Stochastic NonLinear Wave Equations in Local Sobolev Spaces

Existence of weak solutions of stochastic wave equations with nonlinearities of a critical growth driven by spatially homogeneous Wiener processes is established in local Sobolev spaces and local energy estimates for these solutions are proved. A new method to construct weak solutions is employed.


Introduction
Nonlinear wave equations Let us survey the available results in the most important case of a wave equation with polynomial nonlinearities u t t = ∆u − u|u| p−1 + |u| qẆ , u(0) = u 0 , u t (0) = v 0 (1. 2) according to particular ranges of the exponents p, q ∈ (0, ∞): It is known that global weak solutions (weak both in the probabilistic and in the PDE sense) exist provided that (u 0 , v 0 ) is an 0 -measurable [W 1,2 ( d ) ∩ L p+1 ( d )] ⊕ L 2 ( d )-valued random variable, W is a spatially homogeneous Wiener process with bounded spectral correlation Γ (i.e. µ = (2π) d 2 Γ must be a finite measure) and Under (1.3), paths of (u, u t ) take values in [W 1,2 ( d ) ∩ L p+1 ( d )] ⊕ L 2 ( d ) and are weakly continuous in W 1,2 ( d ) ⊕ L 2 ( d ) (see [29]). In the critical case q = p+1 2 , existence of solutions was shown if d ∈ {1, 2} or d ≥ 3 and p ≤ d d−2 (see [30]). Pathwise uniqueness and pathwise norm continuity of solutions in W 1,2 ( d ) ⊕ L 2 ( d ) are known to hold if d ∈ {1, 2} or if d ≥ 3 and p ≤ d+2 d−2 , q ≤ d+1 d−2 , irrespective of (1.3). These results were proved in [29], [30], [31] and [32] in a more general setting (that is, for more general non-linearities). In case d d−2 < p ≤ d+2 d−2 or d d−2 < q ≤ d+1 d−2 some small additional assumptions are needed. In the subcritical case p < d+2 d−2 , q < d+1 d−2 , these results correspond to the present state of art for the deterministic Cauchy problem on d (see [19], [41], [43] and [44]) exactly, whereas there are still some open problems in the (stochastic) critical case p = d+2 d−2 , q = d+1 d−2 (see the discussion in [31]). The aim of the present paper is fivefold. We want to prove 1. existence of weak solutions up to the critical case q = Let us briefly comment on the issues. Ad (1): As mentioned above, existence of weak solutions in the critical case q = p+1 2 is known to hold only in particular cases depending on the dimension d. We will prove that no additional assumption is, in fact, necessary.
Ad (2): To our knowledge, stochastic equations with polynomially growing non-linearities have not been studied in local spaces yet despite it is well known that solutions of wave equations propagate at finite speed and the commonly used restriction to global spaces is therefore unimportant. Nota bene, existence of solutions in global spaces follows trivially from existence of solutions in local spaces by the energy estimate in Theorem 5.2, as demonstrated e.g. in Example 5.5.
As a consequence of the "local" approach to the wave equation (1.1), the second order differential operator in (1.1) need not be uniformly elliptic (as is usually assumed) and mere ellipticity of is sufficient (see (2.1)). In particular, may even decay or explode at infinity, cf. Example 5.4 and 5.5.
The localization of the wave problem is interesting by itself, though it is not very difficult to establish. The main importance of the local approach to the wave equation dwells in our primary interest to prove the subtle existence result in the critical case. We remark at this point that attempts to prove existence of solutions of (1.1) in the critical case while studying the wave equation in global spaces failed (see [29]).
Ad (3): Energy inequalities are a sort of a twin result to any existence theorem in the theory of wave equations as the solutions of the wave equation are, in fact, stationary points of certain Lagrangians and the energy functionals represent their conservation laws (see e.g. Chapter 2 in [42]). On the other hand, energy inequalities also describe basic behaviour of the solution such as the finite speed of propagation mentioned above, long time behaviour of the paths or the conditional dependence on the initial condition (see Theorem 5.2).
Ad (4) & (5): We are not aware of existence results for stochastic wave equations with non-linearities depending on first derivatives of the solution (the velocity and the spatial gradient). This issue is closely related to the fact that we aim at studying systems of stochastic wave equations (1.1). Such generality is not very substantial for the present paper, however, the corresponding results are essential in the newly started research in the field of stochastic wave equations in Riemannian manifolds with possible applications in physical theories and models such as harmonic gauges in general relativity, non-linear σ-models in particle systems, electro-vacuum Einstein equations or Yang-Mills field theory. These models require the target space of the solutions to be a Riemannian manifold (see [18], [42] for deterministic systems and [3], [4] for stochastic ones). For instance, if the unit sphere n−1 is the considered Riemannian manifold, the stochastic geometric wave equation has the form u t t = ∆u + (|∇ x u| 2 − |u t | 2 )u + g(u, u t , ∇ x u)Ẇ , |u| n = 1 where g(p, ·, ·) ∈ T p n−1 , p ∈ n−1 , see e.g. [4]. We do not cover these particular equations here but the present paper is partly intended as a preparation for further applications and as a citation/reference paper for a companion paper on stochastic wave equations in compact Riemannian homogeneous space by Z. Brzeźniak and the author.
Finally, we remark that our proof of the main theorem is based on a new general method of constructing weak solutions of SPDEs, that does not rely on any kind of martingale representation theorem and that might be of interest itself. First applications were done already in [4] and, in the finite-dimensional case, also in [20].
The author wishes to thank Jan Seidler and the referee for a kind helping on the redaction of the paper.

