ǫ needed to cover [0 , T ]. Then,it is easy to deduce from the previous display that log N ( ǫ ) ≍ ǫ − / ( α − , and hence(5.9) lim ǫ ↓ ǫ p log N ( ǫ ) = 0 if and only if α > . Therefore, if 1 < α ≤ t H ( t , x )—and also t W ( t , x )—does not have any continuousmodifications, almost surely. On the other hand, if α >
2, then the integrability condition(5.3) holds manifestly, and hence t H ( t , x ) and t W ( t , x ) both have continuousmodifications. Thus, the sufficiency condition of Theorem 5.3 is also necessary for the presentexample. In addition, when α ≤
2, the random-field solutions H and W are both unboundeda.s. in every open set. This assertion follows from general facts about Gaussian processes;see, for example equation (6.32) of Marcus and Rosen (2006, p. 250). We skip the details. (cid:3) Estimate in the parabolic case.Proposition 5.6.
For every t, ǫ ≥ and ϕ ∈ L L ( R d ) , (5.10) E ( ǫ ; ϕ ) ≤ E (cid:0) | H ( t + ǫ , ϕ ) − H ( t , ϕ ) | (cid:1) ≤ E ( ǫ ; ϕ ) + e t F ( ǫ ; ϕ ) , OCAL TIMES AND SPDES 19 where (5.11) F ( ǫ ; ϕ ) := 1(2 π ) d Z R d (cid:0) ∧ ǫ | Ψ( ξ ) | (cid:1) | ˆ ϕ ( ξ ) | ξ ) dξ. Proof.
Because of density it suffices to prove that the proposition holds for all functions ϕ ∈ S ( R d ) of rapid decrease. By (2.9) and Wiener’s isometry,E (cid:0) | H ( t + ǫ , ϕ ) − H ( t , ϕ ) | (cid:1) = Z t Z R d (cid:12)(cid:12)(cid:0) P ∗ t − s + ǫ ϕ (cid:1) ( y ) − (cid:0) P ∗ t − s ϕ (cid:1) ( y ) (cid:12)(cid:12) dy ds + Z t + ǫt Z R d (cid:12)(cid:12)(cid:0) P ∗ t − s + ǫ ϕ (cid:1) ( y ) (cid:12)(cid:12) dy ds. (5.12)We apply Plancherel’s theorem to find thatE (cid:0) | H ( t + ǫ , ϕ ) − H ( t , ϕ ) | (cid:1) = 1(2 π ) d Z t Z R d (cid:12)(cid:12) e − ( s + ǫ )Ψ( − ξ ) − e − s Ψ( − ξ ) (cid:12)(cid:12) | ˆ ϕ ( ξ ) | dξ ds + 1(2 π ) d Z ǫ Z R d e − s Re Ψ( ξ ) | ˆ ϕ ( ξ ) | dy ds. (5.13)Thus, we can write(5.14) E (cid:0) | H ( t + ǫ , ϕ ) − H ( t , ϕ ) | (cid:1) := T + T (2 π ) d , where T := Z t Z R d e − s Re Ψ( ξ ) (cid:12)(cid:12) − e − ǫ Ψ( ξ ) (cid:12)(cid:12) | ˆ ϕ ( ξ ) | dξ ds, (5.15)and T := Z ǫ Z R d e − s Re Ψ( ξ ) | ˆ ϕ ( ξ ) | dξ ds. (5.16)First we estimate T , viz.,(5.17) Z ǫ e − s Re Ψ( ξ ) ds = ǫ − e − ǫ Re Ψ( ξ ) ǫ Re Ψ( ξ ) . Because(5.18) 12 11 + θ ≤ − e − θ θ ≤
21 + θ for all θ > , it follows that (2 π ) d E ( ǫ ; ϕ ) ≤ T ≤ (2 π ) d E ( ǫ ; ϕ ) . Since T ≥ λ = 1,(5.19) T ≤ e t Z R d (cid:12)(cid:12) − e − ǫ Ψ( ξ ) (cid:12)(cid:12) | ˆ ϕ ( ξ ) | ξ ) dξ. Because | − e − ǫ Ψ( ξ ) | ≤ ∧ ǫ | Ψ( ξ ) | , it follows that T ≤ (2 π ) d exp(2 t ) F ( ǫ ; ϕ ), and hencethe proof is completed. (cid:3) Estimate in the hyperbolic case.Proposition 5.7.
Assume the symmetry condition (2.12) , and let W denote the weak solu-tion to the stochastic wave equation (2.4) . Then, for all t ≥ , ǫ > , and ϕ ∈ L L ( R d ) , (5.20) E (cid:0) | W ( t + ǫ , ϕ ) − W ( t , ϕ ) | (cid:1) ≤ (8 t + 6 ǫ ) E ( ǫ ; ϕ ) . Proof.
