Path properties of the solution to the stochastic heat equation with Lévy noise
aa r X i v : . [ m a t h . P R ] A ug Path properties of the solution to the stochastic heat equationwith Lévy noise
Carsten Chong ∗ , Robert C. Dalang ∗ and Thomas Humeau ∗ Abstract
We consider sample path properties of the solution to the stochastic heat equation, in R d or bounded domains of R d , driven by a Lévy space–time white noise. When viewed as astochastic process in time with values in an infinite-dimensional space, the solution is shownto have a càdlàg modification in fractional Sobolev spaces of index less than − d . Concerningthe partial regularity of the solution in time or space when the other variable is fixed, wedetermine critical values for the Blumenthal–Getoor index of the Lévy noise such that noiseswith a smaller index entail continuous sample paths, while Lévy noises with a larger indexentail sample paths that are unbounded on any non-empty open subset. Our results apply toadditive as well as multiplicative Lévy noises, and to light- as well as heavy-tailed jumps. AMS 2010 Subject Classifications:
Keywords: stochastic PDEs; càdlàg modification; Lévy noise; sample path properties; stable noise ∗ Institut de mathématiques, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, e-mails:carsten.chong@epfl.ch, robert.dalang@epfl.chThis research is partially supported by the Swiss National Foundation for Scientific Research. Introduction
Let
T > and consider, on a stochastic basis (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) satisfying the usual conditions,the stochastic heat equation driven by a Lévy space–time white noise on [0 , T ] × D with Dirichletboundary conditions:(1.1) ∂u∂t ( t, x ) = ∆ u ( t, x ) + σ ( u ( t, x )) ˙ L ( t, x ) , ( t, x ) ∈ (0 , T ) × D ,u ( t, x ) = 0 , for all ( t, x ) ∈ [0 , T ] × ∂D ,u (0 , x ) = u ( x ) , for all x ∈ D , where D is the whole space R d or a bounded domain in R d , σ : R → R is a Lipschitz function, u : ¯ D → R is a bounded continuous initial condition vanishing on ∂D , and ˙ L is a Lévy space–timewhite noise. If D = R d , the boundary conditions on u and u are considered void.A predictable random field u = ( u ( t, x ) : ( t, x ) ∈ [0 , T ] × D ) is called a mild solution to (1.1)if for all ( t, x ) ∈ [0 , T ] × D ,(1.2) u ( t, x ) = V ( t, x ) + Z t Z D G D ( t − s ; x, y ) σ ( u ( s, y )) L (d s, d y ) almost surely, where(1.3) V ( t, x ) = Z D G D ( t ; x, y ) u ( y ) d y , ( t, x ) ∈ [0 , T ] × D , is the solution to the homogeneous version of (1.1).In (1.2) and (1.3), G D denotes the Green’s function of the heat operator on D , which for D = R d equals the Gaussian density(1.4) g ( t, x ) = (4 πt ) − d e − | x | t t > (when t = 0 , we interpret g (0 , x ) as the Dirac delta function δ ( x ) ), while on a bounded domain D with smooth boundary it has the spectral representation(1.5) G D ( t ; x, y ) = X j > Φ j ( x )Φ j ( y ) e − λ j t t > , for all x, y ∈ D , where ( λ j ) j > are the eigenvalues of − ∆ with vanishing Dirichlet boundary conditions, and (Φ j ) j > are the corresponding eigenfunctions forming a complete orthonormal basis of L ( D ) .In the special case where ˙ L is a Gaussian noise, the existence, uniqueness and regularity ofsolutions to Equation (1.1) have been extensively studied in the literature, see e.g. [3, 10, 23, 39]for the case of space–time white noise, [15, 36, 37] for noises that are white in time but coloredin space, and [22] for noises that may exhibit temporal covariances as well. In all cases, the mildsolution to (1.2) is jointly locally Hölder continuous in space and time, with exponents that dependon the covariance structure of the noise.By contrast, suppose that ˙ L is a Lévy space–time white noise without Gaussian part , that is,(1.6) L (d t, d x ) = b d t d x + Z | z | z ˜ J (d t, d x, d z ) + Z | z | > z J (d t, d x, d z )=: L B (d t, d x ) + L M (d t, d x ) + L P (d t, d x ) , where b ∈ R , J is an ( F t ) t ∈ [0 ,T ] -Poisson random measure on [0 , T ] × D × R with intensity d t d x ν (d z ) , and ˜ J is the compensated version of J . Here ν is a Lévy measure , that is, ν ( { } ) = 0 and R R (cid:0) z ∧ (cid:1) ν (d z ) < + ∞ , and we assume that ν is not identically zero. The existence2nd uniqueness of solutions for equations like (1.1) with Lévy noise have been investigated in[1, 2, 11, 12, 31, 34].Already in the linear case with σ ( x ) ≡ , due to the singularity of the Green’s kernel on thediagonal x = y near t = 0 , each jump of the noise creates a Dirac mass for the solution. Evenworse, if ν ( R ) = ∞ , these space–time jump points form a dense subset of [0 , T ] × D . Hence onecannot expect the solution to have any continuity properties jointly in space and time.In this article, we thus take two different viewpoints and consider1. the path properties of t u ( t, · ) as a process with values in an infinite-dimensional space;2. the path properties of the partial maps t u ( t, x ) for fixed x ∈ D , and of x u ( t, x ) for fixed t ∈ [0 , T ] .For each t > , u ( t, · ) may take values in L p ( D ) for some p > almost surely, but since eachatom of the Lévy noise introduces a Dirac delta into the solution, the process t u ( t, · ) cannothave a càdlàg version in such a space (see also [7] or [31, Proposition 9.25]). Instead, one shouldconsider spaces of distributions containing delta functions, such as negative fractional Sobolevspaces H r ( D ) for r < − d (see Sections 2.1.1, 2.2.2 and 2.3.1). If σ = 1 and the noise has a finitesecond moment, the existence of a càdlàg modification in such spaces follows from a result of [24]on maximal inequalities for stochastic convolutions in an infinite-dimensional setting, see also [31,Chapter 9.4.2]. This type of result has also been obtained in the case of additive (possibly colored)Lévy noise in [8, 9, 32]. To our best knowledge, the question of existence of càdlàg versions in thecase of multiplicative noise has only been studied in [21]. For the relation of the results of thispaper to our results, see Remark 2.16.In Section 2 of this paper, we substantially generalize the aforementioned results in the case ofa Lévy space–time white noise (1.6): Without any further assumptions than those required for theexistence of solutions, we prove in Theorems 2.5, 2.15 and 2.19, for both a bounded domain D andthe case D = R d , that t u ( t, · ) has a càdlàg modification in H r ( D ) and H r,loc ( R d ) , respectively,for any r < − d . To this end, we start our analysis by considering the stochastic heat equationon the interval D = [0 , π ] in Section 2.1. Treating this basic case first has the advantage that wecan directly proceed to the main steps of the proof while avoiding the technical difficulties of thegeneral case. Next, in Section 2.2, we demonstrate how the proof for D = [0 , π ] can be directlyextended to the case D = R d , provided σ is bounded and ν has finite second moments. But inorder to cover the general case of Lipschitz continuous σ and heavy-tailed noises, we need to usestopping time techniques from [12] to deal with the (infinitely many) large jumps of the noise,as well as results from the integration theory for general random measures (see the Appendix) tocompensate the absence of finite second moments for d > , due to the singularity of the heatkernel and the small jumps of the noise. Finally, the proof for D = [0 , π ] does not extend tobounded domains in R d with d > because the eigenfunctions are typically no longer uniformlybounded. Instead, the proof we give in Section 2.3 makes use of the fact that in the interior of D ,the Green’s function G D can be decomposed into the Gaussian density g (where we can use theresults of Section 2.2) and a smooth function. With the methods reviewed in the Appendix, wealso obtain sufficient control at the boundary of D .Regarding the partial regularity of t u ( t, x ) and x u ( t, x ) , [34, Section 2] obtained thefollowing result on D = R d : If the Lévy measure ν of L satisfies R R | z | p ν (d z ) < + ∞ for some p < d , then for fixed t , the process x u ( t, x ) has a continuous modification. Similarly, if R R | z | p ν (d z ) < + ∞ for some p < , then there exists a continuous modification of t u ( t, x ) forevery fixed x . Extending the results of [34], our Theorems 3.1 and 3.5, which also apply to boundeddomains, show that it suffices to check whether R [ − , | z | p ν (d z ) < + ∞ is finite, which wouldinclude, for example, α -stable noises with α < d (for spatial regularity) and α < (for temporalregularity). Furthermore, these conditions are essentially sharp as we show in Theorems 3.3 and3.7: If σ ≡ , and if ν has the same behavior near the origin as the Lévy measure of an α -stablenoise, then for d α < d (resp. α < d ), the paths of x u ( t, x ) (resp. t u ( t, x ) )are unbounded on any non-empty open subset of D (resp. [0 , T ] ). Let us remark that the lastconclusion was observed in [29] for an α -stable noise Λ and the (non-Lipschitz) function σ ( x ) = x α with α ∈ (1 , d ) via a connection between the resulting equation and stable super-Brownianmotion (note that our α is β in this reference).In what follows, the letter C , occasionally with subscripts indicating the parameters that itdepends on, denotes a strictly positive finite number whose value may change from line to line. For the interval D = [0 , π ] , the Green’s function G = G D has the explicit representation(2.1) G ( t ; x, y ) := G D ( t ; x, y ) = 2 π X k > sin( kx ) sin( ky ) e − k t t > . The existence and uniqueness of mild solutions to (1.1) in this case basically follow from [11].
Proposition 2.1.
Let σ : R → R be a Lipschitz function, u : [0 , π ] → R be continuous with u (0) = u ( π ) = 0 , and L be a pure jump Lévy white noise as in (1.6) . Furthermore, define (2.2) τ N = inf { t ∈ [0 , T ] : J ([0 , t ] × [0 , π ] × [ − N, N ] c ) = 0 } , N ∈ N , with the convention inf ∅ = + ∞ . Then ( τ N ) N > is an increasing sequence of stopping times suchthat τ N > and τ N = + ∞ for large values of N . In addition, up to modifications, (1.1) has amild solution u satisfying (2.3) sup ( t,x ) ∈ [0 ,T ] × [0 ,π ] E [ | u ( t, x ) | p t τ N ] < + ∞ , for any < p < and N ∈ N . Furthermore, up to modifications, this solution is unique amongall predictable random fields that satisfy (2.3) .Proof. Since [0 , π ] is a bounded interval, almost surely, there is only a finite number of jumpslarger than N in [0 , T ] × [0 , π ] . This immediately implies the statements about ( τ N ) N > . Next,by [3, (B.5)], we know that G ( t ; x, y ) Cg ( t, x − y ) for any ( t, x, y ) ∈ [0 , T ] × [0 , π ] , with g asin (1.4). Consequently, (1) to (4) of Assumption B of [11] are satisfied, and we can apply [11,Theorem 3.5] to obtain the existence of a unique mild solution to (1.1) satisfying (2.3) for all p ∈ (0 , . In order to extend this to all p ∈ (2 , , we notice that the only step in the proofof [11, Theorem 3.5] that uses p is the moment estimate (6.9) given in [11, Lemma 6.1(2)]with respect to the martingale part L M . We now elaborate how this estimate can be extended toexponents < p < . For predictable processes φ and φ , we can use [28, Theorem 1] to get theupper bound(2.4) E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z π Z | z | N G ( t − s ; x, y ) ( σ ( φ ( s, y )) − σ ( φ ( s, y ))) ˜ J (d s, d y, d z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p C E Z t Z π Z | z | N | G ( t − s ; x, y ) | | φ ( s, y ) − φ ( s, y ) | | z | d s d y ν (d z ) ! p + C E "Z t Z π Z | z | N | G ( t − s ; x, y ) | p | φ ( s, y ) − φ ( s, y ) | p | z | p d s d y ν (d z ) . | G ( t − s ; x, y ) | d s d y , the first term is furtherbounded by C N (cid:18)Z t Z π | G ( t − s ; x, y ) | d s d y (cid:19) p − Z t Z π | G ( t − s ; x, y ) | E [ | φ ( s, y ) − φ ( s, y ) | p ] d s d y Since G ( t ; x, y ) Cg ( t, x − y ) for any ( t, x, y ) ∈ [0 , T ] × [0 , π ] , and R T R R g ( t, x ) d t d x < + ∞ , weobtain the following estimate from (2.4) and the simple inequality | x | | x | + | x | p for all p > : E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z π Z | z | N G ( t − s ; x, y ) ( σ ( φ ( s, y )) − σ ( φ ( s, y ))) ˜ J (d s, d y, d z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p C N Z t Z π ( | G ( t − s ; x, y ) | + | G ( t − s ; x, y ) | p ) E [ | φ ( s, y ) − φ ( s, y ) | p ] d s d y . Since R T g p ( t, x ) d t d x < + ∞ for p < (see e.g. [11, (3.20)]), this is exactly the extension of [11,Lemma 6.1(2)] needed to complete the proof of [11, Theorem 3.5]. H r ([0 , π ]) For any function f ∈ L ([0 , π ]) , we can define its Fourier sine coefficients(2.5) a n ( f ) = r π Z π f ( x ) sin( nx ) d x , n ∈ N . Then, by Parseval’s identity, k f k L ([0 ,π ]) = P n > a n ( f ) . For any r > , we define H r ([0 , π ]) := ( f ∈ L ([0 , π ]) : k f k H r := X n > (cid:0) n (cid:1) r a n ( f ) < + ∞ ) . This is a Hilbert space for the inner product h f, h i H r := P n > (cid:0) n (cid:1) r a n ( f ) a n ( h ) . For r > , wedefine H − r ([0 , π ]) as the dual space of H r ([0 , π ]) , that is, the space of continuous linear functionalson H r ([0 , π ]) . Then H − r ([0 , π ]) is isomorphic to the space of sequences b = ( b n ) n > such that k b k H − r := X n > (cid:0) n (cid:1) − r b n < + ∞ . More precisely, for r > and ˜ f ∈ H − r ([0 , π ]) , the coefficients b n are given by b n = ˜ f (cid:0)q π sin( n · ) (cid:1) .Then, k ˜ f k H − r = k b k H − r and the duality between H − r ([0 , π ]) and H r ([0 , π ]) is given by h b, h i = X n > b n a n ( h ) k b k H − r k h k H r . For example, it is easy to check that δ x ∈ H r ([0 , π ]) for any x ∈ (0 , π ) and r < − . Indeed, δ x (sin( n · )) = sin( nx ) , and for any r < − , k δ x k H r = 2 π X n > (cid:0) n (cid:1) r sin ( nx ) π X n > (cid:0) n (cid:1) r < + ∞ . .1.2 Existence of a càdlàg solution in H r ([0 , π ]) with r < − In order to motivate why we consider fractional Sobolev spaces H r ([0 , π ]) with r < − , we startwith a special case. Suppose that b = 0 and that ν is a symmetric measure with ν ( R ) < + ∞ .Then we can rewrite L = P N T i =1 Z i δ ( T i ,X i ) , where ( T i , X i , Z i ) are the atoms of the Poisson randommeasure J , and u ( t, x ) = N T X i =1 G ( t − T i ; x, X i ) σ ( u ( T i , X i )) Z i . In this case, it suffices to check whether for fixed i > , t G ( t − T i ; · , X i ) is càdlàg in H r ([0 , π ]) .Using the series representation (2.1), we immediately see that the function x G ( t − T i ; x, X i ) belongs to H r ([0 , π ]) if and only if X k > (1 + k ) r sin( kX i ) e − k ( t − T i ) t > T i < + ∞ . This is the case for any r ∈ R if t = T i . However, for t ↓ T i , we have to restrict to r < − . Indeed,for the càdlàg property, the only point where a problem might appear is at t = T i . At this point,the existence of a left limit is obvious since G ( t − T i ; · , X i ) = 0 for any t < T i . For right-continuity,we use the fact that (1 − e − k h ) k ε h ε for any < ε < − − r , so k G ( h ; · , X i ) − G (0; · , X i ) k H r = 2 π X k > (1 + k ) r sin( kX i ) (cid:16) − e − k h (cid:17) π X k > (1 + k ) r k ε h ε Ch ε → as h → . Therefore, t u ( t, · ) is càdlàg in H r ([0 , π ]) . For the general case, we first treat the drift term. Lemma 2.2.
