**E**l e c t ro n
ic

**J**o
ur

n a l o

f
**P**

r o

b a b i_{l i t y}

Vol. 15 (2010), Paper no. 33, pages 1041–1091. Journal URL

http://www.math.washington.edu/~ejpecp/

## Stochastic nonlinear wave equations in local Sobolev

spaces

Martin Ondreját

∗†‡### Abstract

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.

**Key words:**stochastic wave equation.

**AMS 2000 Subject Classification:**Primary 60H15.

Submitted to EJP on March24, 2010, final version accepted June 30, 2010.

∗_{Affiliation: Institute of Information Theory and Automation, Academy of Sciences, Czech Republic}

1

## Introduction

Nonlinear wave equations

*u _{t t}*=

_{A}

*u*+

*f*(

*x*,

*u*,

*u*,

_{t}_{∇}

*) +*

_{x}u*g*(

*x*,

*u*,

*u*,

_{t}_{∇}

*)*

_{x}u*W*˙ (1.1) subject to random excitations have been thoroughly studied recently under various sets of hypothe-ses (see e.g. [6], [7], [8], [10], [12], [13], [14], [22], [23], [25], [26],[27], [28], [29], [30], [32],[35],[36],[37]and references therein) with possible applications in physics (e.g. in relativis-tic quantum mechanics or oceanography) in view.

The random perturbation has been usually modelled by a spatially homogeneous Wiener process
which corresponds to a centered Gaussian random field(*W*(*t*,*x*):*t*_{≥}0, *x*_{∈}R*d*)satisfying

E*W*(*t*,*x*)*W*(*s*,*y*) = (*t*_{∧}*s*)Γ(*x*_{−}*y*), *t*,*s*_{≥}0, *x*,*y* _{∈}R*d*

for some function or even a distributionΓcalled*the spatial correlation*of*W* (see e.g. [37]for
de-tails). The operator_{A} in (1.1) is a second order elliptic differential operator, usually the Laplacian.
(More general elliptic operators are considered only in[29]).

Functions*f* and*g*dependent only on*u*are dealt predominantly and their global Lipschitz continuity
is assumed in most of the papers cited above. Then the Nemytskii operators associated with *f* and

*g*are also globally Lipschitzian and existence (and uniqueness) of solutions to (1.1) may be proved
for rather general spatial correlationsΓ(which may be a distribution, e.g. the standard cylindrical
Wiener process is allowed if the space dimension is one, or at least a continuous function unbounded
at the origin), the state space (to which the pair(*u*,*u _{t}*)belongs) being

*L*2(R

*d*)

_{⊕}

*W*−1,2(R

*d*). IfΓis more regular then solutions live in the so called “energy space”

*W*1,2(R

*d*)

_{⊕}

*L*2(R

*d*) (see e.g. [38] for a discussion of the role of the energy space in the deterministic case).

Locally Lipschitz (or even continuous) real functions *f* and *g* are considered in the papers [10],
[27],[29],[30],[31]and[32].

Various techniques including Lyapunov functions, energy estimates, Sobolev embeddings, Strichartz
inequalities or compactness methods - have been employed to show existence of global solutions
in this case. These methods require the state space to be the energy space*W*1,2(R*d*)_{⊕}*L*2(R*d*)and
the spatial correlationΓto be a bounded function, i.e. the*spectral measureµ*= (2*π*)*d*2Γbis a finite

measure (cf. equality (3.1) below). The assumptions on Γare relaxed in [27]in the case of the
planar domain at a price of assuming *g* bounded and globally Lipschitz while *f*(*u*) = _{−}*u*_{|}*u*_{|}*p*−1,
1_{≤}*p*_{≤}3.

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*

_{|}

*qW*˙,

*u*(0) =

*u*

_{0},

*u*(0) =

_{t}*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

_{F}

_{0}-measurable

[*W*1,2(R*d*)_{∩}*Lp*+1(R*d*)]_{⊕} *L*2(R*d*)-valued random variable, *W* is a spatially homogeneous Wiener
process with bounded spectral correlationΓ(i.e. *µ*= (2*π*)*d*2Γbmust be a finite measure) and

1_{≤}*q<* *p*+1

Under (1.3), paths of(*u*,*u _{t}*)take values in[

*W*1,2(R

*d*)

_{∩}

*Lp*+1(R

*d*)]

_{⊕}

*L*2(R

*d*)and are weakly contin-uous in

*W*1,2(R

*d*)

_{⊕}

*L*2(R

*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(R*d*)_{⊕}*L*2(R*d*)are known to hold if*d*∈ {1, 2}or if*d*≥3 and*p*≤ *d _{d}*+

_{−}2

_{2},

*q*≤

*d*+

_{d}_{−}1

_{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 _{d}*+

_{−}2

_{2},

*q*

*<*

*d*+

_{d}_{−}1

_{2}, these results correspond to the present state of art for the deterministic Cauchy problem

*u _{t t}*= ∆

*u*

_{−}

*u*

_{|}

*u*

_{|}

*p*−1,

*u*(0) =

*u*

_{0},

*u*(0) =

_{t}*v*

_{0}(1.4) on R

*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*= *p*+1

2 independently of the dimension of
the spatial domainR*d*,

2. global weak solutions exist for data in the**local**Sobolev space*W _{l oc}*1,2(R

*d*)

_{⊕}

*L*2

*(R*

_{l oc}*d*)and have trajectories weakly continuous in this local space,

3. solutions in the local Sobolev space satisfy a local energy inequality (Theorem 5.2),

4. include dependence on first derivatives in the non-linearities in the equation (1.1)

5. study systems of stochastic wave equations, i.e. when *f* and *g*areR*n*-valued.