Notation and Conventions
We consider complete filtrations in this paper. We say that a filtration ( t ) on a probability space (Ω, , ) is complete provided that 0 contains all -negligible sets of . We denote by • + the set of all nonnegative real numbers, i.e. + = [0, ∞), • (X ) the Borel σ-algebra on a topological space X , • B R the open ball in d with center at the origin and of radius R, • L p = L p ( d , n ), W k,p = W k,p ( d , n ), • is the Schwartz spaces of complex rapidly decreasing C ∞ -functions on d ( [40]), • is the space of tempered distributions on , i.e. the real dual space to , • ξ → ξ the Fourier transformation on , • C k b the space of k-times continuously differentiable functions on d with bounded derivatives up to order k equipped with the supremum norm of all derivatives up to order k, where X is a Banach space, • a second order elliptic operator where a(x) is a symmetric, strictly positive, (d × d)-real-matrix for every x ∈ d and a is a continuous, W see also Remark 2.1.
• π R various restriction maps to the ball B R , for example the space of continuous linear operators from a topological vector space X to a topological vector space Y and we equip it with the strong σ-algebra, i.e. the σ-algebra generated by the family of maps (X , Y ) B → B x ∈ Y , x ∈ X . If X and Y are Banach spaces then (X , Y ) is equipped with the usual operator norm, • 2 (X , Y ) the space of Hilbert-Schmidt operators from a Hilbert space X to a Hilbert space Y and is equipped with the strong σ-algebra, i.e. the σ-algebra generated by the family of maps 2 • C( + , Z) the space of continuous functions from + to a metric space (Z, ρ) and we equip it with the metric If, in addition, Z is a vector space, C 0 ( + ; Z) = {h ∈ C( + ; Z) : h(0) = 0}, • C w ( + ; X ) the space of weakly continuous functions from + to a locally convex space X and we equip it with the locally convex topology generated by the a family · m,ϕ of pseudonorms defined by a m,ϕ = sup • ζ a symmetric C ∞ -density on d supported in the unit ball and we define • capital bold scripts the conic energy functions, i.e. if a measurable function F : d × n → + , x ∈ d , λ > 0 and T > 0 are given then Remark 2.1. The condition (2.1) is equivalent with the following: given