Assume that u is the unique solution to (1.1) as in Proposition 2.1. Then, F ( t, x ) = V ( t, x ) + Z t Z π G ( t − s ; x, y ) σ ( u ( s, y )) d s d y is jointly continuous in ( t, x ) ∈ [0 , T ] × [0 , π ] . In particular, for every r , the process t F ( t, · ) is continuous in H r ([0 , π ]) .Proof. The continuity of V is standard, and with (2.1), the integral term in F equals π X k > sin( kx ) Z t Z π e − k ( t − s ) sin( ky ) σ ( u ( s, y )) d s d y . Each term in this series is jointly continuous in ( t, x ) . Hence, it suffices to show the uniformconvergence of the series. Using Hölder’s inequality and the fact that u has uniformly boundedmoments of any order p < , we obtain this from E X k > sup ( t,x ) ∈ [0 ,T ] × [0 ,π ] (cid:12)(cid:12)(cid:12)(cid:12) sin( kx ) Z t Z π e − k ( t − s ) sin( ky ) σ ( u ( s, y )) d s d y (cid:12)(cid:12)(cid:12)(cid:12) C X k > E " sup ( t,x ) ∈ [0 ,T ] × [0 ,π ] (cid:18)Z t e − k ( t − s ) d s (cid:19) (cid:18)Z t Z π | σ ( u ( s, y )) | d s d y (cid:19) C X k > (cid:18)Z T e − k s d s (cid:19) (cid:18)Z T Z π E [ | σ ( u ( s, y )) | ] d s d y (cid:19) C X k > k − < + ∞ . t F ( t, · ) in H r ([0 , π ]) , it suffices to show the continuityin L ([0 , π ]) because L ([0 , π ]) ֒ → H r ([0 , π ]) (that is, L ([0 , π ]) is continuously embedded in H r ([0 , π ]) ). The continuity in L ([0 , π ]) in turn follows from the fact that F is uniformly con-tinuous on the compact domain [0 , T ] × [0 , π ] . Proposition 2.3.
Let L be a pure jump Lévy white noise, and let σ be a bounded and Lipschitzfunction. Let u be the mild solution to the stochastic heat equation (1.1) . Then, for any r < − ,the stochastic process ( u ( t, · )) t ∈ [0 ,T ] has a càdlàg version in H r ([0 , π ]) .Proof. For N ∈ N , consider the truncated noise(2.6) L N (d t, d x ) = b N d t d x + Z | z | N z ˜ J (d t, d x, d z ) , b N = b − Z < | z | N z ν (d z ) , as well as the mild solution u N to (1.1) driven by L N , that is,(2.7) u N ( t, x ) = V ( t, x ) + Z t Z π G ( t − s ; x, y ) σ ( u N ( s, y )) L N (d s, d y ) , ( t, x ) ∈ [0 , T ] × [0 , π ] . Then, by definition, we have L N = L and therefore also u = u N on the event { T τ N } ,where τ N was defined in (2.2). Since almost surely, τ N = + ∞ for sufficiently large N , we havestationary convergence of the processes u N in (2.7) to u (that is, almost surely, for large enough N , u N ( t, x ) = u ( t, x ) for all ( t, x ) ∈ [0 , T ] × [0 , π ] ). As we are interested in sample path propertiesof the mild solution to (1.1), and these properties are identical to those of u N for sufficiently large N , it is enough to consider u N instead of u in the following. The value of the parameter N hasno importance in our study, so we take N = 1 for simplicity and drop the dependency in N .Therefore, it suffices to consider the solution to the integral equation(2.8) u ( t, x ) = V ( t, x ) + b Z t Z π G ( t − s ; x, y ) σ ( u ( s, y )) d s d y + Z t Z π G ( t − s ; x, y ) σ ( u ( s, y )) L M (d s, d y ) , in other words, to assume that all jumps of L are bounded by . Furthermore, by Lemma 2.2, itonly remains to consider the process(2.9) u M ( t, x ) = Z t Z π G ( t − s ; x, y ) σ ( u ( s, y )) L M (d s, d y ) . For this purpose, we need to calculate the Fourier sine coefficients defined in (2.5). To lightenthe notations, in what follows, we will denote these coefficients by a k ( t ) . Then, by definition, a k ( t ) = r π Z π Z t Z π Z | z | sin( kx ) G ( t − s ; x, y ) σ ( u ( s, y )) z ˜ J (d s, d y, d z ) ! d x , k > . We want to exchange the stochastic integral and the Lebesgue integral, and because all involvedterms are square-integrable, Theorem A.3 with p = 2 allows us to do so. Therefore,(2.10) a k ( t ) = Z t Z π Z | z | r π (cid:18)Z π sin( kx ) G ( t − s ; x, y ) d x (cid:19) σ ( u ( s, y )) z ˜ J (d s, d y, d z )= r π e − k t Z t Z π σ ( u ( s, y )) sin( ky ) e k s L M (d s, d y ) .
7e gather some moment estimates for the family of integrals(2.11) I ba ( k ) := Z ba Z π sin( ky ) e k s σ ( u ( s, y )) L M (d s, d y ) , where a < b T . Since σ is bounded, we can estimate the second and fourth moments of I ba ( k ) using [28, Theorem 1]:(2.12) E h I ba ( k ) i C Z ba Z π sin ( ky ) e k s E h | σ ( u ( s, y )) | i d s d y C Z ba e k s d s = C e k b − e k a k , E h I ba ( k ) i C E "(cid:18)Z ba Z π sin ( ky ) e k s | σ ( u ( s, y )) | d s d y (cid:19) + Z ba Z π sin ( ky ) e k s E h | σ ( u ( s, y )) | i d s d y ! C ( e k b − e k a ) k + e k b − e k a k ! , where C also depends on R | z | z ν (d z ) and R | z | z ν (d z ) , both of which are finite.Also, for a < b c < d T , still assuming that σ is bounded,(2.13) E (cid:20)(cid:16) I ba ( k ) I dc ( j ) (cid:17) (cid:21) = E "(cid:18)Z dc Z π I ba ( k ) sin( jy ) e j s σ ( u ( s, y )) L M (d s, d y ) (cid:19) C Z dc Z π E h I ba ( k ) σ ( u ( s, y )) i sin ( jy ) e j s d s d y C (cid:18)Z ba e k s d s (cid:19) (cid:18)Z dc e j s d s (cid:19) = C e k b − e k a k e j d − e j c j , where we used the fact that I ba ( k ) is F c -measurable, and (2.12) in the second inequality.We will use [18, Chapter III, §4, Theorem 1] to show the existence of a càdlàg version of t u M ( t, · ) . By [18, Chapter III, §2, Theorem 1], u has a separable version, which is, becauseof [11, Theorem 4.7] and [3, Lemma B.1], continuous in L (Ω) , and therefore t u M ( t, · ) iscontinuous in L (Ω) as a process with values in L ([0 , π ]) (and thus in H r ([0 , π ]) since r < − ).Then it suffices to show that for any t ∈ [0 , T ] , u ( t, · ) ∈ H r ([0 , π ]) , and that for some δ > , E (cid:2) k u M ( t + h, · ) − u M ( t, · ) k H r k u M ( t − h, · ) − u M ( t, · ) k H r (cid:3) Ch δ for any h ∈ (0 , . By (2.12), we have E (cid:2) a k ( t ) (cid:3) CT , so for r < − , we have X k > (1 + k ) r a k ( t ) < + ∞ almost surely, and u ( t, · ) ∈ H r ([0 , π ]) is proved. Next, k u M ( t ± h, · ) − u M ( t, · ) k H r = X k > (1 + k ) r ( a k ( t ± h ) − a k ( t )) ,
8o using (2.10), a k ( t + h ) − a k ( t ) = − r π e − k t h (1 − e − k h ) I t ( k ) − e − k h I t + ht ( k ) i ,a j ( t − h ) − a j ( t ) = − r π e − j ( t − h ) h ( e − j h − I t − h ( j ) + e − j h I tt − h ( j ) i . Therefore, using the classical inequality ( a + b ) a + b ) ,(2.14) k u M ( t + h, · ) − u M ( t, · ) k H r k u M ( t − h, · ) − u M ( t, · ) k H r π X k,j > (1 + k ) r (1 + j ) r ( | A ( j, k ) | + | A ( j, k ) | + | A ( j, k ) | + | A ( j, k ) | ) C X k,j > (1 + k ) r (1 + j ) r (cid:0) A ( j, k ) + A ( j, k ) + A ( j, k ) + A ( j, k ) (cid:1) , for some constant C , where A ( j, k ) := e − k t e − j ( t − h ) (1 − e − k h )( e − j h − I t ( k ) I t − h ( j ) ,A ( j, k ) := e − k t e − j ( t − h ) (1 − e − k h ) e − j h I t ( k ) I tt − h ( j ) ,A ( j, k ) := e − k t e − j ( t − h ) e − k h ( e − j h − I t + ht ( k ) I t − h ( j ) ,A ( j, k ) := e − k t e − j ( t − h ) e − k h e − j h I t + ht ( k ) I tt − h ( j ) . We treat each of the four terms separately. A ( j, k ) : E (cid:2) A ( j, k ) (cid:3) = e − k t e − j ( t − h ) (1 − e − k h ) ( e − j h − E h ( I t ( k ) I t − h ( j )) i Ce − k t e − j ( t − h ) (1 − e − k h ) ( e − j h − × E h ( I t − h ( k ) I t − h ( j )) + ( I tt − h ( k ) I t − h ( j )) i =: ˜ A ( j, k ) + ˜ A ( j, k ) . By (2.13), we can write E h ( I tt − h ( k ) I t − h ( j )) i C e j ( t − h ) − j e k t − e k ( t − h ) k Ce j ( t − h ) − e − j ( t − h ) j e k t − e − k h k C e k t e j ( t − h ) j h , where we used − e − k h k h and − e − j ( t − h ) in the last inequality. Since (1 − e − k h ) and (1 − e − j h ) j h , we deduce that(2.15) ˜ A ( j, k ) Ch . Also, by the Cauchy-Schwarz inequality,(2.16) E h ( I t − h ( k ) I t − h ( j )) i E h I t − h ( k ) i E h I t − h ( j ) i .