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
operatorA in (1.1) need not be*uniformly*elliptic (as is usually assumed) and mere ellipticity ofA

is sufficient (see (2.1)). In particular,_{A} may even decay or explode at infinity, cf. Example 5.4 and
5.5.

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 sphereS*n*−1is 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*W*˙,

_{|}

*u*

_{|}

_{R}

*n*=1

where *g*(*p*,_{·},_{·})_{∈}*Tp*S*n*−1, *p*∈S*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
cita-tion/reference paper for a companion paper on stochastic wave equations in compact Riemannian
homogeneous space by Z. Brze´zniak and the author.

Finally, we remark that our proof of the main theorem is based on a new general method of con-structing 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.

2

## Notation and Conventions

We consider complete filtrations in this paper. We say that a filtration(_{F}* _{t}*)on a probability space

(Ω,F,P)is complete provided thatF0contains allP-negligible sets ofF. We denote by

• R_{+}the set of all nonnegative real numbers, i.e. R_{+}= [0,∞),

• B(*X*)the Borel*σ*-algebra on a topological space*X*,

• *B _{R}*the open ball inR

*d*with center at the origin and of radius

*R*,

• *Lp*=*Lp*(R*d*,R*n*),*Wk*,*p*=*Wk*,*p*(R*d*,R*n*),

• *L*_{loc}*p* =*L*_{loc}*p* (R*d*,R*n*)and*W*_{loc}*k*,*p*=*W*_{loc}*k*,*p*(R*d*,R*n*)equipped with the metrics

(*u*,*v*)_{7→}
∞

X

*j*=1
1

2*j* min{1,k*u*−*v*k*Lp*(*Bj*)}, (*u*,*v*)7→
∞

X

*j*=1
1

• H*Rk*=*W*
*k*+1,2_{(}_{B}

*R*)⊕*Wk*,2(*BR*),H*R*:=H*R*0,

• H*k*=*Wk*+1,2_{⊕}*Wk*,2,_{H} :=_{H}0,

• Hloc=*W*
1,2
loc ⊕*L*

2 loc,

• D=_{D}(R*d*,R*n*)is the space ofR*n*-valued compactly supported*C*∞-functions,

• S is the Schwartz spaces of complex rapidly decreasing *C*∞-functions onR*d* ([40]),

• S′is the space of tempered distributions on_{S}, i.e. the real dual space to_{S},

• *ξ*7→*ξ*bthe Fourier transformation on_{S}′,

• *Ck _{b}* the space of

*k*-times continuously differentiable functions onR

*d*with bounded derivatives up to order

*k*equipped with the supremum norm of all derivatives up to order

*k*,

• *Cγ*([*a*,*b*],*X*)the Banach space of*X*-valued*γ*-Hölder continuous functions on[*a*,*b*]with the norm

k*h*_{k}=sup_{{k}*h*(*t*)_{k}* _{X}* :

*t*

_{∈}[

*a*,

*b*]

_{}}+sup

k*h*(*t*)_{−}*h*(*s*)_{k}_{X}

(*t*_{−}*s*)*γ* :*a*≤*s<t*≤*b*

where*X* is a Banach space,

• A a second order elliptic operator

A =

*d*

X

*k*=1
*d*

X

*l*=1
*∂*
*∂x _{k}*

**a***kl*(*x*)

*∂*
*∂x _{l}*

where**a**(*x*)is a symmetric, strictly positive,(*d*×*d*)-real-matrix for every *x*∈R*d* and**a**is a
continu-ous,*W _{l oc}*1,∞(R

*d*)-valued function such that

inf¦*t*−2sup¦**a**(*x*)*y*_{R}*d*:|*x*|_{R}*d<t*,|*y*|_{R}*d*=1

©

:*t>*0©=0, (2.1)

see also Remark 2.1.

• *πR*various restriction maps to the ball*BR*, for example

*πR*:*L*2loc∋*v*7→*v*|*BR*∈*L*

2_{(}_{B}

*R*) or *πR*:Hloc∋*z*7→*z*|*BR*∈ H*R*

• L(*X*,*Y*)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_{L}(*X*,*Y*)_{∋}*B*_{7→} *B x*_{∈}*Y*, *x* _{∈}*X*. If *X* and *Y* are Banach spaces then _{L}(*X*,*Y*)is equipped
with the usual operator norm,

• L2(*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 mapsL2(*X*,*Y*)∋

*B*_{7→}*B x*_{∈}*Y*, *x* _{∈}*X*,

• *C*(R+,*Z*)the space of continuous functions fromR+to a metric space(*Z*,*ρ*)and we equip it with

the metric

(*a*,*b*)_{7→}
∞

X

*j*=1
1

2*j*min{1, sup
*t*∈[0,*j*]

*ρ*(*a*(*t*),*b*(*t*))_{}}.