Spatially homogeneous Wiener process
Given a stochastic basis (Ω, , ( t ) t≥0 , ), an -valued process W = W t t≥0 is called a spatially homogeneous Wiener process with a spectral measure µ that we assume to be positive, symmetric and to satisfy µ( d ) < ∞ throughout the paper, provided that • W ϕ := W t ϕ t≥0 is a real ( t )-Wiener process, for every ϕ ∈ , • W t (aϕ + ψ) = aW t (ϕ) + W t (ψ) almost surely for all a ∈ , t ∈ + and ϕ, ψ ∈ , Remark 3.1. "Spatial homogeneity" refers to the fact that the process W can be represented as a centered ( t )-adapted Gaussian random field µ is a bounded continuous function. The reader is referred to [5], [12], [36] and [37] for further details and examples of spatially homogeneous Wiener processes. Let us denote by H µ ⊆ the reproducing kernel Hilbert space of the -valued random vector W (1), see e.g. [11]. Then W is an H µ -cylindrical Wiener process. Moreover, see [5] and [37], if we denote by L 2 (s) ( d , µ) the subspace of L 2 ( d , µ; ) consisting of all ψ such that ψ = ψ (s) , where ψ (s) (·) = ψ(−·), then we have the following result: The following lemma states that, under some assumptions, H µ is a function space and that multiplication operators are Hilbert-Schmidt from H µ to L 2 (see [30] for a proof). In that case, we can calculate the Hilbert-Schmidt norm explicitly.
Then the reproducing kernel Hilbert space H µ is continuously embedded in C b ( d ), the multiplication operator m g = {H µ ξ → g · ξ ∈ L 2 (D)} is Hilbert-Schmidt and there exists a constant c such that whenever D ⊆ d is Borel and g ∈ L 2 (D).
Remark 3.4. A stochastic integral with respect to a spatially homogeneous Wiener process is understood in the classical way, see e.g. [11], [36] or [37].

The main result
Assume that R>0 are real numbers such that lim R→∞ α r,R = 0 for every r > 0, and, for every y ∈ n , r > 0 and R > 0, hold for almost every w ∈ d .
Remark 5.3. Let us observe that Theorem 5.2 is a sort of extension of Theorem 5.1 that claims that the particular solution constructed in Theorem 5.1 satisfies the infinite number of qualitative properties (i.e. given whichever entries G, L etc.) in Theorem 5.2. We however cannot exclude, at this moment, the possibility of existence of a weak solution of (1.1) for which (5.5) is not satisfied.

Examples
let Θ be a Borel probability measure on loc such that holds for Θ-almost every (u, v) ∈ l oc and let µ be a finite spectral measure. Then the equation has a weak solution z = (u, v) with weakly loc -continuous paths, with the initial distribution Θ and holds a.s. where W is a spatially homogeneous Wiener process with spectral measure µ. It is straightforward to verify that the hypotheses i) -xi) at the beginning of Section 5 hold if we put p+1 + 1 and α r,R = sup t≥R t p ( t p+1 p+1 + 1) −1 . Example 5.5 (Global space). The equation is a particular case of (5.6) with α = β i = γ i = 0, A = 1 and B = b d+1 , so we know, by Example 5.4, which we develop here further, that if a i , b i : d+1 → , i ∈ {0, . . . , d + 1} are bounded measurable functions continuous in the last variable, Θ is a Borel probability measure on such that u ∈ L p+1 holds for Θ-almost every (u, v) ∈ and µ is a finite spectral measure, that the equation (5.8) has a weak solution z = (u, v) with weakly l oc -continuous paths, with the initial distribution Θ and W is a spatially homogeneous Wiener process with spectral measure µ. Notwithstanding, the estimate (5.7) can be further strengthened to sup by applying Theorem 5.2 on the function G(x, y) = | y| p+1 /(p + 1) + | y| 2 /2, L(x) = x and λ = 1 (which satisfy the assumptions of Section 5). In particular, paths of the solution z are not only weakly l oc -continuous, but weakly -continuous a.s.
Proof. The assumptions of Theorem 5.2 are satisfied by a direct verification so if we define then, for a fixed ρ > 0 independent of z, R and T , holds for every T > 0 and t ∈ [0, T ) by Theorem 5.2. Letting T → ∞, we obtain by the Fatou lemma. Thus whence we get (5.9).