9y (2.12) and subadditivity of the square root,(2.17) E h I t − h ( k ) i C e k ( t − h ) − k + e k ( t − h ) − k ! ! Ce k t e − k h − e − k t k + e − k h − e − k t k ! ! Ce k t (cid:18) k + 12 k (cid:19) . Let < δ < , to be chosen later. Then, multiplying each term by (1 − e − k h ) and using (1 − e − k h ) k h for the first term, and (1 − e − k h ) = (1 − e − k h ) + δ (1 − e − k h ) − δ k δ h + δ for the second term of the sum, we get(2.18) (1 − e − k h ) E h I t − h ( k ) i Ce k t (cid:16) h + k δ h + δ (cid:17) . A similar calculation yields(2.19) (1 − e − j h ) E h I t − h ( j ) i Ce j ( t − h ) (cid:16) h + j δ h + δ (cid:17) . Then, we combine (2.16), (2.18) and (2.19) to obtain(2.20) ˜ A ( j, k ) C (cid:16) h + j δ k δ h δ (cid:17) . Therefore, (2.15) and (2.20) give E (cid:2) A ( j, k ) (cid:3) C (cid:16) h + j δ k δ h δ (cid:17) .A ( j, k ) : We treat this term in a similar way to A ( j, k ) : E (cid:2) A ( j, k ) (cid:3) = e − k t e − j ( t − h ) (1 − e − k h ) e − j h E (cid:2) ( I t ( k ) I tt − h ( j )) (cid:3) Ce − k t e − j ( t − h ) (1 − e − k h ) e − j h E h ( I t − h ( k ) I tt − h ( j )) + ( I tt − h ( k ) I tt − h ( j )) i =: B ( j, k ) + B ( j, k ) . In the same way as for the term ˜ A ( j, k ) , we get(2.21) B ( j, k ) Ch . We use the Cauchy-Schwarz inequality to deal with the term B ( j, k ) : B ( j, k ) Ce − k t e − j ( t − h ) (1 − e − k h ) e − j h E (cid:2) I tt − h ( k ) (cid:3) E (cid:2) I tt − h ( j ) (cid:3) . As in (2.17), we get E (cid:2) I tt − h ( j ) (cid:3) Ce j t − e − j h j + − e − j h j ! ! Ce j t (cid:16) h + √ h (cid:17) , and similarly E (cid:2) I tt − h ( k ) (cid:3) Ce k t (cid:16) h + √ h (cid:17) . Also, for < δ < , since (1 − e − k h ) k δ h δ ,(2.22) B ( j, k ) Ck δ h δ (cid:16) h + √ h (cid:17) Ck δ h δ .
10y (2.21) and (2.22), E (cid:2) A ( j, k ) (cid:3) Ck δ h δ .A ( j, k ) : By (2.13), E (cid:2) A ( j, k ) (cid:3) = e − k t e − j ( t − h ) e − k h ( e − j h − E h ( I t + ht ( k ) I t − h ( j )) i Ce − k t e − j ( t − h ) e − k h ( e − j h − e j ( t − h ) − j e k ( t + h ) − e k t k C ( e − j h − − e − j ( t − h ) j − e − k h k C ( e − j h − j − e − k h k . Then, since (1 − e − j h ) j h and − e − k h k h , we get E (cid:2) A ( j, k ) (cid:3) Ch .A ( j, k ) : Again, by (2.13), E (cid:2) A ( j, k ) (cid:3) = e − k t e − j ( t − h ) e − k h e − j h E h ( I t + ht ( k ) I tt − h ( j )) i Ce − k t e − j ( t − h ) e − k h e − j h e j t − e j ( t − h ) j e k ( t + h ) − e k t k C − e − j h j − e − k h k . Therefore, as for the previous term we get E (cid:2) A ( j, k ) (cid:3) Ch . Then, for every r < − , we can pick < δ < such that r + δ < − . Then, E (cid:2) k u M ( t + h, · ) − u M ( t, · ) k H r k u M ( t − h, · ) − u M ( t, · ) k H r (cid:3) Ch δ , so we deduce that ( u M ( t, · )) t > has a càdlàg version in H r ([0 , π ]) for any r < − . Remark 2.4.
The result of Proposition 2.3 is in fact valid for any predictable random field u whose Fourier sine coeficients can be written in the form a k ( u ( t, · )) = Ce − k t Z t Z π sin( ky ) e k s Z ( s, y ) L (d s, d y ) , where Z is another predictable and bounded random field. For unbounded σ , we deduce the result from Proposition 2.3 via an approximation argument. Theorem 2.5.
Let u be the mild solution to the stochastic heat equation (1.1) constructed inProposition 2.1. Then, for any r < − , the process ( u ( t, · )) t ∈ [0 ,T ] has a càdlàg version in H r ([0 , π ]) . Remark 2.6.
The constraint r < − in Theorem 2.5 is optimal. This follows from the discussionat the beginning of Section 2.1.2 and the fact that the Dirac delta distribution δ a , a ∈ (0 , π ) , doesnot belong to H s ([0 , π ]) for any s > − . roof of Theorem 2.5. By the argument given at the beginning of the proof of Proposition 2.3and by Lemma 2.2, we only need to consider u M as defined in (2.9). Let σ n ( u ) = σ ( u ) | u | n . Wedefine u Mn ( t, x ) = Z t Z π G ( t − s ; x, y ) σ n ( u ( s, y )) L M (d s, d y ) . As in (2.10), the Fourier sine coefficients of t u M ( t, · ) − u Mn ( t, · ) are given by a k,n ( t ) = r π Z t Z π sin( ky ) e − k ( t − s ) ( σ ( u ( s, y )) − σ n ( u ( s, y ))) L M (d s, d y ) . Therefore, for any t ∈ [0 , T ] , k u M ( t, · ) − u Mn ( t, · ) k H r = X k > (1 + k ) r a k,n ( t ) . (2.23)Then, using e − k ( t − s ) = 1 − R ts k e − k ( t − r ) d r and Theorem A.3 and (A.3) with p = 2 , we canrewrite a k,n ( t ) = r π (cid:18)Z t Z π sin( ky ) σ ( n ) ( s, y ) L M (d s, d y ) − Z t Z π sin( ky ) (cid:18)Z ts k e − k ( t − r ) d r (cid:19) σ ( n ) ( s, y ) L M (d s, d y ) (cid:19) = r π (cid:18)Z t Z π sin( ky ) σ ( n ) ( s, y ) L M (d s, d y ) − Z t k e − k ( t − r ) (cid:18)Z r Z π sin( ky ) σ ( n ) ( s, y ) L M (d s, d y ) (cid:19) d r (cid:19) , where σ ( n ) ( s, y ) := σ ( u ( s, y )) − σ n ( u ( s, y )) . Therefore,(2.24) | a k,n ( t ) | C sup r ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:12)Z r Z π sin( ky ) σ ( n ) ( s, y ) L M (d s, d y ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) Z t k e − k ( t − r ) d r (cid:19) C sup r ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:12)Z r Z π sin( ky ) σ ( n ) ( s, y ) L M (d s, d y ) (cid:12)(cid:12)(cid:12)(cid:12) , where C does not depend on k . So by Doob’s inequality, we deduce that(2.25) E " sup t ∈ [0 ,T ] a k,n ( t ) C Z T Z π sin ( ky ) E h σ n ) ( s, y ) i d s d y . By (2.3), (2.23) and (2.25), it follows from dominated convergence that for any r < − , E " sup t ∈ [0 ,T ] k u ( t, · ) − u n ( t, · ) k H r C X k > (1 + k ) r Z T Z π sin ( ky ) E h σ n ) ( s, y ) i d s d y → as n → + ∞ . Therefore, sup t ∈ [0 ,T ] k u ( t, · ) − u n ( t, · ) k H r → in L (Ω) as n → + ∞ , and there isa subsequence ( n k ) k > such that sup t ∈ [0 ,T ] k u ( t, · ) − u n k ( t, · ) k H r → almost surely as k → + ∞ .This means that u n k ( t, · ) converges to u ( t, · ) in H r ([0 , π ]) uniformly in time for any r < − . Since σ n k is bounded, t u n k ( t, · ) has a càdlàg version in H r ([0 , π ]) by Proposition 2.3 and Remark 2.4.Therefore, t u ( t, · ) has a càdlàg version in H r ([0 , π ]) for any r < − .12 .2 The stochastic heat equation on R d In [12], the first author proved the existence of a solution to the stochastic heat equation on R d under assumptions on the driving noise that are general enough to include the case of α -stablenoises. More specifically, suppose that D = R d in (1.1) and that the following hypotheses hold: (H) There exists < p < d and p ( d − p ) < q p such that Z | z | | z | p ν (d z ) + Z | z | > | z | q ν (d z ) < + ∞ . If p < , we assume that b := b − R | z | z ν (d z ) = 0 .In contrast to the situation on a bounded domain, we can no longer use the stopping times τ N in (2.2) (with [0 , π ] replaced by R d ) to localize equation (1.1). Indeed, since R d is unbounded,the d t d x ν (d z ) -measure of [0 , T ] × R d × [ − N, N ] c will in general be infinite for any N ∈ N . Inparticular, on any time interval [0 , ε ] where ε > , we already have infinitely many jumps ofarbitrarily large size, which implies that τ N = 0 almost surely for all N ∈ N .Therefore, instead of using the stopping times (2.2), the idea is to use truncation levels thatincrease with the distance to the origin. More precisely, let h : R d → R be the function h ( x ) =1 + | x | η , for some η to be chosen later, and define for N ∈ N ,(2.26) τ N = inf { t ∈ [0 , T ] : J ([0 , t ] × { ( x, z ) : | z | > N h ( x ) } ) > } . For every N > , we can now introduce a truncation of L by(2.27) L N (d t, d x ) = b d t d x + Z | z | z ˜ J (d t, d x, d z ) + Z < | z | Let σ be Lipschitz continuous, u be bounded and continuous, and L be a Lévywhite noise as in (1.6) satisfying (H) for some p, q > . Then, if we choose dq < η < − d ( p − p − q ,we have τ N > for every N > and almost surely, τ N = + ∞ for large N (recall the convention inf ∅ = + ∞ ). Moreover, for any N > , there exists a solution u N to (2.28) such that for someconstant C N < + ∞ , we have (2.29) sup ( t,x ) ∈ [0 ,T ] × [ − R,R ] d E [ | u N ( t, x ) | p ] E − d ( p − p ∨ , p ∨ (cid:16) C N R η ( p − q ) p ∨ (cid:17) < + ∞ , R > , where E α,β ( z ) = P k > z k Γ( αk + β ) for α, β > and z ∈ R are the Mittag–Leffler functions.Furthermore, we have u N ( t, x ) = u N +1 ( t, x ) on { t τ N } , and the random field u defined by u ( t, x ) = u N ( t, x ) on { t τ n } , is a mild solution to (1.1) on D = R d .Proof. The result is a direct application of [12, Theorem 3.1] except for the moment property(2.29). The finiteness of the left-hand side is included in the cited theorem for p < d . In thecase d = 1 and < p < , the only thing we need is an extension of [12, Lemma 3.3(2)], which canbe obtained by combining the arguments given in the proof of [12, Lemma 3.3(2)] and the proof of13roposition 2.1. Indeed, for predictable φ and φ , proceeding as in the proof of [12, Lemma 3.3]but using the moment inequalities of [28, Theorem 1], we see that E (cid:20)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z R g ( t − s, x − y ) ( σ ( φ ( s, y )) − σ ( φ ( s, y ))) L N (d s, d y ) (cid:12)(cid:12)(cid:12)(cid:12) p (cid:21) C Z t Z R ( g ( t − s, x − y ) + g p ( t − s, x − y )) E [ | φ ( s, y ) − φ ( s, y ) | p ] h ( y ) p − q d s d y . In order to obtain the bound involving the Mittag–Leffler functions, observe from the calcula-tions between the last display on page 2272 and equation (3.13) of [12] that for every fixed N > ,there exists C N < + ∞ such that sup ( t,x ) ∈ [0 ,T ] × [ − R,R ] d E [ | u N ( t, x ) | p ] X n > C nN (cid:16) R nη ( p − q ) + Γ (cid:16) nη ( p − q )2 (cid:17)(cid:17) Γ (cid:0) − d ( p − (cid:1) n Γ (cid:0) − d ( p − n (cid:1) p ∨ X n > C nN Γ (cid:16) nη ( p − q )2 (cid:17) Γ (cid:0) − d ( p − n (cid:1) p ∨ + X n > C nN R nη ( p − q ) Γ (cid:0) − d ( p − n (cid:1) ! p ∨ . The first series converges for our choice of η . Furthermore, for all a > , we have by Stirling’sformula that Γ(1 + ax ) p ∨ > C Γ( axp ∨ ) when x > . Hence, sup ( t,x ) ∈ [0 ,T ] × [ − R,R ] d E [ | u N ( t, x ) | p ] C N + X n > (cid:0) C N R η ( p − q ) p ∨ (cid:1) n Γ (cid:16) p ∨ + (2 − d ( p − n p ∨ (cid:17) E − d ( p − p ∨ , p ∨ (cid:16) C N R η ( p − q ) p ∨ (cid:17) , which is (2.29). The proof of Proposition 2.7 heavily relies on the stopping times τ N introduced in (2.26). Theseare “centered” around the origin in the sense that large jumps are permitted if they occur farenough from x = 0 . As a consequence, even if the initial condition is constant, it does not follow apriori from Proposition 2.7 that the solution u to (1.1) is stationary in space. On the other hand,of course, choosing to center τ N around the origin is completely arbitrary. So in this section, weshow that the solution constructed in Proposition 2.7 remains the same if we take other spatialreference points for τ N , from which the stationarity of the solution in space will follow.To this end, let dq < η < − d ( p − p − q and define the family of stopping times τ aN by τ aN := inf { t ∈ [0 , T ] : J ([0 , t ] × { ( x, z ) : | z | > N h ( x − a ) } ) > } , N ∈ N , a ∈ R d . In particular, τ N is the same as τ N defined in (2.26). Since the intensity measure of J is invariantunder translation in the space variable, τ aN has the same law as τ N , and the conclusions of Propo-sition 2.7 are valid for τ aN . In particular, for any N > , almost surely τ aN > , and τ aN = + ∞ forlarge N . Furthermore, by definition, on the event { t τ aN } , L (d t, d x ) = L aN (d t, d x ) , where L aN (d t, d x ) := b d t d x + Z | z | z ˜ J (d t, d x, d z ) + Z < | z | Nh ( x − a ) z J (d t, d x, d z ) . Proposition 2.8. Let σ : R → R be Lipschitz continuous, u be bounded and continuous, and L be a Lévy white noise as in (1.6) fulfilling the assumption (H) with p, q > . Then for any N ∈ N and a ∈ R d , there exists a mild solution u aN to (1.1) with noise L aN instead of L such that (2.29) also holds for u aN . Moreover, for a, b ∈ R d , N ∈ N and ( t, x ) ∈ [0 , T ] × R d , we have (2.30) u aN ( t, x ) t τ aN ∧ τ bN = u bN ( t, x ) t τ aN ∧ τ bN a.s. roof. The first part is proved in the same way as Proposition 2.7. For (2.30), we observe that L aN = L bN on { t < τ aN ∧ τ bN } . Then we use the construction of the solutions u aN and u bN via aPicard iteration scheme as in the proof of [12, Theorem 3.1] and show that at each step of thescheme,(2.31) u a,nN ( t, x ) t τ aN ∧ τ bN = u b,nN ( t, x ) t τ aN ∧ τ bN a.s.For n = 0 , we clearly have u a, N ( t, x ) = u b, N ( t, x ) = u ( x ) . Now if (2.31) holds for some n > , then u a,n +1 N ( t, x ) t τ aN ∧ τ bN = t τ aN ∧ τ bN Z t Z R d g ( t − s, x − y ) σ (cid:0) u a,nN ( s, y ) (cid:1) L aN (d s, d y )= t τ aN ∧ τ bN Z t Z R d g ( t − s, x − y ) σ (cid:16) u b,nN ( s, y ) (cid:17) s τ aN ∧ τ bN L aN (d s, d y )= t τ aN ∧ τ bN Z t Z R d g ( t − s, x − y ) σ (cid:16) u b,nN ( s, y ) (cid:17) s τ aN ∧ τ bN L bN (d s, d y )= t τ aN ∧ τ bN Z t Z R d g ( t − s, x − y ) σ (cid:16) u b,nN ( s, y ) (cid:17) L bN (d s, d y )= u b,n +1 N ( t, x ) t τ aN ∧ τ bN . Since u a,nN ( t, x ) → u aN ( t, x ) and u b,nN ( t, x ) → u bN ( t, x ) as n → + ∞ in L p (Ω) , we deduce (2.30).In [15, Definition 5.1], the second author introduced the property (S) for a stochastic processand a martingale measure, which is a sort of stationarity property in the space variable. In ourcase, the noise is not necessarily a martingale measure, but we can use a similar definition: Definition 2.9. We say the family of random fields u aN has property (S) if the law of the process (cid:16)(cid:16) u aN ( t, a + x ) , ( t, x ) ∈ [0 , T ] × R d (cid:17) ; (cid:16) L aN ([0 , t ] × ( a + B )) , ( t, B ) ∈ [0 , T ] × B b ( R d ) (cid:17)(cid:17) , does not depend on a . Lemma 2.10. If u ( x ) ≡ u is constant, then the family ( u aN : a ∈ R d ) has property (S).Proof. Similarly to the proof of Proposition 2.8, it is enough to show property (S) for the Picard it-erates u a,nN for each n > . For n = 0 , we obviously have u a, N ( t, x + a ) = u = u , N ( t, x ) . So property(S) for u a,nN follows from the fact that the law of (cid:0) L aN ([0 , t ] × ( a + B )) , ( t, B ) ∈ [0 , T ] × B b ( R d ) (cid:1) does not depend on a . Next, assume that u a,nN has the property (S). Since u a,n +1 N ( t, x ) = u + Z t Z R d g ( t − s, x − y ) σ (cid:0) u a,nN ( s, y ) (cid:1) L aN (d s, d y ) , we can use the same argument as in [15, Lemma 18], since the proof only relies on the fact that L has a law that is invariant under translation in the space variable. Theorem 2.11. If u ( x ) ≡ u is constant, for any a ∈ R , the random field ( u ( t, a + x ) : ( t, x ) ∈ [0 , T ] × R d ) has the same law as the random field ( u ( t, x ) : ( t, x ) ∈ [0 , T ] × R d ) .Proof. By (2.30), u aN ( t, a + x ) t τ aN ∧ τ N = u N ( t, a + x ) t τ aN ∧ τ N almost surely. Taking the stationarylimit as N → + ∞ , we get that u a ( t, a + x ) = u ( t, a + x ) almost surely for any ( t, x ) ∈ [0 , T ] × R d .Also, by the property (S) of the family of random fields ( u aN : a ∈ R d ) (see Lemma 2.10), the randomfield ( u aN ( t, a + x ) : ( t, x ) ∈ [0 , T ] × R d ) has the same law as the random field ( u N ( t, x ) : ( t, x ) ∈ [0 , T ] × R d ) . Again, taking the stationary limit as N → + ∞ , we get that the random field ( u a ( t, a + x ) : ( t, x ) ∈ [0 , T ] × R d ) has the same law as the random field ( u ( t, x ) : ( t, x ) ∈ [0 , T ] × R d ) .Therefore, the random field ( u ( t, a + x ) : ( t, x ) ∈ [0 , T ] × R d ) has the same law as the randomfield ( u ( t, x ) : ( t, x ) ∈ [0 , T ] × R d ) . 15 .2.2 Existence of a càdlàg solution in H r,loc ( R d ) with r < − d In the following, we want to establish a regularity result for the paths of the mild solution to(1.1) in the case D = R d , analogous to Theorem 2.5 which concerns D = [0 , π ] . Since D = R d is unbounded, and the solution may not decay in space (see Theorem 2.11), we consider the mildsolution u : t u ( t, · ) as a distribution-valued process in a local fractional Sobolev space, andprove that it has a càdlàg version in this space.Recall to this end the Schwartz space S ( R d ) of smooth functions ϕ : R d → R such that sup x ∈ R d (cid:12)(cid:12) x α ϕ ( β ) ( x ) (cid:12)(cid:12) < + ∞ for any multi-indices α, β ∈ N d , equipped with the topology inducedby the semi-norms P | α | , | β | p sup x ∈ R d (cid:12)(cid:12) x α ϕ ( β ) ( x ) (cid:12)(cid:12) for p ∈ N . Here, ϕ ( β ) ( x ) = ∂ β x . . . ∂ β d x d ϕ ( x ) if β = ( β , . . . , β d ) . Its topological dual is called the space of tempered distributions and is de-noted by S ′ ( R d ) . The classical Fourier transform F ( ϕ )( ξ ) := R R d e − iξ · x ϕ ( x ) d x with ξ ∈ R d and ϕ ∈ S ( R d ) can be extended by duality to f ∈ S ′ ( R d ) : hF ( f ) , ϕ i := h f, F ( ϕ ) i , ϕ ∈ S ( R d ) . Definition 2.12. The (local) fractional Sobolev space of order r ∈ R is defined by H r ( R d ) := n f ∈ S ′ ( R d ) : ξ (cid:0) | ξ | (cid:1) r F ( f )( ξ ) ∈ L ( R d ) o(cid:18) H r, loc ( R d ) := n f ∈ S ′ ( R d ) : (cid:16) ∀ θ ∈ C ∞ c ( R d ) : θf ∈ H r ( R d ) (cid:17)o (cid:19) . The topology on H r ( R d ) is induced by the norm k f k H r ( R d ) := (cid:13)(cid:13)(cid:13) (1 + | · | ) r F ( f )( · ) (cid:13)(cid:13)(cid:13) L ( R d ) , and we have f n → f in H r,loc ( R d ) if θf n → θf in H r ( R d ) as n → + ∞ for any θ ∈ C ∞ c ( R d ) . We now proceed to studying the regularity of u : t u ( t, · ) in H r,loc ( R d ) . As in the case of abounded interval in dimension one, the drift part is easy to handle. Lemma 2.13. Let Z be a bounded measurable random field and F ( t, x ) := Z t Z R d g ( t − s, x − y ) Z ( s, y ) d s d y . Then the process t F ( t, · ) is continuous in H r,loc ( R d ) for any r .Proof. Since g ∈ L ([0 , T ] × R d ) and Z is bounded, it follows from [6, Corollary 3.9.6] that thesample paths of F are jointly continuous in ( t, x ) almost surely. Therefore, t F ( t, · ) is continuousin H ,loc ( R d ) = L loc ( R d ) , hence also in H r,loc ( R d ) for any r .Next, we consider the situation where σ is a bounded function. Already in this restricted case,the unboundedness of space and the possibility of having infinitely many large jumps require amore careful analysis of the different parts of the solution. Proposition 2.14. Let σ be a bounded and Lipschitz function, and u be the mild solution to thestochastic heat equation (1.1) constructed in Proposition 2.7 under hypothesis (H) . Then, for any r < − d , the stochastic process ( u ( t, · )) t ∈ [0 ,T ] has a càdlàg version in H r, loc ( R d ) .Proof. Since the mild solution u N to the truncated equation (2.28) agrees with the mild solution u to the stochastic heat equation (1.1) on { t τ N } (see the last statement of Proposition 2.7),the sample path properties of u and u N are the same, and we can restrict to the study of the16egularity of the sample paths of u N . Furthermore, there is no loss of generality if we take N = 1 .Therefore, we suppose that u ( t, x ) = V ( t, x ) + Z t Z R d g ( t − s, x − y ) σ ( u ( s, y )) L (d s, d y ) , where L is the truncated noise from (2.27) with N = 1 . We use the decomposition(2.32) u ( t, x ) = V ( t, x ) + u , ( t, x ) + u , ( t, x ) + u , ( t, x ) + u , ( t, x ) + u ( t, x ) , where for A > and Z ( s, y ) := σ ( u ( s, y )) , u , ( t, x ) := Z t Z R d g ( t − s, x − y ) Z ( s, y ) y ∈ [ − A, A ] d L M (d s, d y ) ,u , ( t, x ) := Z t Z R d g ( t − s, x − y ) Z ( s, y ) y / ∈ [ − A, A ] d L M (d s, d y ) ,u , ( t, x ) := Z t Z R d g ( t − s, x − y ) Z ( s, y ) y ∈ [ − A, A ] d L P (d s, d y ) ,u , ( t, x ) := Z t Z R d g ( t − s, x − y ) Z ( s, y ) y / ∈ [ − A, A ] d L P (d s, d y ) ,u ( t, x ) := b Z t Z R d g ( t − s, x − y ) Z ( s, y ) d s d y , where L M is defined in (1.6), and L P is the noise obtained by applying the truncation (2.27) with N = 1 to L P from (1.6). It is clear that V is jointly continuous in ( t, x ) , and the same holds for u as pointed out in the proof of Lemma 2.13. Furthermore, on [0 , T ] × [ − A, A ] d , the noise L P consists of only finitely many jumps. So upon a change of the drift b and increasing the truncationlevel for L M from to the largest size of these jumps (which clearly does not affect the argumentsbelow), we may assume that u , = 0 . The remaining terms are now treated separately. u , ( t, x ) : By definition of the Fourier transform, we have for ϕ ∈ S ( R d ) , (cid:10) F (cid:0) u , ( t, · ) (cid:1) , ϕ (cid:11) = (cid:10) u , ( t, · ) , F ( ϕ ) (cid:11) = Z R d u , ( t, x ) F ( ϕ )( x ) d x = Z R d (cid:18)Z t Z R d g ( t − s, x − y ) Z ( s, y ) y ∈ [ − A, A ] d L M (d s, d y ) (cid:19) F ( ϕ )( x ) d x . Permuting the stochastic integral and the Lebesgue integral (because R | z | | z | p ν (d z ) < + ∞ , g ∈ L p ([0 , T ] × R d ) and Z is bounded, this is possible by Theorem A.3 together with the estimate(A.3)) yields(2.33) (cid:10) F ( u , ( t, · )) , ϕ (cid:11) = Z t Z [ − A, A ] d (cid:18)Z R d F ( ϕ )( x ) g ( t − s, x − y ) d x (cid:19) Z ( s, y ) L M (d s, d y )= Z t Z [ − A, A ] d (cid:18)Z R d e − iξ · y − ( t − s ) | ξ | ϕ ( ξ ) d ξ (cid:19) Z ( s, y ) L M (d s, d y )= Z R d Z t Z [ − A, A ] d e − iξ · y − ( t − s ) | ξ | Z ( s, y ) L M (d s, d y ! ϕ ( ξ ) d ξ , which implies that F ( u , ( t, · ))( ξ ) is given by(2.34) a ξ ( t ) := e −| ξ | t Z t Z R d e − iξ · y e s | ξ | Z ( s, y ) y ∈ [ − A, A ] d L M (d s, d y ) . (cid:13)(cid:13) u , ( t ± h, · ) − u , ( t, · ) (cid:13)(cid:13) H r ( R d ) = Z R d (1 + | ξ | ) r | a ξ ( t ± h ) − a ξ ( t ) | d ξ . Since the function t e −| ξ | t is continuous, and the stochastic integral in a ξ ( t ) exists in L (Ω) , t a ξ ( t ) is continuous in L (Ω) . Furthermore, E h | a ξ ( t ) | i C − e − | ξ | t | ξ | C , for some constant C that does not depend on ξ , so by the dominated convergence theorem (whichapplies since r < − d ), E h(cid:13)(cid:13) u , ( t + h, · ) − u , ( t, · ) (cid:13)(cid:13) H r ( R d ) i → , as h → , and the process t u , ( t, · ) is continuous in L (Ω) as a process with values in H r ( R d ) .In order to apply [18, Chapter III, §4, Theorem 1] to deduce the existence of a càdlàg modifi-cation of t u , ( t, · ) in H r ( R d ) for any r < − d , it remains to prove E h k u , ( t + h, · ) − u , ( t, · ) k H r ( R d ) k u , ( t − h, · ) − u , ( t, · ) k H r ( R d ) i Ch δ for some δ > . Upon defining, similar to (2.11), I ba ( ξ ) := Z ba Z R d Z R e − iξ · y e s | ξ | Z ( s, y ) z | z | y ∈ [ − A, A ] d ˜ J (d s, d y, d z ) for a < b T and ξ ∈ R d , the proof is identical to that of Proposition 2.3 for the equation ona bounded interval if we make the following replacements: [0 , π ] ←→ [ − A, A ] d , k ←→ ξ , sin( ky ) ←→ e − iξ · y .u , ( t, x ) : If f : R d → R is a smooth function, then for a, b ∈ R d with a i b i for all i d ,(2.35) f ( b ) = f ( a ) + d X i =1 X k < ··· 18e see from this expression that u , is jointly continuous in ( t, x ) . By the argument at the end ofthe proof of Lemma 2.13, we deduce that t u , ( t, · ) [ − A,A ] d is continuous in H r ( R d ) for r . u , ( t, x ) : This