• *C _{w}*(R

_{+};

*X*)the space of weakly continuous functions fromR

_{+}to a locally convex space

*X*and we equip it with the locally convex topology generated by the a family

_{k · k}

_{m}_{,}

*of pseudonorms defined by*

_{ϕ}k*a*_{k}_{m}_{,}* _{ϕ}*= sup

*t*∈[0,

*m*]|

*ϕ*(*a*(*t*))_{|}, *m*_{∈}N, *ϕ*∈*X*∗,

• *ζ*a symmetric*C*∞-density onR*d* supported in the unit ball and we define
*ζm*(*x*) =*mdζ*(*mx*), *x* ∈R*d*,

• Z =*C _{w}*(R+,

*W*1,2)×

_{l oc}*Cw*(R+,

*L*2

*),*

_{l oc}• capital bold scripts the conic energy functions, i.e. if a measurable function *F* :R*d*_{×}R*n* _{→} R_{+},

*x* _{∈}R*d*,*λ >*0 and*T>*0 are given then

**F***λ*,*x*,*T*(*t*,*u*,*v*) =

Z

*B*(*x*,*T*−*λt*)
(

1 2

*d*

X

*k*=1
*d*

X

*l*=1

**a*** _{kl}*¬

*u*,

_{x}_{k}*u*¶

_{x}_{l}R*n*+

1
2|*v*|

2

R*n*+*F*(*y*,*u*)

)

*d y* (2.2)

is defined for*t*∈[0,*T _{λ}*]and(

*u*,

*v*)

_{∈ H}

*.*

_{l oc}*Remark*2.1*.* The condition (2.1) is equivalent with the following: given *R>*0, there exists *T* *>*0
such that the cylinder{(*t*,*x*)_{∈}R+×R*d*:*t* ∈[0,*R*],|*x*|R*d*≤*R*}is contained in a centered backward
cone_{{}(*t*,*x*)_{∈}R+×R*d*:|*x*|+*tλT* ≤*T*}where

*λT* = sup
*w*∈*B*(0,*T*)k

**a**(*w*)_{k}12. (2.3)

3

## Spatially homogeneous Wiener process

Given a stochastic basis(Ω,_{F},(_{F}* _{t}*)

_{t}_{≥}

_{0},P), an

_{S}′-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*(R

*d*)

*<*∞throughout the paper, provided that

• *Wϕ*:= *W _{t}ϕ_{t}*

_{≥}

_{0}is a real(

_{F}

*)-Wiener process, for every*

_{t}*ϕ*∈ S,

• *W _{t}*(

*aϕ*+

*ψ*) =

*aW*(

_{t}*ϕ*) +

*W*(

_{t}*ψ*)almost surely for all

*a*

_{∈}R,

*t*

_{∈}R

_{+}and

*ϕ*,

*ψ*∈ S,

• E_{{}*W _{t}ϕ*1

*Wtϕ*2}=

*t*〈

*ϕ*b1,

*ϕ*b2〉

*L*2

_{(}

_{µ}_{)}for all

*t*≥0 and

*ϕ*

_{1},

*ϕ*

_{2}∈ S.

*Remark* 3.1*.* “Spatial homogeneity” refers to the fact that the process *W* can be represented as a
centered(_{F}* _{t}*)-adapted Gaussian random field(

_{W}(

*t*,

*x*):

*t*≥0,

*x*∈R

*d*)so that

P

*Wtϕ*=

Z

R*d*

*ϕ*(*x*)_{W}(*t*,*x*)*d x*

=1, *ϕ*∈ S

and

E[_{W}(*t*,*x*)_{W}(*s*,*y*)] =min_{{}*t*,*s*}Γ(*x*− *y*), *t*,*s*≥0, *x*,*y* ∈R*d*

whereΓ = (2*π*)−*d*2*µ*bis a bounded continuous function. The reader is referred to[5],[12],[36]and

Let us denote by *H _{µ}*

_{⊆ S}′ the reproducing kernel Hilbert space of the

_{S}′-valued random vector

*W*(1), see e.g. [11]. Then *W* is an *Hµ*-cylindrical Wiener process. Moreover, see [5]and[37], if
we denote by*L*2_{(}_{s}_{)}(R*d*,*µ*)the subspace of*L*2(R*d*,*µ*;C)consisting of all*ψ*such that*ψ*=*ψ*(s), where

*ψ*(s)(·) =*ψ*(−·), then we have the following result:

### Proposition 3.2.

*Hµ* = {d*ψµ*: *ψ*∈*L*(2s)(R

*d*_{,}_{µ}_{)}

},

〈d*ψµ*,*ϕµ*Ó〉*H _{µ}* =

Z

R*d*

*ψ*(*x*)*ϕ*(*x*)*dµ*(*x*), *ψ*,*ϕ*∈*L*_{(}2_{s}_{)}(R*d*,*µ*).

The following lemma states that, under some assumptions, *Hµ* is a function space and that
multi-plication operators are Hilbert-Schmidt from *Hµ* to *L*2 (see[30]for a proof). In that case, we can
calculate the Hilbert-Schmidt norm explicitly.

**Lemma 3.3.** *Assume thatµ*(R*d*)*<*∞*. Then the reproducing kernel Hilbert space Hµ* *is continuously*

*embedded in C _{b}*(R

*d*)

*, the multiplication operator m*=

_{g}_{{}

*H*

_{µ}_{∋}

*ξ*7→

*g*

_{·}

*ξ*∈

*L*2(

*D*)

_{}}

*is Hilbert-Schmidt*

*and there exists a constant*

**c**

*such that*

k*m _{g}*

_{k}

_{L}

2(*Hµ*,*L*2(*D*))=**c**k*g*k*L*2(*D*) (3.1)

*whenever D*_{⊆}R*d* *is Borel and g*_{∈}*L*2(*D*)*.*

*Remark*3.4*.* A stochastic integral with respect to a spatially homogeneous Wiener process is
under-stood in the classical way, see e.g.[11],[36]or[37].