Guideline through the paper
The present work generalizes the state of art in the five directions mentioned in the Introduction. Let us, though, illustrate the progress with respect to the most related paper [29] on the example of the equation Here, the nonnegative function F is just the potential of − f , i.e. f + ∇F = 0 (provided it exists) and its purpose is to control the growth of the norm of the solutions in the energy space -see the apriori estimates (8.4) and (10.4) which generalize the conservation of energy law in the theory of deterministic wave equations (see e.g. [38]).
If the equation (6.1) is scalar, existence of a weak solution of (6.1) was proved, independently of the dimension d, in [29] provided that, roughly speaking, f and g are continuous, the primitive function F to − f is positive and i.e. only the case of subcritically growing polynomial nonlinearities q < (p + 1)/2 was covered for the equation (1.2) (cf. the hypothesis (1.3)). The condition (6.2) was induced in [29] by the method of proof in global spaces and it was not surprising because it just accompanied the Strauss hypothesis lim |t|→∞ | f (t)|/F (t) = 0 on the drift in the deterministic equation (see [44]) by the expected hypothesis lim |t|→∞ |g(t)| 2 /F (t) = 0 on the diffusion. However, the subtle arguments in Section 11 based on the local nature of the equation show that mere eventual boundedness of |g| 2 /F is sufficient for the proofs to go through and, consequently, the critical case q = (p + 1)/2 for the equation (1.2) is covered.
The proofs of both Theorem 5.1 and Theorem 5.2 are based on a refined stochastic compactness method (adapted from [17]) which consists in the following: a sequence of solutions of suitably constructed approximating equations is shown to be tight in the path space of weakly continuous vector-valued functions . This space is not metrizable hence the Jakubowski-Skorokhod theorem [21] is applied (instead of the Skorokhod representation theorem) to model the Prokhorov weak convergence of laws as an almost sure convergence of processes on a fixed probability space to a limit process which is, eventually, proved to be the desired weak solution. Apart from the Jakubowski-Skorokhod theorem, another novelty of the method relies in the fact that the identification of the limit process with the solution is not done via a martingale representation theorem (which is not available in our setting anyway) but by a few tricks with quadratic variations (see Sections 9 and 10).
Let us briefly comment on the structure of the proofs: The stochastic Cauchy problem (1.1) is first reduced (by a localization in Section 7) to the global Hilbert space W 1,2 ( d )⊕ L 2 ( d ) where an apriori estimate (8.4) independent of the localization is proved. The stochastic compactness method is then applied in two steps: First, existence of a weak solution with W 1,2 ( d )⊕ L 2 ( d )-continuous paths is established in Section 9 for sub-linearly growing Lipschitz functions f = ( f i ), g = (g i ) using the apriori estimate (8.4), where the nonlinearities are simply mollified by smooth densities. This step is not trivial since the Nemytski operators associated to f and g are not "locally Lipschitz" on the state space W 1,2 ( d ) ⊕ L 2 ( d ) -yet, the stochastic compactness method is employed in its standard form (see e.g. [2], [17] or [29]).
Subsequently, a refined apriori estimate (10.4) adapted to finely approximated nonlinearities is established in Section 10 and the full strength of the stochastic compactness method based on the Jakubowski-Skorokhod theorem is carried out in Section 11 which is also the core of the paper. The technicalities are caused mainly by the local space setting of the problem. Finally, Theorem 5.2 is proved collaterally in Section 11.6.

Localization of the operator
Our differential operator must be localized first in order general theorems on generating C 0semigroups can be applied. For, let us define the C 1 -function h : We may verify quite easily that φ m is diffeomorphic on {x : |x| < 11+ 57 16 } and also on {x : 11+ 57 then we have the following result which concerns a realization of the differential operator m in L 2 on the Sobolev space W 2,2 , states its funcional-analytic properties and allows to introduce a matrix infinitesimal generator of a wave C 0 -group. • the graph norm on Dom m is equivalent with the W 2,2 -norm, • the operator • the graph norm on Dom m is equivalent with the W 2,2 ⊕ W 1,2 -norm.
Proof. The operator m is selfadjoint e.g. by a consequence of Theorem 4 in Section 1.6 in [24], the operator 0 The equivalence of the graph norm on Dom m with the W 2,2 -norm follows from Theorem 5 in Section 1.6 in [24].
We close this section by introducing the localized conic energy function relative to F and to the operator m analogously to (2.2). Toward this end, given x ∈ d , T > 0, λ > 0 and a measurable function F : d × n → + , we define Observe that the integrand in (7.3) coincides with the integrand of (2.2) on the centered ball B m , hence the localized conic energy function F m,λ,x,T relative to F and to the operator m is really a "localization" of the conic energy function F λ,x,T relative to F and to the operator .