process takes into account only the jumps that are far away from x , but that canbe arbitrarily large. We can write u , as a sum: u , ( t, x ) = X i > g ( t − T i , x − X i ) Z ( T i , X i ) Z i X i / ∈ [ − A, A ] d , < | Z i | < | X i | η , T i t . We first observe that each term of this sum is jointly continuous in ( t, x ) ∈ [0 , T ] × [ − A, A ] d almostsurely. We show that this sum converges uniformly in ( t, x ) ∈ [0 , T ] × [ − A, A ] d . Choose A largeenough such that T < A d . Because | x − X i | > A , Lemma 3.6 below shows that the maximum ofthe function t g ( t, x − X i ) is attained at t = T : sup t T,x ∈ [ − A,A ] d g ( t − T i , x − X i ) sup x ∈ [ − A,A ] d CT − d e − | x − Xi | T CT − d e − | pA ( Xi ) − Xi | T , where p A is the projection on the convex set [ − A, A ] d . Then, for β = 1 ∧ q ,(2.37) E " X i > sup t T,x ∈ [ − A,A ] d (cid:12)(cid:12)(cid:12) g ( t − T i , x − X i ) Z ( T i , X i ) Z i X i / ∈ [ − A, A ] d , < | Z i | < | X i | η , T i t (cid:12)(cid:12)(cid:12) ! β CT βd E " X i > (cid:12)(cid:12)(cid:12)(cid:12) e − | pA ( Xi ) − Xi | T Z i X i / ∈ [ − A, A ] d , < | Z i | , T i T (cid:12)(cid:12)(cid:12)(cid:12) ! β CT βd E "X i > (cid:12)(cid:12)(cid:12)(cid:12) e − | pA ( Xi ) − Xi | T Z i X i / ∈ [ − A, A ] d , < | Z i | , T i T (cid:12)(cid:12)(cid:12)(cid:12) β C Z T Z y / ∈ [ − A, A ] d Z | z | > | z | β e − β | pA ( y ) − y | T d s d y ν (d z ) < + ∞ . Therefore, the sum defining u , converges uniformly in ( t, x ) ∈ [0 , T ] × [ − A, A ] d , and u , is jointlycontinuous. Thus, t u , ( t, · ) [ − A,A ] d is continuous in H r ( R d ) for every r .Since A can be chosen arbitrarily large, the assertion of the proposition follows.In order to pass from bounded to unbounded nonlinearities σ , the basic strategy remains thesame as in the proof of Theorem 2.5. However, it was crucial in that proof that the solution havea finite second moment. Unfortunately, in dimensions d > , the mild solution u to (1.1) has nofinite second moments as a result of the singularity of g . And it is easy to convince oneself thattaking powers p < instead of does not combine well with the k · k H r ( R d ) -norms. Instead, inthe proof we propose below, the idea is to consider an equivalent probability measure Q (whichobviously does not affect the path properties of u ) under which the solution has a finite secondmoment. Although L might not be a Lévy noise under Q anymore, it follows from the theory ofintegration against random measures, which we briefly recall in the Appendix, that there exists aparticularly clever choice of Q such that we have sufficient control on the second moments of bothintegrands and integrators under Q . Theorem 2.15. If u is the mild solution to the stochastic heat equation (1.1) constructed underthe assumptions of Proposition 2.7, then, for any r < − d , the stochastic process ( u ( t, · )) t ∈ [0 ,T ] hasa càdlàg version in H r, loc ( R d ) .Proof. We first consider the case p > in assumption (H) . As in Proposition 2.14, we can supposethat u is the solution to (2.28) with N = 1 , and use the decomposition (2.32) with A > . The19erms V and u , can be dealt with as in Proposition 2.14. For the remaining terms, we usedifferent arguments. u , ( t, x ) : Let σ n ( u ) = σ ( u ) | u | n and define u , n as in (2.32) but with Z replaced by σ n ( u ) . Then, u , ( t, x ) − u , n ( t, x ) = Z t Z y ∈ [ − A, A ] d g ( t − s, x − y ) ( σ ( u ( s, y )) − σ n ( u ( s, y ))) L M (d s, d y ) , and k u , ( t, · ) − u , n ( t, · ) k H r ( R d ) = Z R d (1 + | ξ | ) r (cid:12)(cid:12) F (cid:0) u , ( t, · ) − u , n ( t, · ) (cid:1) ( ξ ) (cid:12)(cid:12) d ξ . (2.38)Writing σ ( n ) ( s, y ) = σ ( u ( s, y )) − σ n ( u ( s, y )) , we obtain F ( u , ( t, · ) − u , n ( t, · ))( ξ ) = Z t Z [ − A, A ] d e − iξ · y e − ( t − s ) | ξ | σ ( n ) ( s, y ) L M (d s, d y ) as in (2.34). With similar calculations as in (2.24), but using Theorem A.3 with < p < d ,one can show that(2.39) sup t ∈ [0 ,T ] (cid:12)(cid:12) F ( u , ( t, · ) − u , n ( t, · ))( ξ ) (cid:12)(cid:12) C sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z [ − A, A ] d e − iξ · y σ ( n ) ( s, y ) L M (d s, d y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where C does not depend on ξ . With notation from the Appendix, the fact that σ ( u ) ∈ L ,p ( L M , P ) implies that there exists a probability measure Q that is equivalent to P such that the process σ ( u ) belongs to L , ( L M , Q ) , see Theorem A.4. Consequently, using the notation in (A.1), we deducefrom (2.38) that(2.40) E Q " sup t ∈ [0 ,T ] k u , ( t, · ) − u , n ( t, · ) k H r ( R d ) C Z R d (1 + | ξ | ) r E Q sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z y ∈ [ − A, A ] d e − iξ · y σ ( n ) ( s, y ) L M (d s, d y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ξ C Z R d (1 + | ξ | ) r k e − iξ · ( · ) σ ( n ) k L M , , Q d ξ C k σ ( n ) k L M , , Q Z R d (1 + | ξ | ) r d ξ . The last integral is finite because r < − d . Moreover, σ ( n ) ( ω, s, y ) → , pointwise in ( ω, s, y ) , and isbounded by σ ( u ( ω, s, y )) , which belongs to L , ( L M , Q ) by assumption. Hence, by Theorem A.1,the left-hand side of (2.40) converges to as n → + ∞ . As before, we may extract a subsequencethat converges uniformly in [0 , T ] almost surely with respect to Q , and hence P . We now deducethat t u , ( t, · ) has a càdlàg modification because the processes u , n have càdlàg modificationsby Proposition 2.14. u , ( t, x ) : The proof is identical to the corresponding part in Proposition 2.14, provided we canstill apply Theorem A.3 in (2.36). In order to justify this, observe that E "Z u Z R d Z | z | | ∂ x k ...x ki ∂ t g ( u − s, c k ( − A , r ) − y ) σ ( u ( s, y )) z | p y / ∈ [ − A, A ] d d s d y ν (d z ) C Z u Z R d | ∂ x k ...x ki ∂ t g ( u − s, c k ( − A , r ) − y ) | p E − d ( p − p ∨ , p ∨ (cid:16) C | y | η ( p − q ) p ∨ (cid:17) y / ∈ [ − A, A ] d d s d y C Z u Z R d | ∂ x k ...x ki ∂ t g ( u − s, c k ( − A , r ) − y ) | p exp (cid:16) C | y | η ( p − q )2 − d ( p − (cid:17) P ( | y | ) y / ∈ [ − A, A ] d d s d y 20y (2.29) and [19, Theorems 4.3 and 4.4] with some polynomial P . Next, for every multi-index α ∈ N d , it is easily verified by induction that ∂ α g ( t, x ) takes the form Q ( t − , x ) exp( − | x | t ) forsome polynomial Q . So if l denotes the degree of Q , we have for every t ∈ [0 , T ] and | x | > lT , | ∂ α g ( t, x ) | C (1 + t − l + | x | l ) e − | x | t C (1 + T − l + | x | l ) e − | x | T =: ˜ Q ( x ) e − | x | T . Hence, as | c k ( − A , r ) − y | > A for y / ∈ [ − A, A ] d , we obtain for sufficiently large A that theexpectation in the penultimate display is bounded by C Z R d | ˜ Q ( c k ( − A , r ) − y ) | p e − p | c k ( − A ,r ) − y | T exp (cid:16) C | y | η ( p − q )2 − d ( p − (cid:17) P ( | y | ) y / ∈ [ − A, A ] d d y C Z R d | ˜ Q ( c k ( − A , r ) − y ) | p e − p | c k ( − A ,r ) − y | T exp (cid:16) C | y | η ( p − q )2 − d ( p − (cid:17) P ( | y | ) d y = C Z R d | ˜ Q ( y ) | p e − p | y | T exp (cid:16) C | c k ( − A , r ) − y | η ( p − q )2 − d ( p − (cid:17) P ( | c k ( − A , r ) − y | ) d y . Since | c k ( − A , r ) | √ dA , | x − y | b b − ( | x | b + | y | b ) for b > , and P ( | x − y | ) C (1 + | x | m + | y | m ) where m is the degree of P , one can find another polynomial ˜ P such that the last integral is furtherbounded by C Z R d | ˜ Q ( y ) | p e − p | y | T exp (cid:16) C | y | η ( p − q )2 − d ( p − (cid:17) ˜ P ( y ) d y , which is independent of r , and finite because it is possible by assumption (H) to choose η > dq such that η ( p − q )2 − d ( p − < is satisfied. Theorem A.3 is therefore applicable by (A.3). u , ( t, x ) : The argument remains the same as in Proposition 2.14, except that we have to replacethe final bound in (2.37) by C Z T Z y / ∈ [ − A, A ] d Z | z | > | z | β e − β | pA ( y ) − y | T (cid:18) E − d ( p − p ∨ , p ∨ (cid:16) C | y | η ( p − q ) p ∨ (cid:17)(cid:19) βp d s d y ν (d z ) , which is finite by an argument similar to the one for u , . u ( t, x ) : Consider the decomposition u ( t, x ) = u , ( t, x ) + u , ( t, x ) where(2.41) u , ( t, x ) = b Z t Z y ∈ [ − A, A ] d g ( t − s, x − y ) σ ( u ( s, y )) d s d y ,u , ( t, x ) = b Z t Z y / ∈ [ − A, A ] d g ( t − s, x − y ) σ ( u ( s, y )) d s d y . If u , n is the process obtained from u , by replacing σ ( u ( s, y )) by σ n ( u ( s, y )) , then, as in (2.39), sup t ∈ [0 ,T ] (cid:12)(cid:12) F ( u ( t, · ) − u n ( t, · ))( ξ ) (cid:12)(cid:12) C sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t Z R d e − iξ · y σ ( n ) ( s, y ) y ∈ [ − A, A ] d d s d y (cid:12)(cid:12)(cid:12)(cid:12) C Z T Z R d | σ ( n ) ( s, y ) | y ∈ [ − A, A ] d d s d y . Consequently, we have sup t ∈ [0 ,T ] k u , ( t, · ) − u , n ( t, · ) k H r ( R d ) = sup t ∈ [0 ,T ] Z R d (1 + | ξ | ) r (cid:12)(cid:12) F ( u , ( t, · ) − u , n ( t, · ))( ξ ) (cid:12)(cid:12) d ξ C Z T Z R d | σ ( n ) ( s, y ) | y ∈ [ − A, A ] d d s d y Z R d (1 + | ξ | ) r d ξ . u is the solution to (2.28) with N = 1 , the expectation of the left-hand side tends to as n → + ∞ by (2.29) and the dominated convergence theorem. Hence, t u , ( t, · ) inherits thecàdlàg sample paths of u , n , see Lemma 2.13. Concerning u , , the continuity of ( t, x ) u , ( t, x ) on [0 , T ] × [ − A, A ] d is shown in the same way as for u , . Instead of the stochastic Fubini theorem,one can use the ordinary Fubini theorem because Z t d u Z x k − A d r k . . . Z x ki − A d r k i (cid:18)Z u Z R d | ∂ x k ...x ki ∂ t g ( u − s, c k ( − A , r ) − y ) σ ( u ( s, y )) | y / ∈ [ − A, A ] d d s d y (cid:19) < + ∞ almost surely. This is verified by showing that the expectation of the integral in brackets is finiteand uniformly bounded in u and r k , . . . , r k i . This concludes the proof for p > .For < p < , we have to modify the proof in the following way. Because L has drift b = 0 and summable jumps by the assumption R | z | | z | p ν (d z ) < + ∞ , we can write u in the same formas (2.32) with L M (d t, d x ) replaced by R | z | z J (d t, d x, d z ) and u = 0 . An inspection of theproof above shows that the arguments for V , u , , u , remain valid, and in principle also for u , and u , if changing the order of integration in (2.33) and (2.36), respectively, is permitted. Thejustification is comparable to the situation for p > ; one only has to use (A.4) instead of (A.3): Z t Z R d Z | z | (cid:18)Z R d |F ( ϕ )( x ) | g ( t − s, x − y ) d x (cid:19) p E [ σ ( u ( s, y )) | p ] | z | p y ∈ [ − A, A ] d d s d y ν (d z ) C Z T Z R d (cid:18)Z R d g ( t − s, x − y ) d x (cid:19) p y ∈ [ − A, A ] d d s d y = CT (4 A ) d < + ∞ . Remark 2.16. The paper [21] studies the existence of càdlàg modifications in certain Banachspaces of solutions to a class of stochastic PDEs driven by Poisson random measures. Example 2.3in [21] particularizes to the case of the stochastic heat equation with a multiplicative Lévy space–time white noise. However, this example contains an error since the measure ν in the first displayon p. 1502 is not a Lévy measure (it is infinite on sets of the form {| x | > δ } for all sufficiently smallvalues of δ , contradicting Remark 3.1 in [21]). After private communication with the author, itseems that this example could be rewritten for the case of a bounded domain, but cannot be extendedto the case where D = R d (because the stopping times in (2.2) with [0 , π ] replaced by R d are almost surely for all N ∈ N , cf. the discussion at the beginning of