4

## Solution

**Definition 4.1.** *Let f*1, . . . ,*fd* :R*d*_{×}R*n* _{→}R*n*×*n* *be Borel matrix-valued functions, let fd*+1,*gd*+1 :

R*d*_{×}R*n*_{→}R*nbe Borel functions,µa given finite spectral measure on*R*d,*(Ω,_{F},(_{F}* _{t}*),P)

*a completely*

*filtered probability space with a spatially homogeneous*(

_{F}

*)*

_{t}*-Wiener process W with spectral measure*

*µ. An* (_{F}* _{t}*)

*-adapted process z*= (

*u*,

*v*)

*with weakly continuous paths in*H

*l oc*

*is a solution of (1.1)*

*where, for*(*x*,*y*,*z*) = (*x*,*y*,*z*_{0}, . . . ,*z _{d}*)

_{∈}R

*d*

_{×}R

*n*

_{×}Q

*d*

_{i}_{=}

_{0}R

*n,*

*f*(*x*,*y*,*z*) =

*d*

X

*i*=0

*fi*(*x*,*y*)*zi*+*fd*+1(*x*,*y*), *g*(*x*,*y*,*z*) =
*d*

X

*i*=0

*gi*(*x*,*y*)*zi*+*gd*+1(*x*,*y*) (4.1)

*provided that*

P

Z *T*

0

n

k*f*(_{·},*u*(*s*),*v*(*s*),_{∇}* _{x}u*(

*s*))

_{k}

*1*

_{L}_{(}

_{B}*T*)+k*g*(·,*u*(*s*),*v*(*s*),∇*xu*(*s*))k

2
*L*2_{(}_{B}

*T*)

o
*ds*

*holds for every T>*0*and*

〈*u*(*t*),*ϕ*〉 = _{〈}*u*(0),*ϕ*〉+

Z *t*

0

〈*v*(*s*),*ϕ*〉*ds* (4.3)

〈*v*(*t*),*ϕ*〉 = _{〈}*v*(0),*ϕ*〉+

Z *t*

0

〈*u*(*s*),_{A}*ϕ*〉*ds*+

Z *t*

0

*f*(_{·},*u*(*s*),*v*(*s*),_{∇}* _{x}u*(

*s*)),

*ϕds*

+

Z *t*

0

〈*g*(_{·},*u*(*s*),*v*(*s*),_{∇}*u*(*s*))*dW _{s}*,

*ϕ*〉

*holds for every t*≥0*a.s. wheneverϕ*∈ D*.*

*Remark* 4.2*.* The assumption (4.2) guarantees existence of integrals in (4.3). Let us verify the
convergence of the stochastic integral in (4.3). For, let us denote*ρ*=*g*(_{·},*u*,*v*,_{∇}*u*), let*ξj*=*ψ*Ô*jµ*be
an ONB in*Hµ* (i.e. by Proposition 3.2,(*ψj*)is an ONB in *L*2_{(}_{s}_{)}(R*d*,*µ*)) and let*T* *>*0 be such that the
support of*ϕ* is contained in*B _{T}*. Then

k〈*ρ*(*s*)_{·},*ϕ*〉k2_{L}_{2}(*H _{µ}*,R) =
X

*j*

_{〈}* _{ρ}*(

*s*)

*ξj*,

*ϕ*〉

2

= (2*π*)−*d*X

*j*

Z

R*d*

Z

R*d*

*e*−*i*〈*x*,*y*〉_{〈}*ρ*(*s*,*x*),*ϕ*(*x*)_{〉}_{R}*nψ _{j}*(

*x*)

*d xµ*(

*d y*)

2

= (2*π*)−*d*

Z

R*d*

Z

R*d*

*e*−*i*〈*x*,*y*〉_{〈}*ρ*(*s*,*x*),*ϕ*(*x*)_{〉}_{R}*nd x*

2
*µ*(*d y*)

≤ *c*_{◦}_{〈}*ρ*(*s*,*x*),*ϕ*〉R*n*

2

*L*1≤*c*_{◦}
* _{ϕ}*2

*L*2_{(}_{B}

*T*)

* _{ρ}*(

*s*,

*x*)2

*2*

_{L}_{(}

_{B}*T*)

where*c*_{◦}= (2*π*)−*d _{µ}*

_{(}

_{R}

*d*

_{)}

_{and, by (4.2),}R

*t*

0k*ξ*7→ 〈*ρ*(*s*)*ξ*,*ϕ*〉k
2

L2(*Hµ*,R)*ds<*∞for all*t>*0 a.s. Hence,

the stochastic integral in (4.3) is well defined e.g. by[11].

5

## The main result

Assume that

i) *fi*,*gi*:R*d*×R*n*→R*n*×*n*,*i*∈ {0, . . . ,*d*},
ii) *fd*+1,*gd*+1:R*d*_{×}R*n*_{→}R*n*,

iii) *F*:R*d*_{×}R*n*_{→}R_{+}

are measurable functions,*κ*∈R_{+},(*αr*,*R*)*r*,*R>*0 are real numbers such that lim*R*→∞*αr*,*R*=0 for every

*r>*0, and, for every *y*_{∈}R*n*,*r>*0 and*R>*0,

iv) *F*(*w*,_{·})_{∈}*C*1(R*n*),

vi) _{|}*f*0(*w*,*y*)_{|}2+_{|}*g*0(*w*,*y*)_{|}2_{≤}*κ*,

vii)

*n*

X

*j*=1
*n*

X

*k*=1

**a**−12(*w*)

*f _{jk}*1(

*w*,

*y*)

.. .