By density, it suffices to prove the proposition for all functions ϕ ∈ S ( R d ) of rapiddecrease. Henceforth, we choose and fix such a function ϕ .In accord with (2.21) we write(5.21) E (cid:0) | W ( t + ǫ , ϕ ) − W ( t , ϕ ) | (cid:1) := T + T , where(5.22) T := 1(2 π ) d Z t Z R d (cid:12)(cid:12)(cid:12) sin (cid:16)p Ψ( ξ ) ( r + ǫ ) (cid:17) − sin (cid:16)p Ψ( ξ ) r (cid:17)(cid:12)(cid:12)(cid:12) Ψ( ξ ) | ˆ ϕ ( ξ ) | dξ dr, and(5.23) T := 1(2 π ) d Z t + ǫt Z R d sin (cid:16)p Ψ( ξ ) r (cid:17) Ψ( ξ ) | ˆ ϕ ( ξ ) | dξ dr. We estimate T first: The argument that led to (3.16) also leads to the following inequality:(5.24) Z R d sin (cid:16)p Ψ( ξ ) r (cid:17) Ψ( ξ ) | ˆ ϕ ( ξ ) | dξ ≤ Z R d (cid:18) r ∧ ξ ) (cid:19) | ˆ ϕ ( ξ ) | dξ. Because R ǫǫ ( r ∧ a ) dr ≤ ǫ ( ǫ ∧ a ) for all a, ǫ > T ≤ ǫ (2 π ) d Z R d (cid:18) ǫ ∧ ξ ) (cid:19) | ˆ ϕ ( ξ ) | dξ ≤ ǫ E ( ǫ ; ϕ ) . (5.25)The estimate for T is even simpler to derive: Because | sin α − sin β | is bounded aboveby the minimum of 4 and 2[1 − cos( β − α )], Z R d (cid:12)(cid:12)(cid:12) sin (cid:16)p Ψ( ξ ) ( r + ǫ ) (cid:17) − sin (cid:16)p Ψ( ξ ) r (cid:17)(cid:12)(cid:12)(cid:12) Ψ( ξ ) | ˆ ϕ ( ξ ) | dξ ≤ Z R d − cos (cid:16)p Ψ( ξ ) ǫ (cid:17) Ψ( ξ ) | ˆ ϕ ( ξ ) | dξ. (5.26) OCAL TIMES AND SPDES 21
This and the elementary inequality 1 − cos x ≤ x / − cos (cid:16)p Ψ( ξ ) ǫ (cid:17) Ψ( ξ ) ≤ ǫ ∧ ξ ) ) . Consequently,(5.28) 1(2 π ) d Z R d (cid:12)(cid:12)(cid:12) sin (cid:16)p Ψ( ξ ) ( r + ǫ ) (cid:17) − sin (cid:16)p Ψ( ξ ) r (cid:17)(cid:12)(cid:12)(cid:12) Ψ( ξ ) | ˆ ϕ ( ξ ) | dξ ≤ E ( ǫ ; ϕ ) , whence T is at most t times the right-hand side of the preceding. This and (5.25) togetheryield the proof. (cid:3) Proof of Theorem 5.3.
We start with the weak solution H to the stochastic heat equation.Throughout, ϕ ∈ L L ( R d ) is held fixed.If the integrability condition (5.3) holds, then according to Lemma 5.6, for all T > t,ǫ ≥ ≤ t ≤ t + ǫ ≤ T E (cid:0) | H ( t + ǫ , ϕ ) − H ( t , ϕ ) | (cid:1) E ( ǫ ; ϕ ) ≤ e T + 1 < ∞ . Since ǫ E ( ǫ ; ϕ ) is nondecreasing, a direct application of Gaussian-process theory impliesthat { H ( t , ϕ ) } t ∈ [0 ,T ] has a continuous modification provided that(5.30) Z + p E ( ǫ ; ϕ ) + F ( ǫ ; ϕ ) ǫ p log(1 /ǫ ) dǫ < ∞ . See Lemma 6.4.6 of Marcus and Rosen (2006, p. 275). A standard measure-theoretic ar-gument then applies to prove that t H ( t , ϕ ) has a continuous modification. A similarargument works for the weak solution W to the stochastic wave equation (2.4), but we appealto Proposition 5.6 in place of 5.7. Thus, the first portion of our proof will be completed,once we prove that condition (5.3) implies (5.30).Let us write Z + p E ( ǫ ; ϕ ) + F ( ǫ ; ϕ ) ǫ p log(1 /ǫ ) dǫ ≤ Z + p E ( ǫ ; ϕ ) ǫ p log(1 /ǫ ) dǫ + Z + p F ( ǫ ; ϕ ) ǫ p log(1 /ǫ ) dǫ := I + I . (5.31)Let us consider I first. We multiply and divide the integrand of I by the square root of g (1 /ǫ ), and then and apply the Cauchy–Schwarz inequality to obtain the following: I ≤ (cid:18)Z + E ( ǫ ; ϕ ) g (1 /ǫ ) ǫ dǫ (cid:19) / (cid:18)Z + ǫ log(1 /ǫ ) g (1 /ǫ ) dǫ (cid:19) / = const · (cid:18)Z + E ( ǫ ; ϕ ) g (1 /ǫ ) ǫ dǫ (cid:19) / . (5.32) Note that Z /e E ( ǫ ; ϕ ) g (1 /ǫ ) ǫ dǫ = 1(2 π ) d Z R d | ˆ ϕ ( ξ ) | dξ Z /e g (1 /ǫ )1 + ǫ Re Ψ( ξ ) dǫ ! = 1(2 π ) d Z Re Ψ ≥ e | ˆ ϕ ( ξ ) | dξ ( · · · ) + 1(2 π ) d Z Re Ψ
On the other hand, we can change variables and appeal to Proposition 5.2 and deduce that I ≤ Z ∞ N g ( u ) u du ∼ g ( N ) N as N → ∞ .(5.37)We obtain the following upon setting N := Re Ψ( ξ ) and combining inequalities (5.34)–(5.36)above:(5.38) (5.3) = ⇒ I < ∞ . OCAL TIMES AND SPDES 23