A local energy inequality
In this technical section we shall establish a backward cone energy estimate that, on one hand, makes it possible to find uniform bounds for a suitable approximating sequences of processes that will later on yield a solution by invoking a compactness argument, and on the other hand, imply finite propagation property of solutions of (1.1).
Proposition 8.1. Let m ∈ , T > 0, let U be a separable Hilbert space and W a U-cylindrical Wiener process. Let α and β be progressively measurable processes with values in L 2 and 2 (U, L 2 ) respectively such that  is an adapted process with continuous paths in such that holds a.s. for every t ≥ 0 and every ϕ ∈ . Assume that F : d × n → + is such that where φ m was defined in (7.1), and consider the conic energy function F = F m,λ,x,T for F (see (7.3)). Assume also that a nondecreasing function L : Proof. By density, (8.2) holds for every ϕ ∈ W 2,2 and if we test with ϕ = 2 ( − m ) −2 ψ for > 0 and ψ ∈ L 2 then and the integrals converge in Dom m = W 2,2 whose norms are equivalent by Proposition 7.1. If iii) sup |D γ y F j (w, y)| : |z| ≤ r, y ∈ n < ∞ for every multiindex γ and r > 0 holds for every y ∈ d and r > 0. We may thus apply the Ito formula (see [11]) on L(F j (z )) to obtain for every 0 ≤ r < t ≤ T a.s. We may, in fact, find the functions F j satisfying i)-iii) even so that Thus for every ω ∈ Ω. Moreover, by the Gauss theorem, so, after applying (8.6) and letting j → ∞ in (8.5), for every 0 ≤ r < t ≤ T a.s. by the Lebesgue dominated convergence theorem and the convergence theorem for stochastic integrals (see e.g. Proposition 4.1 in [33]).
for every ω ∈ Ω so we get the result from (8.7) by the Lebesgue dominated convergence theorem and a convergence result for stochastic integrals (e.g. Proposition 4.1 in [33]).