*f _{jk}d*(

*w*,

*y*)

2

R*d*

+

**a**−12(*w*)

*g*1* _{jk}*(

*w*,

*y*)

.. .

*gd _{jk}*(

*w*,

*y*)

2

R*d*

≤*κ*,

viii) *gd*+1(*w*,*y*)2+_{∇}* _{y}F*(

*w*,

*y*) +

*fd*+1(

*w*,

*y*)2

_{≤}

*κF*(

*w*,

*y*), ix)

_{|}

*fd*+1(

*w*,

*y*)

_{| ≤}

*κF*(

*w*,

*y*),

x) **1**[|*w*|≤*r*]∩[|*y*|≥*R*]|*fd*+1(*w*,*y*)| ≤*αr*,*RF*(*w*,*y*),
xi) k*F*max(·,*r*)k*L*1_{(}_{B}

*r*)*<*∞where

*F*_{max}(*x*,*r*) = sup

|*y*|≤*r*

*F*(*x*,*y*) (5.1)

hold for almost every*w*∈R*d*.

**Theorem 5.1**(Existence)**.** *Letµbe a finite spectral measure and let*Θ*be a Borel probability measure*
*on*_{H}_{l oc}*such that*

Θ_{{}(*u*,*v*):_{∈ H}* _{l oc}*:

_{k}

*F*

_{max}(

_{·},

_{|}

*u*(

_{·})

_{|}+1)

_{k}

*1*

_{L}_{(}

_{B}*r*)*<*∞}=1, *r>*0. (5.2)

*Then there exists a completely filtered stochastic basis* (Ω,_{F},(_{F}* _{t}*),P)

*with a spatially homogeneous*

(_{F}* _{t}*)

*-Wiener process W with spectral measure*

*µand an*(

_{F}

*)*

_{t}*-adapted process z*= (

*u*,

*v*)

*with weakly*

*continuous paths in*H

*l oc*

*which is a solution of the equation (1.1) in the sense of Definition 4.1 and*P[

*z*(0)

_{∈}

*A*] = Θ(

*A*)

*for every A*

_{∈ B}(

_{H}

*)*

_{l oc}*.*

**Theorem 5.2**(Energy estimate)**.** *Letµ,*Θ*,κand*(Ω,_{F},(_{F}* _{t}*),P,

*W*,

*z*)

*be the same as in Theorem 5.1,*

*let (5.2) hold and letκ*˜∈R+

*. Let G*:R

*d*×R

*n*→R+

*be a measurable function such that*

• *G*(*w*,_{·})_{∈}*C*1(R*n*)*,*

• |*gd*+1(*w*,*y*)_{|}2+_{|∇}* _{y}G*(

*w*,

*y*) +

*fd*+1(

*w*,

*y*)

_{|}2

_{≤}

*κ*˜

*G*(

*w*,

*y*)

*, y*

_{∈}R

*n*

*hold for almost every w*∈R*d* *and let*

Θ_{{}(*u*,*v*):∈ H*l oc*:k*G*max(·,|*u*(·)|+1)k*L*1_{(}_{B}

*r*)*<*∞}=1, *r>*0

*where*

*G*_{max}(*w*,*r*) = sup

|*y*|≤*r*

*G*(*w*,*y*), *r>*0.

*Then there exists a constantρ*∈R_{+} *depending only on the numbersµ*(R*d*)*and*max_{{}*κ*, ˜*κ*}*such that,*
*given x*_{∈}R*d, T* *>*0*,*

*λ*≥ sup
*w*∈*B*(*x*,*T*)k

**a**(*w*)_{k}12, (5.3)

*a non-decreasing function L*∈*C*(R+)∩*C*2(0,∞)*such that*

*the estimate*

E

¨

**1*** _{A}*(

*z*(0)) sup

*r*∈[0,

*t*]

*L*(**G**(*r*,*z*(*r*)))

«

≤4*eρt*E**1*** _{A}*(

*z*(0))

*L*(

**G**(0,

*z*(0))) (5.5)

*holds with the convention*0_{· ∞}=0*for every A*_{∈ B}(_{H}* _{l oc}*)

*and t*

_{∈}[0,

*T/λ*]

*where*

**G**=

**G**

_{λ}_{,}

_{x}_{,}

_{T}*is the*

*conic energy function for G defined as in (2.2).*

*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.

5.1

## Examples

*Example* 5.4 (Local space)*.* Let *α* ∈ (_{−∞}, 2), let *a _{i}*,

*b*: R

_{i}*d*+1

_{→}R,

*i*

_{∈ {}0, . . . ,

*d*

_{}}be bounded measurable functions continuous in the last variable, let

*A*:R

*d*→R+,

*B*:R

*d*+1→Rbe measurable

functions such that*B*is continuous in the last variable, let*B*2_{≤}*ρA*for some*ρ*≥0, let*A*be locally
integrable, let

0*<p*, 0≤*q*≤ *p*+1

2 , *βi*≤
*α*

2, *γi*≤
*α*

2, *i*∈ {1, . . . ,*d*},
letΘbe a Borel probability measure on_{H}* _{l oc}* such that

Z

*Br*

*A*(*x*)_{|}*u*(*x*)_{|}*p*+1*d x<*∞, *r>*0

holds forΘ-almost every(*u*,*v*)_{∈ H}* _{l oc}*and let

*µ*be a finite spectral measure. Then the equation

*u _{t t}* = (1+

_{|}

*x*

_{|})