We now look at I . Let us recall the definition of F ( ǫ ; ϕ ) and write F ( ǫ ; ϕ ) = 1(2 π ) d (cid:20)Z | Ψ |≤ /e ( · · · ) dξ + Z /e ≤| Ψ |≤ /ǫ ( · · · ) dξ + Z | Ψ | > /ǫ ( · · · ) dξ (cid:21) = const · [ I + I + I ] . (5.39)Because I ≤ ǫ E (1 ; ϕ ) whenever ǫ < /e ,(5.40) Z /e √ I ǫ p log(1 /ǫ ) dǫ ≤ const · p E (1 ; ϕ ) . An application of Cauchy–Schwarz inequality yields Z /e √ I ǫ p log(1 /ǫ ) dǫ = Z /e (cid:18)Z /e< | Ψ |≤ /ǫ | Ψ( ξ ) | | ˆ ϕ ( ξ ) | ξ ) dξ (cid:19) / ǫ p log(1 /ǫ ) dǫ ≤ const · Z R d Z / Ψ( ξ )0 | Ψ( ξ ) | | ˆ ϕ ( ξ ) | ξ ) ǫ log(1 /ǫ ) dǫ dξ ! / ≤ const · p E (1 ; ϕ ) . (5.41)We multiply and divide the integrand below by the square root of g (1 /ǫ ) and apply theCauchy–Schwarz inequality in order to obtain Z /e √ I ǫ p log(1 /ǫ ) dǫ ≤ Z /e Z | Ψ | > /ǫ | ˆ ϕ ( ξ ) | g (1 /ǫ ) ǫ (1 + Re Ψ( ξ )) dξ dǫ ! / ≤ const · (cid:18)Z R d g (Ψ( ξ ))(1 + Re Ψ( ξ )) | ˆ ϕ ( ξ ) | dξ (cid:19) / . (5.42)In the last inequality we have changed the order of integration and used Proposition 5.2.Taking into account inequalites (5.36)–(5.42), we obtain(5.43) (cid:18)Z R d g (1 + Ψ( ξ ))(1 + Re Ψ( ξ )) | ˆ ϕ ( ξ ) | dξ (cid:19) / < ∞ = ⇒ I < ∞ . Inequalities (5.38) and (5.43) together with (5.31) imply that(5.44) Z + p E ( ǫ ; ϕ ) + F ( ǫ ; ϕ ) ǫ p log(1 /ǫ ) dǫ < ∞ , provided that (5.3) holds. This proves the first assertion of the theorem.Suppose γ >
0, where(5.45) γ := ind E ( • ; ϕ ) , for brevity. Then by definition, E ( ǫ ; ϕ ) ≤ ǫ γ + o (1) as ǫ ↓
0. Another standard result fromGaussian analysis, used in conjunction with Proposition 5.6 proves that H has a H¨older-continuous modification with H¨older exponent ≤ γ/ W is analogous.For the remainder of the proof we consider only the weak solution H to the stochastic heatequation (2.3), and write H t := H ( t , ϕ ) for typographical ease. If γ >
0, then, Proposition5.6 and elementary properties of normal laws together imply that(5.46) inf t ≥ k H t + ǫ − H t k L (P) ≥ ǫ γ + o (1) for infinitely-many ǫ ↓ . Consequently, for all δ ∈ (0 , q > γ/ T > S >
0, and t ∈ [ S , T ]—all fixed—the followingholds for infinitely-many values of ǫ ↓ ( sup r ∈ [ S,T ] | H r + ǫ − H r | ≥ ǫ q ) ≥ P n | H t + ǫ − H t | ≥ δ k H t + ǫ − H t k L (P) o = 1 − r π Z δ exp( − x / dx. (5.47)The inequality exp( − x / ≤ ( sup r ∈ [ S,T ] | H r + ǫ − H r | ≥ (1 + o (1)) ǫ q for infinitely many ǫ ↓ ) ≥ − r π δ. Since q > γ/ ( lim sup ǫ ↓ sup r ∈ [ S,T ] | H r + ǫ − H r | ǫ q = ∞ ) ≥ − r π δ. Let δ ↓ ǫ ↓ sup r ∈ [ S,T ] | H r + ǫ − H r | ǫ q = ∞ a.s.This proves that any q > γ/ H . Moreover, in the case that γ = 0, we find that (5.50) holds a.s. for all q >
0. Thus,it follows that with probability one, H has no H¨older-continuous modification in that case.The proof is now complete. (cid:3) Spatial and joint continuity: Proofs of Theorems 2.5 and 2.6
Proof of Theorem 2.5.
We begin by proving the portion of Theorem 2.5 that relates to thestochastic heat equation and its random-field solution { H ( t , x ); t ≥ , x ∈ R } ; Theorem 2.1guarantees the existence of the latter process. OCAL TIMES AND SPDES 25
Throughout we can—and will—assume without loss of generality that x H ( t , x ) iscontinuous, and hence so is the local time of ¯ X in its spatial variable. As we saw, during thecourse of the proof of Theorem 2.4, this automatically implies the condition (4.6), which weare free to assume henceforth.According to Proposition 3.1, specifically Corollary 3.2,(6.1) E ( t ; δ x − δ y ) ≤ E (cid:0) | H ( t , x ) − H ( t , y ) | (cid:1) ≤ E ( t ; δ x − δ y ) . Since | ˆ δ x ( ξ ) − ˆ δ y ( ξ ) | = 2[1 − cos( ξ ( x − y ))], it follows from this and (4.1) that(6.2) h ( | x − y | ) ≤ E (cid:0) | H ( t , x ) − H ( t , y ) | (cid:1) ≤ h ( | x − y | ) . This implies that the critical H¨older exponent of z H ( t , z ) is almost surely equal toone-half of the following quantity:(6.3) ind h := lim inf ǫ ↓ log h ( ǫ )log ǫ . We will not prove this here, since it is very similar to the proof of temporal H¨older continuity(Theorem 5.3). Consequently,(6.4) z H ( t , z ) has a H¨older-continuous modification iff ind h > . Among other things, this implies the equivalence of parts (1) and (2) of the theorem.Let Z ( t , x ) denote the local time of ¯ X at spatial value x at time t ≥