Section 2.2). Let D be a C ∞ -regular domain of R d , where d > , that is, we assume that D is a bounded openset whose boundary ∂D is a smooth ( d − -dimensional manifold, and whose closure ¯ D has thesame boundary ∂ ¯ D = ∂D . For the stochastic heat equation (1.1) on such a domain D , we assume: (H’) There exists < p < d such that R | z | | z | p ν (d z ) < + ∞ .As in the case of an interval (Section 2.1), the stopping times(2.42) τ N = inf { t ∈ [0 , T ] : J ([0 , t ] × D × [ − N, N ] c ) = 0 } are almost surely strictly positive and equal to + ∞ for large N .22 roposition 2.17. Let D be a C ∞ -regular domain, σ : R → R be a Lipschitz function and let L be a pure jump Lévy white noise as in (1.6) such that (H’) is satisfied. Then there exists apredictable mild solution u to (1.1) such that for all < p < d , (2.43) sup ( t,x ) ∈ [0 ,T ] × D E [ | u ( t, x ) | p t τ N ] < + ∞ . Furthermore, up to modifications, the solution is unique among all predictable random fields thatsatisfy (2.43) .Proof. By [16, Corollary 3.2.8], G D ( t ; x, y ) Ct − d e − | x − y | t , so [11, Theorem 3.5] applies.As in the proof of Proposition 2.3, the stopping times τ N allow us to ignore the big jumps forthe analysis of path properties of the solution. So we only need to consider(2.44) L N (d t, d x ) = b N d t d x + Z | z | N z ˜ J (d t, d x, d z ) , where b N := b − R < | z | N z ν (d z ) , and the corresponding mild solutions to(2.45) u N ( t, x ) = V ( t, x ) + Z t Z D G D ( t − s ; x, y ) σ ( u N ( s, y )) L N (d s, d y ) . For simplicity, we take N = 1 in the following, so that our equation becomes(2.46) u ( t, x ) = V ( t, x ) + b Z t Z D G D ( t − s ; x, y ) σ ( u ( s, y )) d s d y + Z t Z D G D ( t − s ; x, y ) σ ( u ( s, y )) L M (d s, d y ) . H r ( D ) The operator − ∆ on D with vanishing Dirichlet boundary conditions admits a complete orthonor-mal system in L ( D ) of smooth eigenfunctions (Φ j ) j > , with eigenvalues ( λ j ) j > . Then we havethe following properties (see for example [39, Chapter V, p. 343]): X j > (1 + λ j ) r < + ∞ , for any r < − d , (2.47) k Φ j k L ∞ ( D ) C (1 + λ j ) α , for any α > d . (2.48)The Green’s function G D has the representation (1.5) and we have the decomposition(2.49) f ( x ) = X j > a j ( f )Φ j ( x ) , x ∈ D , for every f ∈ L ( D ) where a j ( f ) = h f, Φ j i L ( D ) . For r > , we now define H r ( D ) := ( f ∈ L ( D ) : k f k H r := X j > (1 + λ j ) r a j ( f ) < + ∞ ) , which becomes a Hilbert space with the inner product h f, h i H r := P j > (1 + λ j ) r a j ( f ) a j ( h ) . Wedenote by H − r ( D ) the topological dual space of H r ( D ) , which turns out to be isomorphic to thespace of sequences b = ( b n ) n > such that k b k H − r := X j > (1 + λ j ) − r b j < + ∞ . 23n fact, with b j = ˜ f (Φ j ) for ˜ f ∈ H − r ( D ) , we have k ˜ f k H − r = k b k H − r and the pairing between H − r ( D ) and H r ( D ) is given by h b, h i = X j > b j a j ( h ) k b k H − r k h k H r . We need the following technical lemma, for which we could not find a reference in the literature. Lemma 2.18. For r , the restriction of H r ( R d ) to D is continuously embedded in H r ( D ) .Proof. For m ∈ N , let H m ( D ) = { u : u ( α ) ∈ L ( D ) for all | α | m } , with an upper index m , bethe “usual” Sobolev spaces as in [27, p. 3]. For real r > , let m be the smallest even integer with m > r . Following [27, Chapitre 1, (9.1)], we define, with a superscript index, H r ( D ) := (cid:2) H m ( D ) , L ( D ) (cid:3) − rm , where the right-hand side is the notation of [27, Chapitre 1, Définition 2.1] for interpolation spaces.Furthermore, define H r ( D ) for r > as the closure in H r ( D ) of the set of smooth functions withcompact support in D , see [27, Chapitre 1, (11.1)]. Similarly, as in [20, Definition 8.1], let H rB ( D ) for r > be the closed subspace of H r ( D ) such that its elements are equal to zero on ∂D .By the definition of interpolation spaces, there exists for each θ ∈ [0 , some self-adjointpositive operator Λ in L ( D ) with domain H mB ( D ) such that (cid:2) H mB ( D ) , L ( D ) (cid:3) θ = dom (cid:16) Λ − θ (cid:17) . The notion of domain is as in [27, Chapitre 1, p. 12], and the power in this case is to be understoodas the spectral power of the operator Λ . By [27, Chapitre 1, Remarque 2.3], dom (Λ − θ ) coincideswith dom ( ˜Λ − θ ) for any other self-adjoint positive operator ˜Λ in L ( D ) with domain H mB ( D ) . Inparticular, we can choose ˜Λ = ( − ∆) m , where ∆ is the Dirichlet Laplacian, and the power m , aninteger because m is even, is to be understood as the composition of partial differential operators.Then, from [20, Théorème 8.1], we deduce that (cid:2) H mB ( D ) , L ( D ) (cid:3) θ = H m (1 − θ ) B ( D ) , and with the choice θ = 1 − rm further that(2.50) dom (cid:16) ˜Λ rm (cid:17) = H rB ( D ) . Let f ∈ L ( D ) be as in (2.49). Then, see e.g. [30, (2.12)], we have that ˜Λ rm f = P j > µ rm j a j ( f )Φ j ,where µ j = λ m j is the j th eigenvalue of ˜Λ . The previous sum converges in L ( D ) if and only if P j > λ rj | a j ( f ) | < + ∞ , so together with (2.50), we obtain H r ( D ) = dom (cid:16) ˜Λ rm (cid:17) = H rB ( D ) . Therefore, by [20, Théorème 8.1] and the discussion that follows, we have H r ( D ) ֒ → H r ( D ) .If r , we have by [30, (2.13)] H r ( D ) = ( H r ( D ) if r < ,H ( D ) if r = , where H ( D ) is the Lions–Magenes space satisfying H ( D ) ֒ → H ( D ) by [27, Chapitre 1,Théorème 11.7]. In addition, for any r > , H r ( D ) ֒ → H r ( D ) and thus H r ( D ) ֒ → H r ( D ) .Next, by [27, Chapitre 1, Théorèmes 9.1, 9.2 and (7.1)], there exists a constant C such that anyfunction u ∈ H r ( D ) is the restriction of a function ˜ u ∈ H r ( R d ) to D with k ˜ u k H r ( R d ) C k u k H r ( D ) .Therefore, H r ( D ) ֒ → H r ( D ) ֒ → H r ( R d ) | D for any r > , and by duality, we have H r ( R d ) | D ֒ → H r ( D ) ֒ → H r ( D ) for r . 24 .3.2 Existence of a càdlàg solution in H r ( D ) with r < − d Theorem 2.19. The mild solution u to (1.1) constructed in Proposition 2.17 has a càdlàg modi-fication in H r ( D ) for any r < − d . In contrast to the case D = [0 , π ] , the eigenfunctions of − ∆ on a general domain D in R d maynot be uniformly bounded, see (2.48). Thus, the proof of Theorem 2.5 does not extend to higherdimensions. Instead, we use [17, Theorem 1] to write(2.51) G D ( t ; x, y ) = g ( t, x − y ) + H ( t ; x, y ) , where g is the heat kernel on R d and H is a function such that for any ε > , ( t, x, y ) H ( t ; x, y ) is smooth on [0 , T ] × D × B cε ( ∂D ) , where B ε ( ∂D ) is the ε -neighborhood of ∂D . Away from theboundary ∂D , g can be dealt with as in Theorem 2.15, and H is smooth and therefore easilyhandled. In order to control the behavior close to the boundary, the change of measure technique(see the Appendix and also the proof of Theorem 2.15) is again fruitful. Proof of Theorem 2.19. We may assume that u satisfies (2.46). For ε > , split u ( t, x ) into thesum of three terms: u ε ( t, x ) := Z t Z D g ( t − s, x − y ) σ ( u ( s, y )) y ∈ B cε ( ∂D ) L (d s, d y ) ,u ε ( t, x ) := Z t Z D H ( t − s ; x, y ) σ ( u ( s, y )) y ∈ B cε ( ∂D ) L (d s, d y ) ,u ε ( t, x ) := Z t Z D G D ( t − s ; x, y ) σ ( u ( s, y )) y ∈ B ε ( ∂D ) L (d s, d y ) . By (2.43), the same proof as in Theorem 2.15 for u , and u shows that t u ε ( t, · ) has acàdlàg version in H r,loc ( R d ) for r < − d , where the spatial variable takes values in the whole space R d rather than just D . Thus, as a process with x ∈ D , it has a càdlàg version in H r ( D ) for r < − d by Lemma 2.18. Regarding u ε , since ( t, x, y ) H ( t ; x, y ) is smooth on [0 , T ] × D × B cε ( ∂D ) , we canmimic the part of the proof of Theorem 2.15 concerning u , in order to get that ( t, x ) u ε ( t, x ) isjointly continuous, and in fact, uniformly continuous since D is bounded. In particular, t u ε ( t, · ) is continuous in H r ( D ) for any r . For the last term u ε , we want to show that it converges to in H r ( D ) , uniformly in t ∈ [0 , T ] . As a first step, we have (cid:13)(cid:13) u ε ( t, · ) (cid:13)(cid:13) H r = X k > (1 + λ k ) r ( a εk ( t )) , where a εk ( t ) := R D Φ k ( x ) u ε ( t, x ) d x . As in (2.24) and (2.39), one can then show that | a εk ( t ) | C sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t Z D Φ k ( y ) σ ( u ( s, y )) y ∈ B ε ( ∂D ) L (d s, d y ) (cid:12)(cid:12)(cid:12)(cid:12) . Next, let Φ( x ) := (cid:0) P k > (1 + λ k ) r Φ k ( x ) (cid:1) , which belongs to L ( D ) by (2.47) since k Φ k k L ( D ) = 1 and r < − d . Hence, assuming without loss of generality that p in (H’) satisfies p < d ,we have by Lemma A.2: k Φ σ ( u ) k L,p k Φ σ ( u ) k L B ,p + k Φ σ ( u ) k L M ,p | b | (cid:13)(cid:13)(cid:13)(cid:13)Z T Z D | Φ( x ) σ ( u ( t, x )) | d t d x (cid:13)(cid:13)(cid:13)(cid:13) L p + C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) Z T Z D Z | z | | Φ( x ) σ ( u ( t, x )) z | J (d t, d x, d z ) (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p Z T Z D Φ( x ) k σ ( u ( t, x )) k L p d t d x + C (cid:18)Z T Z D Φ p ( x ) E [ | σ ( u ( t, x )) | p ] d t d x (cid:19) p C ( k Φ k L ( D ) + k Φ k L p ( D ) ) < + ∞ . Thus, by Theorem A.4, there exists an equivalent probability measure Q such that(2.52) k Φ σ ( u ) k L, , Q < + ∞ . Furthermore, the Doob–Meyer decomposition of L under Q is given by L = L B, Q + L M, Q , where L B, Q (d t, d x ) = b Q ( t, x ) d t d x = b + Z | z | ( Y ( t, x, z ) − ν (d z ) ! d t d x ,L M, Q (d t, d x ) = Z | z | z ˜ J Q (d t, d x, d z ) , ˜ J Q (d t, d x, d z ) = J (d t, d x, d z ) − Y ( t, x, z ) d t d x ν (d z ) with some predictable random function Y , see [13, Theorem 3.6]. By [4, Theorem 4.14], we deducethat k Φ σ ( u ) k L B, Q , , Q < + ∞ and k Φ σ ( u ) k L M, Q , , Q < + ∞ . As a consequence,(2.53) E Q "(cid:18)Z T Z D Φ( x ) | σ ( u ( t, x )) b Q ( t, x ) | d t d x (cid:19) < + ∞ and(2.54) E Q "Z T Z D Z | z | (Φ( x ) σ ( u ( t, x )) Y ( t, x, z )) d t d x ν (d z ) < + ∞ . We obtain E Q h(cid:13)(cid:13) u ε ( t, · ) (cid:13)(cid:13) H r i = X k > (1 + λ k ) r E Q [( a εk ( t )) ] C X k > (1 + λ k ) r k Φ k σ ( u ) B ε ( ∂D ) k L, , Q C X k > (1 + λ k ) r (cid:16) k Φ k σ ( u ) B ε ( ∂D ) k L B, Q , , Q + k Φ k σ ( u ) B ε ( ∂D ) k L M, Q , , Q (cid:17) . For the first term in the parenthesis, we use the Cauchy–Schwarz inequality to obtain X k > (1 + λ k ) r k Φ k σ ( u ) B ε ( ∂D ) k L B, Q , , Q = X k > (1 + λ k ) r E Q "(cid:18)Z T Z D | Φ k ( x ) σ ( u ( t, x )) x ∈ B ε ( ∂D ) b Q ( t, x ) | d t d x (cid:19) = E Q (cid:20) X k > (1 + λ k ) r Z T Z D Z T Z D | Φ k ( x )Φ k ( y ) σ ( u ( t, x )) σ ( u ( s, y )) x,y ∈ B ε ( ∂D ) × b Q ( t, x ) b Q ( s, y ) | d t d x d s d y (cid:21) E Q (cid:20)Z T Z D Z T Z D | Φ( x )Φ( y ) σ ( u ( t, x )) σ ( u ( s, y )) x,y ∈ B ε ( ∂D ) b Q ( t, x ) b Q ( s, y ) | d t d x d s d y (cid:21) = E Q "(cid:18)Z T Z D | Φ( x ) σ ( u ( t, x )) x ∈ B ε ( ∂D ) b Q ( t, x ) | d t d x (cid:19) → ε → by (2.53) and dominated convergence. Similarly, (2.54) implies that X k > (1 + λ k ) r k Φ k σ ( u ) B ε ( ∂D ) k L M, Q , , Q C X k > (1 + λ k ) r E Q "Z T Z D Z | z | (Φ k ( x ) σ ( u ( t, x )) Y ( t, x, z )) x ∈ B ε ( ∂D ) d t d x ν (d z ) = C E Q "Z T Z D Z | z | (Φ( x ) σ ( u ( t, x )) Y ( t, x, z )) x ∈ B ε ( ∂D ) d t d x ν (d z ) → as ε → . Altogether, there exists a subsequence of u ε ( t, · ) that converges almost surely to in H r ( D ) for r < − d , uniformly in t ∈ [0 , T ] , which completes the proof. In Section 2, we have