*α*∆

*u*+

a_{0}(*x*,*u*)*u _{t}*+

*d*

X

*i*=1

(1+_{|}*x*_{|})*βi _{a}*

*i*(*x*,*u*)

*∂u*

*∂xi* −

*A*(*x*)*u*_{|}*u*_{|}*p*−1

+

b_{0}(*x*,*u*)*ut*+
*d*

X

*i*=1

(1+_{|}*x*|)*γi _{b}*

*i*(*x*,*u*)

*∂u*

*∂x _{i}* +

*B*(

*x*,

*u*)|

*u*|

*q*

* _{W}*˙

_{(5.6)}

has a weak solution *z* = (*u*,*v*) with weakly H*l oc*-continuous paths, with the initial distribution Θ
and

sup
*t*∈[0,*r*]

Z

*Br*

*A*(*x*)_{|}*u*(*t*,*x*,*ω*)_{|}*p*+1*d x<*∞, *r* *>*0 (5.7)

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

*F*(*x*,*y*) =*A*(*x*)|*y*|*p*+1

*p*+1 +1

and*αr*,*R*=sup*t*≥*Rtp*(
*tp*+1

*p*+1+1)

−1_{.}

*Example*5.5 (Global space)*.* The equation

*u _{t t}* = ∆

*u*+

*a*

_{0}(

*x*,

*u*)

*u*+

_{t}*d*

X

*i*=1

*a _{i}*(

*x*,

*u*)

*u*

_{x}_{i}_{−}

*u*

_{|}

*u*

_{|}

*p*−1

+

b_{0}(*x*,*u*)*ut*+
*d*

X

*i*=1

*bi*(*x*,*u*)*uxi*+*bd*+1(*x*,*u*)|*u*|

*q*

is a particular case of (5.6) with*α*=*βi*=*γi* =0,*A*=1 and*B*=*bd*+1, so we know, by Example 5.4,
which we develop here further, that if*ai*,*bi* :R*d*+1→R, *i*∈ {0, . . . ,*d*+1}are bounded measurable
functions continuous in the last variable,

1≤*q*≤ *p*+1

2 ,

Θis a Borel probability measure on_{H} such that*u*_{∈}*Lp*+1holds forΘ-almost every(*u*,*v*)_{∈ H} and
*µ* is a finite spectral measure, that the equation (5.8) has a weak solution*z* = (*u*,*v*) with weakly

H*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
*t*∈[0,*r*] k

*z*(*t*)_{k}_{H} +_{k}*u*(*t*)_{k}* _{L}p*+1

*<*∞,

*r>*0, a.s. (5.9)

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
weaklyH*l oc*-continuous, but weaklyH-continuous a.s.

*Proof.* The assumptions of Theorem 5.2 are satisfied by a direct verification so if we define

*HR*={(*u*,*v*)∈ H :k(*u*,*v*)kH +k*u*k*Lp*+1≤*R*}, *R>*0

then, for a fixed*ρ >*0 independent of*z*,*R*and*T*,

E**1**_{H}

*R*(*z*(0))sup

*r*≤*t*

k

*z*(*r*)_{k}2_{HT−t}

2 +

k*u*(*r*)_{k}*p*+1

*Lp*+1_{(}_{B}

*T−t*)

*p*+1

≤ 4*eρt*
Z

*HR*

k

(*u*,*v*)_{k}2

HT

2 +

k*u*k*p _{L}*+

*p*+11

_{(}

_{B}*T*)

*p*+1

*d*Θ

≤ 4*eρt*

*R*2

2 +

*Rp*+1
*p*+1

=:*C _{t}*

_{,}

_{R}holds for every*T* *>*0 and*t*∈[0,*T*)by Theorem 5.2. Letting*T*→ ∞, we obtain

E**1*** _{H}_{R}*(

*z*(0))sup

*r*≤

*t*

k*z*(*r*)k2H

2 +

k*u*(*r*)_{k}*p*+1

*Lp*+1
*p*+1

_{≤}*C _{t}*

_{,}

_{R}by the Fatou lemma. Thus

P

**1***HR*(*z*(0))sup

*r*≤*t*

k*z*(*r*)k

2

H

2 +

k*u*(*r*)_{k}*p*+1

*Lp*+1
*p*+1

*<*∞, *R>*0

=1

whence we get (5.9).

6

## 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

in the global space *W*1,2(R*d*)_{⊕}*L*2(R*d*). 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

lim

|*t*|→∞

|*f*(*t*)_{|}+_{|}*g*(*t*)_{|}2

*F*(*t*) =0, (6.2)

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 _{Z} = *C _{w}*(R

_{+},

*W*1,2)

_{l oc}_{×}

*C*(R

_{w}_{+},

*L*2

*). 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).*

_{l oc}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(R*d*)_{⊕}*L*2(R*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(R*d*)_{⊕}*L*2(R*d*)-continuous paths is established in Section
9 for sub-linearly growing Lipschitz functions *f* = (*fi*), *g* = (*gi*) 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(R*d*)_{⊕}

*L*2(R*d*)- yet, the stochastic compactness method is employed in its standard form (see e.g. [2],[17]
or[29]).