Linear growth + Global space case
We first prove existence of weak solutions for a localized equation with regular nonlinearities. The proof is based on a compactness method: local energy estimates yield tightness of an approximating sequence of solutions. This sequence converges, on another probability space, to a limit due to the Jakubowski-Skorokhod theorem and finally, it is shown, that this limit is the desired weak solution of the localized equation (9.1).
Lemma 9.1. Let µ be a finite spectral measure on d , let m ∈ , let ν be a Borel probability measure supported in a ball in , The proof of Lemma 9.1 will be carried out in a sequence of lemmas. For, let us introduce the mappings f k : → and g k : Lemma 9.2. For every k ∈ , there exists a completely filtered stochastic basis (Ω k , k , k ) with a spatially homogeneous ( k t )-Wiener process W k with spectral measure µ and an ( k t )-adapted process z k = (u k , v k ) with -continuous paths such that ν is the law of z k (0) under k and Moreover, for every p ∈ [2, ∞), there exists a constant K (1) p,l = K p,l , k, l ∈ (9.2) and, if q ∈ (1, ∞) and γ > 0 are such that γ + 1 q < 1 2 then there exists a constant K (2) q,l = K (2) l,m,γ,q, f ,g,a (2,m) ,c,ν such that Proof. The mappings f k : → and g k : are Lipschitz on bounded sets and have at most linear growth hence there exists a completely filtered stochastic basis (Ω k , k , k ) with a spatially homogeneous ( k t )-Wiener process W k with spectral measure µ and an ( k t )-adapted process z k with -continuous paths such that ν is the law of z k (0) under k and by e.g. [11] extended in the sense of Theorem 12.1 in Chapter V.2.12 in [39] whose generalization to SPDE is possible and can be proved in the same way as in [39]) since 〈 m z, z〉 hence the square norm of the local solution z k 2 Dom (I− m ) 1 2 ⊕L 2 cannot explode in finite time and so z k is a global solution in the sense of (4.3) by the Chojnowska-Michalik theorem (see [9] or Theorem 12 in [33]).
1 2 , with the notation F T = F m,λ 0 ,0,T and for t ∈ [0, T /λ 0 ] by Lemma 3.3 where K depends only on p, c, m, a, g, f and for t ∈ + by the Lebesgue dominated convergence theorem and a convergence result for stochastic integrals (e.g. Proposition 4.1. in [33]) where Hence, applying the exponential on both sides, we get 2 )t by the Doob maximal inequality for martingales and the Novikov criterion, where Since F 1 2 ∞ is an equivalent norm on , we have proved (9.2). Next, by the Chojnowska-Michalik theorem (see [9] or Theorem 12 in [33]) almost surely for every t ∈ + , where f (2) k and g (2) k are the second components of f k and g k , respectively, and the integrals converge in L 2 . We get that Proof. It holds, by the Chojnowska-Michalik theorem (see [9] or Theorem 12 in [33]), that almost surely where the integral converges in L 2 . So, if γ ∈ (0, 1) then Hence, if we fix > 0, q ∈ (1, ∞) and γ > 0 such that γ + 1 q < 1 2 and we assume We may now proceed to the proof of Lemma 9.1. Fixing an ONB (e l ) in H µ , by the Jakubowski-Skorokhod theorem A.1 applied to the × C 0 ( + ; dim H µ )-valued sequence (z k , (W k (e l )) l ) k , there exists • a subsequence (k j ), a probability space (Ω, , ) with • a -valued random variable Z, • C 0 ( + ; dim H µ )-valued random variables β j , j ∈ and β such that i) the law of (z k j , (W k j (e l )) l ) under k j coincides with the law of (Z j , β j ) under on Remark 9.4. We point out for completeness that tightness of the sequence (z k , (W k (e l )) l ) k in × C 0 ( + ; dim H µ ) follows from Lemma 9.3 and from the fact that all (W k (e l )) l have the same Radon law on the Polish space C 0 ( + ; dim H µ ) for every k ∈ . Consequently, the sequence (W k (e l )) l is tight in C 0 ( + ; dim H µ ).
Remark 9.5. It should be also noted that the random variables Z j are -valued by the Jakubowski-Skorokhod theorem. However, since z k j and Z j have the same law on and z k j are C( + ; )valued, we conclude that Z j may be assumed to be C( + ; )-valued satisfying the property i) above without loss of generality as C( + ; ) is a measurable subset of by Corollary A.2.
p,l , j, l ∈ (9.7) where K (1) p,l is the same constant as in (9.2 p,l , j, l ∈ by the property i) and sup p,l , j, l ∈ by the Fatou lemma and the property ii). Proof. Let us consider the sequence (ϕ i ) from Corollary C.1, let 0 ≤ s < t, J ∈ , 0 ≤ s 1 ≤ · · · ≤ s J ≤ s, let h 0 : for j ∈ . Then, for every j ∈ , by the property ii) and we conclude that holds for every F s ∈ s whenever s < t, i.e. σ(β(t) − β(s)) is -independent from s . Since s (e l )) have the normal centered distribution with covariance (t − s)I l for every j, l ∈ and 0 ≤ s < t by the property i) preceding Remark 9.4 and the fact that W k j are cylindrical Wiener processes on H µ by Section 3, we conclude that (β 1 (t) − β 1 (s), . . . , β l (t) − β l (s)) has the normal centered distribution with covariance (t − s)I l as well, as β j → β on Ω by the property ii) preceding Remark 9.4. The proof of Lemma 9.9 is thus complete. Corollary 9.10. Let (e l ) be the previously fixed ONB in H µ . Then the cylindrical process is a spatially homogeneous ( t )-Wiener process with spectral measure µ.