established the existence of a version such that t u ( t, · ) has càdlàgpaths in (local) fractional Sobolev spaces. The goal of the current section is to investigate thepartial regularity of the solution, that is, the behavior of the partial functions t u ( t, x ) forfixed x ∈ D and x u ( t, x ) for fixed t ∈ [0 , T ] . In the case where the Lévy noise L has locallyfinite intensity (so L is a compound Poisson noise), it is clear that almost surely, no jump will fallonto a fixed t - or x -section of the solution. Because the Green’s function G D ( t ; x, y ) is smoothoutside { } × { ( x, x ) : x ∈ D } , the partial functions are continuous, and even smooth, in thiscase. However, a general Lévy noise can have infinitely many jumps on any compact subset of [0 , T ] × D , which may even fail to be summable. Still they never lie on a fixed section, but maycome arbitrarily close to it, so its regularity is unclear a priori. As we shall show, the answercritically depends on the Blumenthal–Getoor index of the noise (that is, the smallest p for which R [ − , | z | p ν (d z ) is finite), and both continuous and locally unbounded sample paths may arise.Throughout this section, we consider the stochastic heat equation (1.1) on a bounded C ∞ -regular domain or D = R d , with some Lipschitz continuous σ : R → R and some bounded con-tinuous u : ¯ D → R that is zero on ∂D . Furthermore, let L be a pure-jump Lévy white noise asin (1.6) and u be the mild solution constructed under the hypotheses in Propositions 2.1, 2.7 or2.17, respectively. In particular, if D = R d , we are given p, q > such that (H) is satisfied; andif D is a bounded domain, there exists p > such that (H’) holds. Theorem 3.1. In the setting described above, assume that p < d . Then, for any t ∈ [0 , T ] , theprocess x u ( t, x ) has a continuous modification.Proof. The solution u is the stationary limit of the mild solution u N to the truncated equationdefined in (2.7), (2.28) or (2.45) with noise L N given in (2.6), (2.27) or (2.44), respectively.Therefore, we can suppose that u = u N for some N > , and for simplicity, we only consider N = 1 . We prove the claim using different approaches depending on the value of p . < p < : Notice that < p < can only occur in d = 1 because of the hypothesis p < d .However, we keep the exponent d since we will use similar ideas in the next case. Let A > besuch that x ∈ ( − A, A ) d , and split u into seven parts according to (2.32) and (2.41), with obviouschanges when D is bounded. In this case, we further assume that A > is large enough such that D ⊂ ( − A, A ) d . Clearly, V is jointly continuous, and as shown in the proof of Theorem 2.15, thesame is true for u , [ − A,A ] d , u , [ − A,A ] d and u , in the case D = R d , while they are zero if D is bounded. Furthermore, almost surely, u , consists of finitely many jumps, none of which occur27t time t , so x u , ( t, x ) is smooth because the Green’s function G D ( t ; x, y ) is so for t > . Itremains to consider u , + u , , for which we apply the Kolmogorov continuity criterion. We have E [ | ( u , + u , )( t, x ) − ( u , + u , )( t, z ) | p ]= E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b Z t Z [ − A, A ] d ( G D ( t − s ; x, y ) − G D ( t − s ; z, y )) σ ( u ( s, y )) d s d y + Z t Z [ − A, A ] d ( G D ( t − s ; x, y ) − G D ( t − s ; z, y )) σ ( u ( s, y )) L M (d s, d y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p C E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b Z t Z [ − A, A ] d ( G D ( t − s ; x, y ) − G D ( t − s ; z, y )) σ ( u ( s, y )) d s d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p + E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z [ − A, A ] d ( G D ( t − s ; x, y ) − G D ( t − s ; z, y )) σ ( u ( s, y )) L M (d s, d y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p for any x, z ∈ D . Then, using Hölder’s inequality and (2.3) or (2.29), E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b Z t Z [ − A, A ] d ( G D ( t − s ; x, y ) − G D ( t − s ; z, y )) σ ( u ( s, y )) d s d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p Z t Z [ − A, A ] d | G D ( t − s ; x, y ) − G D ( t − s ; z, y ) | E [ | σ ( u ( s, y )) | p ] d s d y ! × Z t Z [ − A, A ] d | G D ( t − s ; x, y ) − G D ( t − s ; z, y ) | d s d y ! p − C (cid:18)Z t Z D | G D ( t − s ; x, y ) − G D ( t − s ; z, y ) | d s d y (cid:19) p . For D = R , the last term is bounded by C | x − z | p , see [34, Lemme A2]; for D = [0 , π ] , we cantake the power p inside the integral by Hölder’s inequality, and further assume that < p < (by(2.3) there is no harm in taking a larger value of p on bounded domains). Then the upper boundbecomes C | x − z | − p by [3, Lemma B.1(a)]. For the martingale part, we have by [28, Theorem 1]and (2.29), E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z [ − A, A ] d ( G D ( t − s ; x, y ) − G D ( t − s ; z, y )) σ ( u ( s, y )) L M (d s, d y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p C Z t Z [ − A, A ] d Z | z | | z | p | G D ( t − s ; x, y ) − G D ( t − s ; z, y ) | p E [ | σ ( u ( s, y )) | p ] d s d y ν (d z ) C Z t Z D | G D ( t − s ; x, y ) − G D ( t − s ; z, y ) | p d s d y . If D = R d , then according to [34, Lemme A2], this is bounded by C | x − z | p , if p < ,C | x − z | log ( | x − z | ) if p = ,C | x − z | − p if p > ; and if D = [0 , π ] , and we take < p < , then the upper bound we obtain is again C | x − z | − p by [3,Lemma B.1(a)]. Thus, the Kolmogorov continuity criterion (see e.g. [39, Chapter 1, Corollary 1.2])ensures the existence of a continuous modification of u , + u , in the space variable x .28 < p : We use the same decomposition of u as above, except that we replace b by b and L M (d t, d x ) by R | z | z J (d t, d x, d z ) . The proofs for V , u , and u , are not affected by thischange, and up to the modification indicated at the end of the proof of Theorem 2.15, the prooffor u , is not affected, either. Since R | z | | z | ν (d z ) < + ∞ , the small jumps of L are summable and u , is actually a sum of possibly infinitely many terms, each of which is continuous in x becauseno jump occurs exactly at time t . Furthermore, by [16, Corollary 3.2.8], we have for x ∈ D ,(3.1) E " Z t Z [ − A, A ] d Z | z | sup x : | x − x | G D ( t − s ; x, y ) | zσ ( u ( s, y )) | J (d s, d y, d z ) ! p Z t Z [ − A, A ] d Z | z | | z | p sup x : | x − x | G pD ( t − s ; x, y ) E [ | σ ( u ( s, y )) | p ] d s d y ν (d z ) C Z t Z R d sup x : | x − x | g p ( t − s, x − y ) d s d y = C Z t Z | y − x | g p ( s, 0) d s d y + Z t Z | y − x | > (4 πs ) − pd e − p ( | y − x |− s d s d y ! , which is finite because p < d . So the sum defining u , converges locally uniformly in x , whichimplies that x u , ( t, x ) is continuous almost surely. Recalling that b = 0 for p < and D = R d , u , and u , are non-zero in this case only if p = 1 . Then u , ( t, x ) is jointly continuousin ( t, x ) , as shown in the proof of Theorem 2.15. If D is bounded, u , is zero. For u , we usethe approximation sequence u , n defined after (2.41). For each n , the process x u , n ( t, x ) iscontinuous because | u , n ( t, x ) − u , n ( t, x ′ ) | C Z T Z D | G D ( t − s ; x, y ) − G D ( t − s ; x ′ , y ) | d s d y → as x ′ → x , see [34, Lemme A2] for D = R d and the proof of [37, Proposition 5] for bounded D .Hence, it suffices to prove that for fixed t , u , n ( t, x ) converges to u , ( t, x ) , locally uniformly in x .By dominated convergence, this can be reduced to showing Z t Z [ − A, A ] d sup x : | x − x | G D ( t − s ; x, y ) | σ ( u ( s, y )) | d s d y < + ∞ a.s.But this follows from (3.1) together with (2.3), (2.29) and (2.43), respectively, by taking expecta-tion. Remark 3.2. In particular, any (tempered) α -stable noise with α ∈ (0 , d ) (and b = 0 if D = R d and α < ) satisfies the hypothesis of Proposition 3.1. The same holds for (variance-)gammanoises for all d > , inverse Gaussian noises for d = 1 , , and normal inverse Gaussian noisesfor d = 1 (cf. [14]). The next theorem shows that the value d in the previous theorem is essentially optimal. Theorem 3.3. Let σ = 1 and suppose there is δ > such that the Lévy measure of L satisfies (3.2) ν (d z ) = f ( z ) | z | α +1 d z for z ∈ [ − δ, δ ] , where α ∈ [ d , (1 + d ) ∧ and f : [ − δ, δ ] → [0 , + ∞ ) is measurable with f (0) = 0 and (3.3) Z δ − δ | f ( z ) − f (0) || z | α +1 | z | r d z < + ∞ or some < r < d . Then for fixed t ∈ [0 , T ] , the path x u ( t, x ) is unbounded on any non-emptyopen subset of D with probability .Proof. Fix t ∈ [0 , T ] . Since V is continuous in ( t, x ) , it suffices to show the unboundedness of Y ( x ) = Z t Z D G D ( t − s ; x, y ) L (d s, d y ) , x ∈ D . We start with the case where f is constant, that is, L is an α -stable noise. Then ( Y ( x ) : x ∈ D ) is an α -stable process given in the form of [35, (10.1.1)] with E = [0 , T ] × D and control measure d s d y . We shall check that the necessary condition [35, (10.2.14)] for sample path boundedness in[35, Theorem 10.2.3] is not satisfied, in particular that for x ∈ D , and δ such that B x ( δ ) ⊂ D ,(3.4) Z t Z D sup x ∈ B x ( δ ) G D ( t − s, x, y ) ! α d s d y = + ∞ . Indeed, by [38, Theorem 2 and Lemma 9], for any x, y ∈ B x ( δ ) ,(3.5) G D ( t − s, x, y ) > Cg ( t − s, x − y ) , which implies that Z t Z D sup x ∈ B x ( δ ) G αD ( t − s ; x, y ) d s d y > C Z t Z B x ( δ ) π ( t − s )) αd d s d y = + ∞ , and (3.4) is proved. In the case of general f , we write ν (d z ) = ν (d z ) − ν (d z ) + ν (d z ) + ν (d z )= (cid:18) ( f ( z ) − f (0)) + | z | α +1 − ( f ( z ) − f (0)) − | z | α +1 + f (0) | z | α +1 (cid:19) z ∈ [ − δ,δ ] d z + z ∈ [ − δ,δ ] c ν (d z ) , and decompose L accordingly into L − L + L + L such that for i , L i has Lévy measure ν i and is independent of the other three parts. If u i solves the additive heat equation with drivingnoise L i , then by (3.3) and Theorem 3.1, for any fixed t ∈ [0 , T ] , x u ( t, x ) , x u ( t, x ) and x u ( t, x ) each has a continuous version. And since the first part of the proof shows that x u ( t, x ) is unbounded on any open subset of D , the same property holds for x u ( t, x ) . Remark 3.4. Taking f ≡ , Theorem 3.3 shows that for the solution u to the heat equation withan additive α -stable noise where d α < d (since α < , this can only occur for d > ), forany t ∈ [0 , T ] , x u ( t, x ) is unbounded on any non-empty open subset of D . The same holds trueif ν has the form (3.2) with α in the same range and some f that is ( α − d + ε ) -Hölder continuousat . For d > , this includes tempered stable Lévy noise (with stability index in the indicatedrange) and normal inverse Gaussian noises. Theorem 3.5. In the set-up described at the beginning of Section 3, assume that < p < . Thenfor any x ∈ D , the process t u ( t, x ) has a continuous modification. We need the following elementary lemma. Lemma 3.6. If g is the heat kernel (1.4) , then there exists C > such that for every T > , sup t ∈ [0 ,T ] g ( t, x ) = CT − d e − | x | T , if T < | x | d ,C | x | − d , if T > | x | d . roof of Theorem 3.5. Again, by a stopping time argument, it suffices to show the regularity of u N for any N > , as defined in (2.7), (2.28) or (2.45), respectively. We only consider N = 1 ,and decompose u = V + u , + u , + u , + u , + u as in the part “ < p ” of the proof ofTheorem 3.1 (with A > such that x ∈ ( − A, A ) d and D ⊂ ( − A, A ) d if D is bounded). Therewe explained that V , u , [ − A,A ] d and u , [ − A,A ] d are