7

## Localization of the operator

A

Our differential operator _{A} must be localized first in order general theorems on generating *C*0
-semigroups can be applied. For, let us define the*C*1-function*h*:R→[0, 1]by

*h*(*t*) =1, *t*_{∈}(_{−∞}, 1], *h*(*t*) = (2*t*_{−}1)(*t*_{−}2)2, *t*_{∈}[1, 2], *h*(*t*) =0, *t*_{∈}[2,_{∞}),

*φm*(*x*):=*h*(_{|}*x*_{|}*/m*)*x*, *x*_{∈}R*d*, *m*_{∈}N. (7.1)
We may verify quite easily that*φm* is diffeomorphic on_{{}*x* :_{|}*x*_{|}*<* 11+

p

57

16 }and also on{*x* :
11+p57

16 *<*

|*x*_{|} *<* 2_{}} hence the set *B*_{2}_{m}_{∩}[*φm* ∈ *C*] is negligible for the Lebesgue measure whenever so is

*C* ∈ B(R*d*), hence**a**◦*φm* _{is uniformly continuous on}_{R}*d*_{, belongs to}* _{W}*1,∞

_{(}

_{R}

*d*

_{)}

_{where}

*∂*(**a**_{◦}*φm*)

*∂x _{j}* =

**1**

*B*2

*m*

*d*

X

*l*=1
*∂***a**

*∂x _{l}*(

*φ*

*m*_{)}*∂ φ*
*m*
*l*

*∂x _{j}* ,

*j*∈ {1, . . . ,

*d*},

*m*∈N

and if we define

A*m*=

*d*

X

*i*=1
*d*

X

*j*=1
*∂*
*∂x _{i}*

(**a**_{i j}_{◦}*φm*) *∂*

*∂x _{j}*

, *m*_{∈}N (7.2)

then we have the following result which concerns a realization of the differential operator_{A}*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.

**Proposition 7.1.** *For m*_{∈}N*, it holds that*

• *the operator*A*m _{with}*

_{Dom}

_{A}

*m*

_{=}

*2,2*

_{W}*2*

_{is uniformly elliptic, selfadjoint and negative on L}

_{,}• Dom(*I*_{− A}*m*)12 =*W*1,2*with equivalence of the graph norm and the W*1,2*-norm,*

• *the graph norm on*Dom_{A}*m* *is equivalent with the W*2,2*-norm,*

• *the operator*

G*m*= 0 *I*

A*m* 0

, Dom_{G}*m*=*W*2,2_{⊕}*W*1,2

*generates a C*_{0}*-group*(*Sm _{t}* )

*on the space W*1,2

_{⊕}

*L*2

*,*

• *the graph norm on*DomG*m* * _{is equivalent with the W}*2,2

_{⊕}

*1,2*

_{W}

_{-norm.}*Proof.* The operator_{A}*m* is selfadjoint e.g. by a consequence of Theorem 4 in Section 1.6 in[24],
the operator

_{0} _{I}

A*m*−*I* 0

is skew-adjoint in Dom(*I*_{− A}*m*)12 ⊕*L*2 hence generates a unitary *C*_{0}-group by the Stone theorem

(see e.g. Theorem 10.8 in Chapter 1.10 in[34]). The operator

_{0 0}

*I* 0

is bounded on Dom(*I*_{− A}*m*)12 ⊕*L*2 soG*m* generates a*C*

0-group by Theorem 1.1 in Chapter 3.1 in [34].

The equivalence of the graph norm on Dom_{A}*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_{A}*m* analogously to (2.2). Toward this end, given *x* _{∈}R*d*, *T>*0,*λ >*0 and a measurable
function*F*:R*d*×R*n*→R+, we define

**F*** _{m}*,

*λ*,

*x*,

*T*(

*t*,

*u*,

*v*) =

Z

*B*(*x*,*T*−*λt*)
(

1 2

*d*

X

*k*=1
*d*

X

*l*=1

(**a*** _{kl}*◦

*φm*)¬

*u*,

_{x}_{k}*u*¶

_{x}_{l}R*n*+
1
2|*v*|

2

R*n*+*F*(*y*,*u*)

)

*d y* (7.3)

for*t*_{∈}[0,*T _{λ}*]and(

*u*,

*v*)

_{∈ H}

*.*

_{l oc}*Remark*7.2*.* Observe that the integrand in (7.3) coincides with the integrand of (2.2) on the
cen-tered ball *B _{m}*, hence the localized conic energy function

**F**

*,*

_{m}*λ*,

*x*,

*T*relative to

*F*and to the operator

A*m*is really a “localization” of the conic energy function**F**_{λ}_{,}_{x}_{,}* _{T}* relative to

*F*and to the operator

_{A}.

8

## 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*_{∈}N*, T* *>*0*, let U be a separable Hilbert space and W a U-cylindrical Wiener*
*process. Letαandβbe progressively measurable processes with values in L*2*and*L2(*U*,*L*2)*respectively*

*such that*

P

Z *T*

0

n

k*α*(*s*)_{k}* _{L}*2+k

*β*(

*s*)k2

L2(*U*,*L*2)
o

*ds<*∞

=1. (8.1)

*Assume that z*= (*u*,*v*)*is an adapted process with continuous paths in*H *such that*

〈*u*(*t*),*ϕ*〉*L*2 = 〈*u*(0),*ϕ*〉* _{L}*2+
Z

*t*

0

〈*v*(*s*),*ϕ*〉*L*2*ds* (8.2)

〈*v*(*t*),*ϕ*〉*L*2 = 〈*v*(0),*ϕ*〉* _{L}*2+
Z

*t*

0

〈*u*(*s*),_{A}*mϕ*〉*L*2*ds*+
Z *t*

0

〈*α*(*s*),*ϕ*〉*L*2*ds*+
Z *t*

0

〈*β*(*s*)*dW _{s}*,

*ϕ*〉

*L*2

*holds a.s. for every t*≥0*and everyϕ*∈ D*. Assume that F*:R*d*_{×}R*n*_{→}R_{+}*is such that*

*(a) F*(*w*,·)_{∈}*C*1(R*n*)*for every w*∈R*d,*
*(b) F*(_{·},*y*)*measurable for every y*∈R*n*

sup

(

*F*(*w*,*y*)

1+_{|}*y*_{|}2 +

_{∇}* _{y}F*(

*w*,

*y*)

1+_{|}*y*_{|} :|*w*| ≤*r*,*y*∈R

*n*

)

*<*∞, *r>*0.