0. We prove that(1), (2), and (3) are equivalent by proving that (6.4) continues to hold when H is replacedby Z . Fortunately, this can be read off the work of Barlow (1988). We explain the detailsbriefly. Because (4.6) holds, Theorem 5.3 of Barlow (1988) implies that there exists a finiteconstant c > t ≥ I ⊂ R ,(6.5) lim δ ↓ sup a,b ∈ I | a − b | <δ | Z ( t , a ) − Z ( t , b ) | p h ( | a − b | ) log(1 / | b − a | ) ≥ c (cid:18) sup x ∈ I Z ( s , x ) (cid:19) / a.s.Moreover, we can choose the null set to be independent of all intervals I ⊂ R with rationalendpoints. In fact, we can replace I by R , since a Z ( t , a ) is supported by the closureof the range of the process ¯ X up to time t , and the latter range is a.s. bounded since ¯ X iscadlag.By their very definition local times satisfy R ∞−∞ Z ( s , x ) dx = s a.s. Thus, sup x ∈ R Z ( t , x ) > q > ind h ,(6.6) lim δ ↓ sup | a − b | <δ | Z ( t , a ) − Z ( t , b ) || a − b | q = ∞ a.s. That is, there is no H¨older-continuous modification of a Z ( t , a ) of order > ind h . Inparticular, if ind h = 0, then a Z ( t , a ) does not have a H¨older-continuous modification.Define d ( a , b ) := p h ( | a − b | ); it is easy to see that d is a pseudo-metric on R . Accordingto Bass and Khoshnevisan (1992),(6.7) lim sup δ ↓ sup d ( a,b ) <δ | Z ( t , a ) − Z ( t , b ) | R d ( a,b )0 (log N ( u )) / du ≤ (cid:18) sup x Z ( t , x ) (cid:19) / a.s.,where N ( u ) denotes the smallest number of d -balls of radius ≤ u needed to cover [ − , x Z ( t , x ) < ∞ a.s. (Bass and Khoshnevisan, 1992, Theorem 3.1). Consequently,(6.8) sup d ( a,b ) <δ | Z ( t , a ) − Z ( t , b ) | = O (cid:18)Z δ (log N ( u )) / du (cid:19) as δ ↓ . Since h is increasing,(6.9) sup | a − b | <δ | Z ( t , a ) − Z ( t , b ) | = O Z h − ( δ )0 (log N ( u )) / du ! as δ ↓ . According to equations (6.128) and (6.130) of Marcus and Rosen (2006, Lemma 6.4.1, p.271), there exists a finite constant c > u > N ( u ) ≤ cℓ { ( x , y ) ∈ [ − , : h ( | x − y | ) < u/ }≤ const h − ( u/ , (6.10)where ℓ ( A ) denotes the Lebesgue measure of A ⊂ R . [Specifically, we apply Lemma 6.4.1of that reference with their K := [ − ,
1] and their µ := c .] This and (6.9) together implythat with probability one the following is valid: As δ ↓ | a − b | <δ | Z ( t , a ) − Z ( t , b ) | = O (cid:18)Z δ | log u | / h ( du ) (cid:19) = O (cid:0) h ( δ ) | log(1 /δ ) | / (cid:1) + O (cid:18)Z δ h ( u ) u | log u | / du (cid:19) . (6.11)The last line follows from integration by parts. If γ := ind h >
0, then h ( δ ) = o ( δ q ) for allfixed choices of q ∈ (0 , γ/ a Z ( t , a ) is H¨older continuous of any order < γ/
2. Among other things, this implies (6.4) with Z replacing H , whence it follows that(1)–(3) of the theorem are equivalent.The hyperbolic portion of the theorem is proved similarly, but we use Proposition 3.6 inplace of Proposition 3.1 everywhere. (cid:3) OCAL TIMES AND SPDES 27
Proof of Theorem 2.6.
Since β ′′ is positive, Re Ψ( ξ ) → ∞ as | ξ | → ∞ . Therefore, for all ǫ, ϑ > x ∈ R , E ( ǫ ; δ x ) = 1(2 π ) d Z R d dξ (1 /ǫ ) + Re Ψ( ξ ) ≤ const · ǫ + Z { Re Ψ >ϑ } dξ (1 /ǫ ) + Re Ψ( ξ ) . (6.12)If ξ is sufficiently large, then for all γ ∈ ( d , β ′′ ), we can find a constant C γ ∈ (0 , ∞ ) suchthat Re Ψ( ξ ) ≥ C γ | ξ | γ for all ξ ∈ { Re Ψ > ϑ } . Consequently, E ( ǫ ; δ x ) = O (cid:18) ǫ + Z { Re Ψ >ϑ } dξ (1 /ǫ ) + | ξ | γ (cid:19) = O (cid:0) ǫ − ( d/γ ) (cid:1) , (6.13)as ǫ ↓
0. Thus, ind E ( • ; δ x ) ≤ − ( d/β ′′ ), where this index was introduced in (5.4). A similarcalculation shows that ind h ≤ β ′′ − d ; confer with (6.3) for the definition of this quantity.Thus, for all fixed T >
0, Proposition 5.6 and (6.2) together prove the following: For all τ < ( β ′′ − d ) /β ′′ , ζ < β ′′ − d , x, y ∈ R , and s, t ∈ [0 , T ],(6.14) E (cid:0) | H ( t , x ) − H ( s , y ) | (cid:1) ≤ const · (cid:0) | s − t | τ + | x − y | ζ (cid:1) . A two-dimensional version of Kolmogorov’s continuity theorem finishes the proof; see theproof of Theorem (2.1) of Revuz and Yor (1991, p. 25).The proof, in the case of the stochastic wave equation, is similar, but we use Propositions3.6 and 5.6 instead of Propositions 3.1 and 5.7, respectively. (cid:3) Heat equation via generators of Markov processes
We now consider briefly the stochastic heat equation, where the spatial movement is gov-erned by the generator L of a [weakly] Markov process X := { X t } t ≥ that takes values in alocally compact separable metric space F . We assume further that X admits a symmetrizingmeasure m that is Radon and positive. Let us emphasize that m satisfies ( P t f, g ) = ( f, P t g )for all t ≥ f, g ∈ L ( m ), where { P t } t ≥ denotes the transition operators of X , and( ϕ , ϕ ) := R ϕ ϕ dm for all ϕ , ϕ ∈ L ( m ).7.1. The general problem.