Lemma 9.11.
Proof. Fix ϕ ∈ and define the continuous operators  and g (2) k are the second components of f k and g k , respectively. Then, fixing 0 ≤ s < t and with the notation of the proof of Lemma 9.9, by the property i) since, by (9.4), is an L 2 (Ω k j )-integrable martingale in L 2 by (9.2) and Lemma 3.3, and the integrals (expectations) in (9.8)-(9.10) converge by (9.2) and (9.6). Since for every r ∈ + , l ∈ and q > 0 by (9.6), we get (2) (Z(r))e l , ϕ 2 d r = 0 by the property ii) where In particular, the processes Thus Z is a solution of (9.1). Moreover, by the Chojnowska-Michalik theorem (see [9] or Theorem 13 in [33]), dW s holds a.s. for every t ∈ + , hence paths of Z are -continuous almost surely.

General growth + Local space case
In this section, we use the existence result for the localized equation (9.1) and mimic the compactness method of the previous section based on the local energy estimates, tightness of an approximating sequence of solutions, convergence to a limit on another probability space due to the Jakubowski-Skorokhod theorem and final identification of the limit with a solution. The construction-approximation procedure is, however, much more refined this time.

.3) and F = F λ,x,T is the conic energy function for F defined as in (2.2) then
holds for every t ∈ [0, T /λ] and every Ω 0 ∈ 0 .
With the notation λ T defined in (2.3), given r > 0, let T r be the smallest radius of the base of a backward cone that contains (houses) the cylinder [0, r] × B r and, given m ∈ , let r m be the radius of the largest cylinder [0, r] × B r for which the radius of the housing backward cone is not larger than m, i.e. T r ≤ m. We can define these radii by Remark 10.2. Observe that T r < ∞ for every r > 0 and r m ∈ (0, m] satisfy r m ↑ ∞ by (2.1).
Given r > 0, let use define extension operators • E * r maps L p ( d ) continuously to L p (B r ), Then there exists a constantρ ∈ + depending only on R, d, p, κ, r 0 , δ, γ and c (see Section 2 and (10.6)) such that the following holds. If f i , g i : d × n → n×n and f d+1 , g d+1 : d × n → n are measurable functions satisfying the assumptions of Lemma 9.1, m ∈ , z is an -continuous solution of (9.1), F : d × n → + satisfies the assumptions (a)-(c) in Proposition 8.1 and, for every y ∈ n , the inequalities (10.1)-(10.3) hold for a.e. w ∈ B T R∧r m . If, for every y ∈ n , | f d+1 (w, y)| ≤ κF (w, y) for a.e. w ∈ B r m ∧R (10.8) and F r = F λ T r ,0,T r is the conic energy function defined as in (2.2) for the function F then if R > r and Observe that max {c 1 , c 2 , c 3 } can be dominated by a constant c that depends only on d, R and r 0 by Lemma 10.3. Let us prove just the third inequality in case R > r as the other cases are straightforward. For let ϕ ∈ W d * ,2 R . Then If we define the processes where the integral I 2 converges in W 1,2 and the integrals I 3 , I 4 converge in L 2 then by the Chojnowska-Michalik theorem (see [9] or Theorem 13 in [33] holds for every t ∈ [0, r m ∧ R], we get, using (10.9), where Ω 0 = [F r m ∧R (0, z(0)) ≤ δ], the estimate of I 4 follows from the Garsia-Rodemich-Rumsey lemma [16], the Burkholder inequality (see e.g. [33]) and Lemma 3.3. Altogether, ≤ c d,R,r 0 ,c,κ,p,γ,δ by the inequality (10.4).

Compactness
The present section is the actual core of the paper. We list all preliminary results and assumptions prepared and discussed in previous parts of the paper (Section 11.1) and then we carry out, in a few steps, the refined stochastic compactness method as indicated in Section 6. That is to say, we prove tightness of a sequence of solutions of approximating equations (Section 11.2), then we verify the assumptions of the Jakubowski-Skorokhod theorem A.1 and prove that the limit process yielded by this theorem is the solution of (1.1) claimed in Theorem 5.1 (Sections 11.3 -11.8), whereas the proof of Theorem 5.2 is given simultaneously in Section 11.6.