jointly continuous. The term u , is againa finite sum of weighted heat kernels, so t u ( t, x ) is smooth because none of the jumps fallsexactly on a given x ∈ D . Moreover, for u , , we can use [34, Théorème 2.2.2] in the case D = R d because E [ | σ ( u ( t, x )) | p ] = E [ | σ ( u ( t, x )) | p ] is uniformly bounded on [0 , T ] × [ − A, A ] d . The proofalso applies to bounded D because the heat kernel G D is majorized by a multiple of the heatkernel on R d by [16, Corollary 3.2.8]. Finally, since p < and b = 0 if D = R d , u only needsto be considered for bounded D . With < r < d and t ′ < t T , we can use Hölder’sinequality, the moment bound (2.3) or (2.43) and [37, Proposition 5] (and the proof therein) todeduce for every γ ∈ (0 , , E [ | u ( t, x ) − u ( t ′ , x ) | r ] (cid:18)Z t Z D | G D ( t − s ; x, y ) − G D ( t ′ − s ; x, y ) | d s d y (cid:19) r − × Z t Z D | G D ( t − s ; x, y ) − G D ( t ′ − s ; x, y ) | E [ | σ ( u ( s, y )) | r ] d s d y (cid:18)Z t Z D | G D ( t − s ; x, y ) − G D ( t ′ − s ; x, y ) | d s d y (cid:19) r C | t − t ′ | γr Thus, by choosing γ close to , the claim follows from Kolmogorov’s continuity theorem. Theorem 3.7. If σ ≡ and ν satisfies (3.2) for some α ∈ [1 , d ) , δ > , and some measurable f : [ − δ, δ ] → [0 , + ∞ ) with f (0) = 0 and (3.3) for some < r < , then for any x ∈ D , the process t u ( t, x ) is unbounded on any non-empty open interval in [0 , T ] with probability .Proof. The proof is analogous to the proof of Theorem 3.3. It suffices to show that(3.6) Z t t Z D sup t ∈ [ t ,t ] G D ( t − s ; x, y ) ! α d s d y = + ∞ for x ∈ D and t < t T . By [38, Theorem 2 and Lemma 9], it suffices to consider D = R d .In this case, the integral above is bounded from below by Z t t Z R d sup t ∈ [ t ,t ] g ( t − s, x − y ) α d s d y > Z t − t Z R d sup v ∈ [0 ,s ] g ( v, x − y ) α d s d y , so (3.6) follows from Lemma 3.6: Z t t Z R d sup t ∈ [ t ,t ] g ( t − s, x − y ) ! α d s d y > Z t − t Z | x − y | √ ds C | x − y | dα d s d y = + ∞ . Remark 3.8. In contrast to the results on regularity in space, the critical exponent p = 1 fortemporal regularity does not depend on the dimension d . In particular, for all d > , we obtaincontinuity of t u ( t, x ) , x fixed, for any (tempered) α -stable noise with α ∈ (0 , (and b = 0 if D = R d ), any (variance-)gamma noise and inverse Gaussian noise. In the case of additive noise, t u ( t, x ) is almost surely unbounded on any non-empty open subinterval for (tempered) α -stablenoises with α ∈ [1 , d ) and normal inverse Gaussian noises. Appendix: Integration with respect to random measures Just as Lévy processes are special instances of semimartingales, a Lévy noise as in (1.6) is a randommeasure as introduced in [4]. We give a short introduction into the integration theory associated toit, hereby concentrating on the main ideas and results that we need for our purposes. All detailsnot mentioned or explained can be found in [4, 13, 25, 26]. Given a filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) satisfying the usual conditions, consider a Polish space E (e.g., E = D , thespatial domain in (1.1)), and denote by P the σ -field P ⊗ B ( E ) , where P is the usual predictable σ -field. With an abuse of terminology, P -measurable mappings from ˜Ω := Ω × [0 , T ] × E to R areagain called predictable and their collection again denoted by P .Given a sequence ( ˜Ω k ) k > in P satisfying ˜Ω k ↑ ˜Ω , a mapping M : P M := S k > P| ˜Ω k → L p where p ∈ [0 , + ∞ ) is called an L p -random measure if for every sequence ( A i ) i > of pairwisedisjoint sets in P M with S i > A i ∈ P M , we have M ( S i > A i ) = P i > M ( A i ) in L p , and someadditional “ ( F t ) t ∈ [0 ,T ] -adaptedness” conditions are satisfied. In our example of a Lévy noise on [0 , T ] × D , we can take ˜Ω k = D if D is bounded, and ˜Ω k = [ − k, k ] d if D = R d .The stochastic integral of a simple integrand of the form S = P ri =1 a i A i , where r ∈ N , a i ∈ R and A i ∈ P M , is defined in the canonical way by Z T Z E S ( t, x ) M (d t, d x ) := r X i =1 a i M ( A i ) . Denoting by S M the collection of such simple integrands, the extension of the integral to a largersubset of P is carried out using the Daniell mean (A.1) k H k M,p := k H k M,p, P := sup S ∈S M , | S | | H | (cid:13)(cid:13)(cid:13)(cid:13)Z T Z E S ( t, x ) M (d t, d x ) (cid:13)(cid:13)(cid:13)(cid:13) L p , H ∈ P . A predictable process H is called p -integrable with respect to M if there exists a sequence ( S n ) n > of simple integrands with k H − S n k M,p → as n → + ∞ . The collection of p -integrable processes isdenoted by L ,p ( M ) (or L ,p ( M, P ) if we want to emphasize the probability measure). The stochas-tic integral of H with respect to M is then defined as the L p -limit of R T R E S n ( t, x ) M (d t, d x ) (which exists and does not depend on the choice of S n ). In all notions introduced, the prefix p issuppressed if p = 0 . The constructed integral obeys the dominated convergence theorem, see [4,(2.6)]. Theorem A.1. If ( H n ) n > are predictable and converge pointwise to H , and | H n | H for all n > and some H ∈ L ,p ( M ) , then H n , H ∈ L ,p ( M ) and k H − H n k M,p → as n → + ∞ . In this paper, we are particularly interested in the case where M is a linear combination ofrandom measures of one of the following forms:(a) M is a predictable strict random measure , that is, almost every realization of M is a measureon [0 , T ] × E and t R T R E A ( s, y ) [0 ,t ] ( s ) M (d s, d y ) is a predictable process for all A ∈ P M .(b) M (d t, d x ) = R z ∈ R W ( t, x, z ) ˜ J (d t, d x, d z ) , where J is an ( F t ) t ∈ [0 ,T ] -Poisson random measurewith intensity measure ν (d t, d x, d z ) , ˜ J = J − ν , and W ˜Ω k is -integrable with respect to ˜ J (arandom measure on E × R ) in the sense above for every k > .(c) M is a strict random measure of the form M (d t, d x ) = R z ∈ R W ( t, x, z ) J (d t, d x, d z ) where W ˜Ω k is integrable with respect to J for every k > .In these cases, the Daniell mean can be computed (or estimated) explicitly.32 emma A.2. Let k X k L p = E [ | X | p ] for < p < and k X k L p = ( E [ | X | p ]) p be the usual L p -normfor p > .1. In the case (a) above, we have for every < p < + ∞ and H ∈ P , (A.2) k H k M,p = (cid:13)(cid:13)(cid:13)(cid:13)Z T Z E | H ( t, x ) | | M | (d t, d x ) (cid:13)(cid:13)(cid:13)(cid:13) L p , where | M | is the total variation measure of M .2. In the case (b) above, there exist for every p > constants c p , C p > such that for all H ∈ P , c p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z T Z E H ( t, x ) [ M ](d t, d x ) (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p k H k M,p C p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z T Z E H ( t, x ) [ M ](d t, d x ) (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p , where [ M ](d t, d x ) = R z ∈ R W ( t, x, z ) J (d t, d x, d z ) is the quadratic variation measure of M . Inparticular, (A.3) k H k M,p C p (cid:18)Z T Z E Z R k H ( t, x ) W ( t, x, z ) k pL p ν (d t, d x, d z ) (cid:19) p . 3. In the case (c) above, we have for < p and H ∈ P , k H k M,p Z T Z E Z R k H ( t, x ) W ( t, x, z ) k L p ν (d t, d x, d z ) . Proof. For the first statement, the “ ”-part follows from (cid:12)(cid:12)(cid:12)(cid:12)Z T Z E S ( t, x ) M (d t, d x ) (cid:12)(cid:12)(cid:12)(cid:12) Z T Z E | S ( t, x ) | | M | (d t, d x ) Z T Z E | H ( t, x ) | | M | (d t, d x ) for all S with | S | | H | . For the “ > ”-part, observe that the right-hand side of (A.2) equals k H k | M | ,p by dominated convergence. Next, consider the measure µ (d ω, d t, d x ) = M ( ω, d t, d x ) P (d ω ) on P and let D ( ω, t, x ) be its Radon–Nikodym derivative with respect to | µ | (d ω, d t, d x ) . Then D ispredictable, | D | ≡ and | M | ( ω, d t, d x ) = D ( ω, t, x ) M ( ω, d t, d x ) . Hence, sup S ∈S M , | S | | H | (cid:13)(cid:13)(cid:13)(cid:13)Z T Z E S ( t, x ) | M | (d t, d x ) (cid:13)(cid:13)(cid:13)(cid:13) L p = sup S ∈S M , S | H | (cid:13)(cid:13)(cid:13)(cid:13)Z T Z E S ( t, x ) | M | (d t, d x ) (cid:13)(cid:13)(cid:13)(cid:13) L p = sup S ∈S M , S | H | (cid:13)(cid:13)(cid:13)(cid:13)Z T Z E S ( t, x ) D ( t, x ) M (d t, d x ) (cid:13)(cid:13)(cid:13)(cid:13) L p sup S ∈S M , S | H | (cid:13)(cid:13)(cid:13)(cid:13)Z T Z E S ( t, x ) M (d t, d x ) (cid:13)(cid:13)(cid:13)(cid:13) L p k H k M,p , and (A.2) is provedFor the second statement, we observe that t R T R E S ( s, y ) [0 ,t ] ( s ) M (d s, d y ) is a local mar-tingale for all S ∈ S M , see [4, Proposition 4.9(b)]. So the statement for S ∈ S M follows from theBurkholder–Davis–Gundy inequalities. The general case is again a consequence of the dominatedconvergence theorem. Inequality (A.3) can be proved along the lines of the third statement, whichwe now establish. 33et ( T i , X i , Z i ) i > be the points of J in [0 , T ] × E × R . Then, using ( x + y ) p x p + y p for x, y > and < p , we get E "(cid:12)(cid:12)(cid:12)(cid:12)Z T Z E S ( t, x ) M (d t, d x ) (cid:12)(cid:12)(cid:12)(cid:12) p = E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i > S ( T i , X i ) W ( T i , X i , Z i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p E "X i > | S ( T i , X i ) W ( T i , X i , Z i ) | p E (cid:20)Z T Z E Z R | H ( t, x ) W ( t, x, z ) | p J (d t, d x, d z ) (cid:21) = E (cid:20)Z T Z E Z R | H ( t, x ) W ( t, x, z ) | p ν (d t, d x, d z ) (cid:21) , and the proof is complete.With the help of the Daniell mean, one can obtain the following stochastic Fubini theorem. Theorem A.3. Let ( A, A , µ ) be a σ -finite measure space, M be an L p -random measure for some p > , and H be a P ⊗ A -measurable function.1. If p > and R A k H ( · , · , a ) k M,p µ (d a ) < + ∞ , then Z A (cid:18)Z T Z E H ( t, x, a ) M (d t, d x ) (cid:19) µ (d a ) and Z T Z E (cid:18)Z A H ( t, x, a ) µ (d a ) (cid:19) M (d t, d x ) are equal almost surely, and all integrals involved are well defined.2. If < p and M is a random measure as in (c) above, the conclusion of the first partcontinues to hold if (A.4) Z T Z E Z R (cid:13)(cid:13)(cid:13)(cid:13)Z A | H ( t, x, a ) | µ (d a ) | W ( t, x, z ) | (cid:13)(cid:13)(cid:13)(cid:13) L p ν (d t, d x, d z ) < + ∞ . The first part has been proved in [26, Theorem 2] for p = 1 (see also [5] for processes indexed onlyby time), but is obviously also valid for p > by the monotonicity of L p -norms. In particular,Lemma A.2 can be used to verify the integrability assumption. The second part follows from theordinary Fubini theorem ( M is a strict random measure here) together with an argument as inthe proof of third part of Lemma A.2.A last result that we need relates to the possibility of recovering L -integrability from L p -integrability, p < , upon an equivalent change of probability measure. For semimartingales,this result is well known, see [33, Chapter IV, Theorem 34], for example. For random measures,it is proved in [25, Corollary of Theorem 2]. Theorem A.4. If M is an L p -random measure and H ∈ L ,p ( M ) for some p < , then thereexists a probability measure Q that is equivalent to P on F such that d Q d P is bounded, d P d Q ∈ L p − p ( P ) , M is an L -random measure under Q , and H ∈ L , ( M, Q ) . Acknowledgements We thank two anonymous referees for their careful reading of our manuscript.34 eferences [1] Balan, R. M. SPDEs with α -stable Lévy noise: A random field approach. Int. J. Stoch. Anal. ,2014. 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