Consider the stochastic heat equation(7.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ t u ( t , x ) = ( L u )( t , x ) + ˙ w ( t , x ) ,u (0 , x ) = 0 , valid for all t ≥ x ∈ F . Here, the underlying noise w in (7.1) is a Gaussian martingalemeasure on R + × F in the sense of Walsh (1986): w is defined on the filtered probabilityspace (Ω , F , { F t } t ≥ , P) and w t ( ϕ ) := R t R F ϕ ( s , x ) w ( dx ds ) defines an { F t } t ≥ -martingale for ϕ ∈ L ( ds × m ); w can be characterized by the covariance functional for the correspondingWiener integrals:(7.2) E (cid:18)Z f dw · Z g dw (cid:19) = Z ∞ Z F Z F f ( s , x ) g ( s , y ) m ( dx ) m ( dy ) ds, for all f, g ∈ L ( ds × m ).We can follow the description of Walsh (1986) and write the weak form of equation (7.1)as follows: For all ϕ ∈ L ( m ) and t ≥ u ( t , ϕ ) = Z t Z F ( P t − s ϕ )( x ) w ( dx ds ) . Lemma 7.1.
The integral defined by (7.3) is well defined for all ϕ ∈ L ( m ) .Proof. We follow closely the proof of Proposition 2.8, and apply the fact that the semigroup { P s } s ≥ is a contraction on L ( m ). (cid:3) Let Z denote the occupation measure of X ; consult (1.3). The following is the key resultof this section. It identifies an abstract Hilbertian quasi-isometry between the occupation-measure L -norm of X and a similar norm for the solution to the stochastic heat equation(7.1) for L . Theorem 7.2. If u denotes the weak solution to (7.1) , then for all ϕ ∈ L ( m ) and t ≥ , (7.4) t E (cid:0) | u ( t , ϕ ) | (cid:1) ≤ E m (cid:0) | Z ( t , ϕ ) | (cid:1) ≤ t E (cid:0) | u ( t , ϕ ) | (cid:1) . As usual, E m refers to the expectation operator for the process X , started according tothe measure m .The preceding theorem follows from the next formula. Proposition 7.3. If u denotes the weak solution to (7.1) , then for all ϕ ∈ L ( m ) and t ≥ , (7.5) E m (cid:0) | Z ( t , ϕ ) | (cid:1) = 4 Z t E (cid:0) | u ( s/ , ϕ ) | (cid:1) ds. Proof of Proposition 7.3.
Since m is a symmetrizing measure for X , the P m -law of X u is m for all u ≥
0. By the Markov property and Tonelli’s theorem,(7.6) E m (cid:0) | Z ( t , ϕ ) | (cid:1) = 2 Z t Z tu ( P v − u ϕ , ϕ ) dv du. OCAL TIMES AND SPDES 29
We computing the Laplace transform of both sides, viz., Z ∞ e − λt E m (cid:0) | Z ( t , ϕ ) | (cid:1) dt = 2 Z ∞ Z t Z tu e − λt ( P v − u ϕ , ϕ ) dv du dt = 2 Z ∞ Z ∞ u (cid:18)Z ∞ v e − λt dt (cid:19) ( P v − u ϕ , ϕ ) dv du = 2 λ Z ∞ e − λs ( P s ϕ , ϕ ) ds. (7.7)The exchange of the integrals is justified because ( P r ϕ , ϕ ) = k P r/ ϕ k L ( m ) is positive andfinite. Because ϕ ∈ L ( m ), Fubini’s theorem implies that for all λ > Z ∞ e − λt E m (cid:0) | Z ( t , ϕ ) | (cid:1) dt = 2 λ ( R λ ϕ , ϕ ) , where R λ := R ∞ exp( − λs ) P s ds defines the resolvent of { P t } t ≥ .Let T λ denote an independent mean-(1 /λ ) exponential holding time. The preceding displaycan be rewritten as follows:(7.9) E m (cid:0) | Z ( T λ , ϕ ) | (cid:1) = 2 λ ( R λ ϕ , ϕ ) for all λ > . Next we consider the weak solution u to the stochastic heat equation (7.1) by first observingthat E( | u ( t , ϕ ) | ) = R t k P s ϕ k L ( m ) ds . It follows from this, and successive applications ofTonelli’s theorem, that for all β > (cid:0) | u ( T β , ϕ ) | (cid:1) = Z ∞ βe − βt Z t k P s ϕ k L ( m ) ds dt = Z ∞ e − βs k P s ϕ k L ( m ) ds. (7.10)Because k P s ϕ k L ( m ) = ( P s ϕ , ϕ ), we may apply Fubini’s theorem once more, and select β := 2 λ , to find that E (cid:0) | u ( T λ , ϕ ) | (cid:1) = Z ∞ e − λs ( P s ϕ , ϕ ) ds = (cid:18) ϕ , Z ∞ e − λs P s ϕ ds (cid:19) = 12 ( R λ ϕ , ϕ ) . (7.11)The condition of square integrability for ϕ justifies the appeal to Fubini’s theorem. We cancompare (7.9) and (7.11) to find that(7.12) E m (cid:0) | Z ( T λ , ϕ ) | (cid:1) = 4 λ E (cid:0) | u ( T λ , ϕ ) | (cid:1) for all λ > . Define q ( t ) := E m ( | Z ( t , ϕ ) | ), ρ ( t ) := E( | u ( t/ , ϕ ) | ) and ( t ) := 1 for all t ≥
0. Thepreceding shows that the Laplace transform of q is equal to 4 times the product of therespective Laplace transforms of ρ and . Thus, we can invert to find that q = 4 ρ ∗ , whichis another way to state the theorem. (cid:3) Proof of Theorem 7.2.