Assumptions
be measurable functions such that, for a.e. w ∈ d , there is lim m→∞ m π R (z m (0)) ∈ · − Θ π R ∈ · = 0 (11.1) holds for every R > 0 where the norm is taken in the total variation of measures on ( R ) and π R : → R is the restriction operator (see Section 2), where F * = sup F m , ix) given r > 0, r be an extension operator on L 2 (B r ) and on W 1,2 (B r ), i.e.
where r m and E r m were defined in (10.6) and (10.7), (11.4) xii) given r > 0, there exist α R = α r,R for R > 0 such that lim R→∞ α R = 0 and holds for every m ∈ and almost every w ∈ B r .  Remark 11.2. Observe also that, given R > 0, the real valued sequence F m (·, U m (0)) for m ∈ such that r m ≥ R where R,m → 0 by (11.1). Tightness follows from (11.2).

Skorokhod representation
Since Z m are tight in by Lemma 11.3, are tight in + (where T k were defined in (10.6)) by Remark 11.2 and m [Z m (0) ∈ ·] converge to Θ weakly on l oc by Remark 11.1, i.e. Z m (0) are tight in loc by the Prokhorov theorem, fixing an ONB (e l ) in H µ , we may apply Theorem A.1 on the sequence to claim that there exist • a probability space (Ω, , ), • a subsequence m j , • C( + ; )-valued random variables z j = (u j , v j ) defined on Ω, • + -valued random variable ν = (ν k ) k∈ defined on Ω, • a -valued random variable z = (u, v) with σ-compact range defined on Ω such that (i) (Z m j , (W m j (e l )) l ) has the same law under m j as (z j , β j ) under on the space for every j ∈ , (ii) (z j , β j ) converges to (z, β) on Ω in the topology of × C 0 ( + ; dim H µ ), Definition 11.4. We also define, for completness,

Property of β
Let us define Apparently, the filtration ( t ) is complete. The proof of the following Lemma is analogous to the proof of Lemma 9.9.
Corollary 11.6. The cylindrical process is a spatially homogeneous ( t )-Wiener process with spectral measure µ.

Property of u
Lemma 11.7. There is almost surely for every ϕ ∈ .
Proof. If ϕ is supported in B r m j and t ∈ [0, r m j ] then The rest of the proof is analogous with the proof of Lemma 9.8.

Martingale property
Let us remind the reader that the integrals in the following Proposition converge by the assumption v) in Section 11.1 and by (11.11). Proof. Let k ∈ , let ϕ ∈ have support in B k and, throughout this proof, consider only j ∈ such that r m j ≥ T k , i.e. j ≥ j 0 for some j 0 and it holds that Fixing 0 ≤ s < t ≤ k, we consider the sequence (ϕ i ) from Corollary C.1. Let also J ∈ , 0 ≤ s 1 ≤ · · · ≤ s J ≤ s, let

Corollary A.2. Under the assumptions of Theorem A.1, iff Z is a Polish space and b : Z → X is a continuous injection, then b[B] is a Borel set whenever B is Borel in Z.
Proof. Since the map F = ( f 1 , f 2 , . . . ) : X → is a continuous injection, F • b : Z → is also a continuous injection. Let us take a Borel set B ⊂ Z. Since both Z and are Polish spaces, we infer that (F • b) are both homeomorphisms onto closed sets.
Proof. The proof of Lemma B.1 is straightforward.

C A measurability lemma
Let X be a separable Fréchet space (with a countable system of pseudonorms ( · k ) k∈ , let X k be separable Hilbert spaces and i k : X → X k linear mappings such that i k (x) X k = x k , k ≥ 1. Let ϕ k, j ∈ X * k , j ∈ separate points of X k . Then the mappings (ϕ k, j • i k ) k, j∈ generate the Borel σ-algebra on X .
Proof. Denote by σ 0 the σ-algebra generated by the mappings (ϕ k, j • i k ) k, j∈ and denote Then V k is a closed dense subspace in X * k , hence V k = X * k . There exists ψ k, j ∈ X * k such that z X k = sup j∈ |ψ k, j (z)|, x ∈ X k , and so the mapping