Let us choose and fix a measurable function ϕ : F → R such that | ϕ | ∈ L ( m ). The defining isometry for Wiener integrals yields the following identity, whereboth sides are convergent: E( | u ( t , ϕ ) | ) = R t k P s ϕ k L ( m ) ds . Proposition 7.3 implies that(7.13) 2 t E (cid:0) | u ( t/ , ϕ ) | (cid:1) ≤ E m (cid:0) | Z ( t , ϕ ) | (cid:1) ≤ t E (cid:0) | u ( t/ , ϕ ) | (cid:1) . [For the lower bound, we use the bound R t E( | u ( s/ , ϕ ) | ) ds ≥ R tt/ E( | u ( s/ , ϕ ) | ) ds .] Bymonotonicity, E( | u ( t/ , ϕ ) | ) ≤ E( | u ( t , ϕ ) | ), whence follows the announced upper boundfor E m ( | Z ( t , ϕ ) | ).In order to prove the other bound we first write(7.14) Z ( t , ϕ ) = Z ( t/ , ϕ ) + Z ( t/ , ϕ ) ◦ θ t/ , where { θ s } s ≥ denotes the collection of all shifts on the paths of X . We care only aboutdistributional properties. Therefore, by working on an appropriate probability space, we canalways insure that these shifts can be constructed; see Blumenthal and Getoor (1968).We apply the Markov property at time t/
2. Since P m ◦ X − t/ = m , it follows that(7.15) E m (cid:16)(cid:12)(cid:12) Z ( t/ , ϕ ) ◦ θ t/ (cid:12)(cid:12) (cid:17) = E m (cid:0) | Z ( t/ , ϕ ) | ◦ θ t/ (cid:1) = E m (cid:0) | Z ( t/ , ϕ ) | (cid:1) , and hence E m ( | Z ( t , ϕ ) | ) ≤ m ( | Z ( t/ , ϕ ) | ). Consequently,(7.16) E m (cid:0) | Z ( t , ϕ ) | (cid:1) ≤ m (cid:0) | Z ( t/ , ϕ ) | (cid:1) for all t ≥ . This and the first inequality of (7.13) together imply the remaining bound in the statementof the theorem. (cid:3)
The stochastic heat equation in dimension − ǫ . We now specialize the setup ofthe preceding subsection to produce an interesting family of examples: We suppose that F is a locally compact subset of R d for some integer d ≥
1, and m is a positive Radon measureon F , as before. Let { R λ } λ> denote the resolvent of X , and suppose that X has jointlycontinuous and uniformly bounded resolvent densities { r λ } λ> . In particular, r λ ( x , y ) ≥ x, y ∈ F and(7.17) ( R λ f )( x ) = Z F r λ ( x , y ) f ( y ) m ( dy ) , OCAL TIMES AND SPDES 31 for all measurable functions f : F → R + . Recall that X has local times { Z ( t , x ) } t ≥ ,x ∈ F ifand only if for all measurable functions f : F → R + , and every t ≥ Z ( t , f ) = Z F Z ( t , z ) f ( z ) m ( dz ) , valid P x -a.s. for all x ∈ F . Choose and fix some point a ∈ F , and define(7.19) f aǫ ( z ) := B ( a,ǫ ) ( z ) m ( B ( a , ǫ )) for all z ∈ F and ǫ > . Of course, B ( a , ǫ ) denotes the ball of radius ǫ about a , measured in the natural metric of F . Because r λ is jointly continuous, lim ǫ ↓ ( R λ f aǫ )( x ) = r λ ( x , a ), uniformly for x -compacta.Define ϕ ǫ,δ := f aǫ − f aδ , and observe that ϕ ∈ L ( m ). Furthermore,(7.20) lim ǫ,δ ↓ ( R λ ϕ ǫ,δ , ϕ ǫ,δ ) = lim ǫ,δ ↓ { ( R λ f aǫ , f aǫ ) − R λ f aǫ , f aδ ) + ( R λ f aδ , f aδ ) } = 0 . If h ∈ L ( m ), then the weak solution h to (7.1)—where L denotes the L -generator of X —satisfies E (cid:0) | u ( t , h ) | (cid:1) = Z t k P s h k L ( m ) ds ≤ e λt Z ∞ e − λs k P s h k L ( m ) ds = e λt R λ h , h ) . (7.21)Therefore, Theorem 7.2 and (7.20) together imply that { u ( t , f aǫ ) } ǫ> is a Cauchy sequencein L (P) for all t ≥
0. In other words, we have shown that the stochastic heat equation (7.1)has a “random-field solution.”
Example 7.4.
It is now easy to check, using the heat-kernel estimates of Barlow (1998,Theorems 8.1.5 and 8.1.6), that for all d ∈ (0 ,
2) there exists a compact “fractal” F ⊂ R of Hausdorff dimension d such that Brownian motion on F satisfies the bounded/continuousresolvent-density properties here. The preceding proves that if we replace L by the Laplacianon F , then the stochastic heat equation (7.1) has a “random-field solution.” Specifically, thelatter means that for all t ≥ a ∈ F , u ( t , f aǫ ) converges in L (P) as ǫ ↓
0, where f aǫ is defined in (7.19). This example comes about, because the fractional diffusions of Barlow(1998) have local times when the dimension d of the fractal on which they live satisfies d < (cid:3) A semilinear parabolic problem
We consider the semilinear problems that correspond to the stochastic heat equation (2.3).At this point in the development of SPDEs, we can make general sense of nonlinear stochasticPDEs only when the linearized SPDE is sensible. Thus, we assume henceforth that d = 1.We investigate the semilinear stochastic heat equation. Let b : R → R be a measurablefunction, and consider the solution H b to the following SPDE:(8.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ t H b ( t , x ) = ( L H b )( t , x ) + b ( H b ( t , x )) + ˙ w ( t , x ) ,H b (0 , x ) = 0 , where L denotes the generator of the L´evy process X , as before.Equation (8.1) has a chance of making sense only if the linearized problem (2.3) has arandom-field solution H , in which case we follow Walsh (1986) and write the solution H b asthe solution to the following:(8.2) H b ( t , x ) = H ( t , x ) + Z t Z ∞−∞ b ( H b ( s , x − y )) P t − s ( dy ) ds, where the measures { P t } t ≥ are determined from the semigroup of X by P t ( E ) := ( P t E )(0)for all Borel sets E ⊂ R . [This is standard notation.] We will soon see that this randomintegral equation has a “good solution” H b under more or less standard conditions on thefunction b . But first, let us make an observation. Lemma 8.1. If (2.3) has a random-field solution, then the process X has a jointly measur-able transition density { p t ( x ) } t> ,x ∈ R that satisfies the following: For all η > there existsa constant C := C η ∈ (0 , ∞ ) such that for all t > , (8.3) Z t k p s k L ( R ) ds ≤ Ce ηt . Finally, ( t , x ) p t ( x ) is uniformly continuous on [ ǫ , T ] × R for all fixed ǫ, T > .Proof. We can inspect the function y = x exp( − x ) to find that exp( − x ) ≤ (1 + x ) − for all x ≥
0. Consequently,(8.4) Z ∞−∞ e − ǫ Re Ψ( ξ ) dξ ≤ Z ∞−∞ dξ ǫ Re Ψ( ξ ) . The second integral, however, has been shown to be equivalent to the existence of random-field solutions to (2.3); see (3.17). It follows that the first integral in (8.4) is convergent. Weapply the inversion theorem to deduce from this that the transition densities of X are givenby p t ( x ) = (2 π ) − R ∞−∞ exp {− ixξ − t Ψ( ξ ) } dξ , where the integral is absolutely convergent forall t > x ∈ R . Among other things, this formula implies the uniform continuity of OCAL TIMES AND SPDES 33 p t ( x ) away from t = 0. In addition, by Plancherel’s theorem, for all s > k p s k L ( R ) = 12 π Z ∞−∞ (cid:12)(cid:12) e − s Ψ( ξ ) (cid:12)(cid:12) dξ = 12 π Z ∞−∞ e − s Re Ψ( ξ ) dξ. (8.5)Therefore, Lemma 3.3 and Tonelli’s theorem together imply that for all λ > Z t k p s k L ( R ) ds ≤ e t/λ π Z ∞−∞ dξ (1 /λ ) + Re Ψ( ξ ) < ∞ . This completes the proof. (cid:3)
Thanks to the preceding lemma, by (8.1) we mean a solution to the following:(8.7) H b ( t , x ) = H ( t , x ) + Z t Z ∞−∞ b ( H b ( s , x − y )) p t − s ( y ) dy ds. For the following we assume that the underlying probability space (Ω , F , P) is complete.
Theorem 8.2.
Suppose b is bounded and globally Lipschitz, and the stochastic heat equa-tion (2.3) has a random-field solution H ; thus, in particular, d = 1 . Then, there exists amodification of H , denoted still by H , and a process H b with the following properties: (1) H b ∈ L p loc ( R + × R ) for all p ∈ [1 , ∞ ) . (2) With probability one, (8.7) holds for all ( t , x ) ∈ R + × R . (3) For all
T > , J is a.s. bounded and continuous on [0 , T ] × R , where J ( t , x ) := Z t Z ∞−∞ b ( H b ( s , x − y )) p t − s ( y ) dy ds. Remark . Before we proceed with a proof, we make two remarks:(1) It is possible to adapt the argument of Nualart and Pardoux (1994, Proposition1.6) to deduce that the laws of H and H b are mutually absolutely continuous withrespect to one another; see also Dalang and Nualart (2004, Corollary 5.3) and Dalang,Khoshnevisan, and Nualart (2007, Equations (5.2) and (5.3)). A consequence of thismutual absolute continuity is that H b is [H¨older] continuous iff H is.(2) Theorem 8.2 implies facts that are cannot be described by change-of-measure meth-ods. For instance, it has the striking consequence that with probability one, H b and H blow up in exactly the same points ! [This is simply so, because H b − H is locallybounded.] For an example, we mention that the operators considered Example 5.5,when the parameter α there is ≤
2, lead to discontinuous solutions H that blow up(a.s.) in every open subset of R + × R (Marcus and Rosen, 2006, Section 5.3). Inthose cases, H b inherits this property as well. Proof.
We will need the following fact:(8.8) H is continuous in probability . In fact, we prove that H is continuous in L (P).Owing to (4.2), for all t ≥ x, y ∈ R ,(8.9) E (cid:0) | H ( t , x ) − H ( t , y ) | (cid:1) ≤ e t π Z ∞−∞ − cos( ξ | x − y | )1 + Re Ψ( ξ ) dξ. Moreover, according to (3.17), (1 + Re Ψ) − ∈ L ( R ). Therefore, the dominated convergencetheorem implies that for all T > y ∈ R ,(8.10) sup t ∈ [0 ,T ] k H ( t , x ) − H ( t , y ) k L (P) → x → y. Similarly, Theorem 5.3 implies that for all t ≥ s → t sup x ∈ R k H ( t , x ) − H ( s , x ) k L (P) ≤ lim s → t π Z ∞−∞ dξ (1 / | t − s | ) + Re Ψ( ξ ) , which is zero by the dominated convergence theorem. This and (8.10) together imply (8.8).Now we begin the proof in earnest.Throughout, we fix(8.12) Lip b := sup x = y | b ( x ) − b ( y ) || x − y | and λ := 2Lip b . The condition that b is globally Lipschitz tells precisely that Lip b and/or λ are finite.Now we begin with a fixed-point scheme: Set u ( t , x ) := 0, and define, iteratively for allintegers n ≥ u n +1 ( t , x ) := H ( t , x ) + Z t Z ∞−∞ b ( u n ( s , x − y )) p t − s ( y ) dy ds. Consider the processes(8.14) D n +1 ( x ) := Z ∞ e − λt | u n +1 ( t , x ) − u n ( t , x ) | dt for all n ≥ x ∈ R . Also, define r λ to be the λ -potential density of X , given by(8.15) r λ ( z ) := Z ∞ p s ( z ) e − λs ds for all z ∈ R . According to Lemma 8.1, this is well defined.
OCAL TIMES AND SPDES 35
The processes D , D , . . . satisfy the following recursion: D n +1 ( x ) ≤ Lip b λ ( D n ∗ r λ )( x )= 12 ( D n ∗ r λ )( x ) , (8.16)as can be seen by directly manipulating (8.13). We can iterate this to its natural end, anddeduce that(8.17) D n +1 ( x ) ≤ − n − ( D ∗ r λ )( x ) for all n ≥ . Since u ( t , x ) − u ( t , x ) = H ( t , x ) + tb (0),(8.18) D ( x ) ≤ Z ∞ e − λt | H ( t , x ) | dt + | b (0) | λ . Thanks to (8.8), we can always select a measurable modification of ( ω , t , x ) H ( t , x )( ω ),by the separability theory of Doob (1953, Theorem 2.6, p. 61). Therefore, we can apply thepreceding to a Lebesgue-measurable modification of t H ( t , x ) to avoid technical problems(Doob, 1953, Theorem 2.7, p. 62). In addition, it follows from this and convexity that(8.19) Z ∞−∞ k D ( x ) k L (P) e −| x | dx ≤ Z ∞−∞ Z ∞ e − λt −| x | k H ( t , x ) k L (P) dt dx + 2 | b (0) | λ . We can apply Proposition 3.1, with its λ replaced by (4 λ ) − here, to find that k H ( t , x ) k L (P) ≤ e λt/ π Z ∞−∞ dξ (4 λ ) − + Re Ψ( ξ ):= const · e λt/ , (8.20)where the constant depends neither on t nor on n . Consequently,(8.21) Z ∞−∞ k D ( x ) k L (P) e −| x | dx < ∞ . Because λr λ is a probability density on R , it follows that for all integers n ≥ Z ∞−∞ k D n +1 ( x ) k L (P) e −| x | dx ≤ const · − n . Therefore, P ∞ n =0 R ∞−∞ k D n ( x ) k L (P) exp( −| x | ) dx < ∞ . Furthermore, for all T, k > n →∞ Z T Z k − k | u n +1 ( t , x ) − u n ( t , x ) | dx dt ≤ e λT + k lim n →∞ Z k − k D n ( x ) e −| x | dx = 0 a.s.(8.23) Because L ([0 , T ] × [ − k , k ]) is complete, and since (Ω , F , P) is complete, standard argumentsshow that there exists a process u ∞ ∈ L loc ( R + × R ) such that(8.24) lim n →∞ Z T Z k − k | u n ( t , x ) − u ∞ ( t , x ) | dx dt = 0 , almost surely for all T, k >
0. Since b is globally Lipschitz, it follows easily from this thatoutside a single set of P-measure zero,(8.25) u ∞ ( t , x ) = H ( t , x ) + Z t Z ∞−∞ b ( u ∞ ( s , x − y )) p t − s ( y ) dy ds, simultaneously for almost all ( t , x ) ∈ R + × R . An application of Fubini’s theorem impliesthen that the assertion (8.25) holds a.s. for almost all ( t , x ) ∈ R + × R . [Observe the orderof the quantifiers.]Consider the finite Borel measure(8.26) Υ( dt dx ) := e − λt −| x | dt dx, defined on R + × R . Our proof, so far, contains the fact that u ∞ ∈ L (Υ) almost surely.Moreover, thanks to (8.25), if p ∈ [1 , ∞ ) then(8.27) k u ∞ ( t , x ) k L p (P) ≤ k H ( t , x ) k L p (P) + sup z | b ( z ) | t for all ( t , x ) ∈ R + × R . One of the basic properties of centered Gaussian random variables is that their p th momentis proportional to their second moment. Consequently,(8.28) k u ∞ ( t , x ) k L p (P) ≤ const · (cid:16) k H ( t , x ) k L (P) + 1 (cid:17) , which we know to be locally bounded. This and the Tonelli theorem together prove that u ∞ ∈ L p loc ( R + × R ) a.s.We recall a standard fact from classical analysis: If f ∈ L (Υ) , then f is continuous inthe measure Υ. This means that for all δ > ǫ,η → Υ (cid:8) ( t , x ) : (cid:12)(cid:12) f ( t + η , x + ǫ ) − f ( t , x ) (cid:12)(cid:12) > δ (cid:9) = 0 , and follows immediately from standard approximation arguments; see the original classicbook of Zygmund (1935, § u ∞ ∈ L (Υ) a.s. together imply that u ∞ is continuous in measurea.s. Because b is bounded and Lipschitz, the integrability/continuity properties of p t ( x ), asexplained in Lemma 8.1, together imply that ( t , x ) R t R ∞−∞ b ( u ∞ ( s , x − y )) p t − s ( y ) dy ds = R t R ∞−∞ b ( u ∞ ( s , z )) p t − s ( x − z ) dz ds is a.s. continuous, and bounded on [0 , T ] × R , for every OCAL TIMES AND SPDES 37 nonrandom and fixed
T >
0. Now let us define (8.30) H b ( t , x ) := H ( t , x ) + Z t Z ∞−∞ b ( u ∞ ( s , x − y )) p t − s ( y ) dy ds. Thanks to (8.25),(8.31) P { H b ( t , x ) = u ∞ ( t , x ) } = 1 for all ( t , x ) ∈ R + × R .Thus, H b is a modification of u ∞ . In addition, the Tonelli theorem applies to tell us that H b inherits the almost-sure local integrability property of u ∞ . That is, H b ∈ L p loc ( R + × R ) a.s.Moreover, outside a single null set we have(8.32) J ( t , x ) = Z t Z ∞−∞ b ( u ∞ ( s , x − y )) p t − s ( y ) dy ds for all ( t , x ) ∈ R + × R .This proves the theorem. (cid:3) References
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