*Fix x*∈R*d* *andλso that*

*λ*≥ sup
*w*∈*B*(*x*,*T*)k

**a**(*φm*(*w*))_{k}12

*whereφm* _{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*:R_{+}_{→}R*is of C*2*-class on*[0,_{∞})*and put*

*V*(*t*,*z*) = 1

2*L*

′′_{(}_{F}_{(}_{t}_{,}_{z}_{))}X

*l*

*v*,*β*(*t*)*el*

2

*L*2_{(}_{B}_{(}_{x}_{,}_{T}_{−}_{λt}_{))}+

1
2*L*

′_{(}_{F}_{(}_{t}_{,}_{z}_{))}_{k}_{β}_{(}_{t}_{)}_{k}2

L2(*U*,*L*2(*B*(*x*,*T*−*λt*)))

+ *L*′(**F**(*t*,*z*))_{〈}*v*,∇*yF*(·,*u*) +*α*(*t*)〉*L*2_{(}_{B}_{(}_{x}_{,}_{T}_{−}_{λt}_{))} (8.3)

*for t*∈[0,*T _{λ}*]

*and z*= (

*u*,

*v*)

_{∈ H}

*where*(

*el*)

*is any ONB in U. Then*

*L*(**F**(*t*,*z*(*t*))) _{≤} *L*(**F**(*r*,*z*(*r*))) +

Z *t*

*r*

*V*(*s*,*z*(*s*))*ds*

+

Z *t*

*r*

*L*′(**F**(*s*,*z*(*s*)))_{〈}*v*(*s*),*β*(*s*)*dW _{s}*〉

*L*2

_{(}

_{B}_{(}

_{x}_{,}

_{T}_{−}

_{λs}_{))}(8.4)

*is satisfied for every*0_{≤}*r<t*_{≤} *T _{λ}*

*almost surely.*

*Proof.* By density, (8.2) holds for every*ϕ*∈*W*2,2and if we test with*ϕ*=*ǫ*2(*ǫ*− A*m*)−2*ψ*for*ǫ >*0
and*ψ*∈*L*2then

*uǫ*(*t*) = *uǫ*(0) +

Z *t*

0

*vǫ*(*s*)*ds*

*vǫ*(*t*) = *vǫ*(0) +

Z *t*

0

A*muǫ*(*s*) +*αǫ*(*s*)

*ds*+

Z *t*

0

*βǫ*(*s*)*dWs*

holds for every*t*_{≥}0 a.s. where

*uǫ*= *ǫ*2(*ǫ*− A*m*)−2*u*, *vǫ*= *ǫ*2(*ǫ*− A*m*)−2*v*, *zǫ*= (*uǫ*,*vǫ*)

*αǫ*= *ǫ*2(*ǫ*− A*m*)−2*α*, *βǫξ*= *ǫ*2(*ǫ*− A*m*)−2[*βξ*], *ξ*∈ *U*

and the integrals converge in Dom_{A}*m* =*W*2,2 whose norms are equivalent by Proposition 7.1. If

*Fj*:R*d*×R*n*→R+satisfies

i) *Fj*(*w*,*y*) =0 for every*w*_{∈}R*d* and_{|}*y*_{|}*>* *j*,
ii) *Fj*(*w*,_{·})_{∈}*C*∞(R*n*)for every*w*_{∈}R*d*,
iii) sup¦|*DγyFj*(*w*,*y*)|:|*z*| ≤*r*,*y*∈R*n*

©

then

**F***j*=**F**_{m}j_{,}_{λ}_{,}_{x}_{,}_{T}_{∈}*C*1,2[0,*T/λ*]_{×}*W*2,2⊕*W*2,2

as

k*ϕ*k*L*2_{(}_{∂}_{B}_{(}_{y}_{,}_{r}_{))}≤2k*ϕ*k* _{L}*2k∇

*ϕ*k

*2,*

_{L}*ϕ*∈

*W*1,2

holds for every *y* _{∈}R*d* and *r* *>* 0. We may thus apply the Ito formula (see[11]) on *L*(**F***j*(*z _{ǫ}*))to
obtain

*L*(**F***j*(*t*,*zǫ*(*t*)))−*L*(**F***j*(*r*,*zǫ*(*r*))) =

− *λ*

Z *t*

*r*

*L*′(**F***j*(*s*,*zǫ*(*s*)))k*Fj*(·,*uǫ*(*s*))k*L*1_{(}_{∂}_{B}_{(}_{x}_{,}_{T}_{−}_{λs}_{))}*ds*

− *λ*

2

Z *t*

*r*

*L*′(**F***j*(*s*,*zǫ*(*s*)))

*d*
X

*i*=1
*d*

X

*k*=1

(**a**_{ik}_{◦}*φm*)

*∂uǫ*(*s*)

*∂x _{i}* ,

*∂uǫ*(*s*)

*∂x _{k}*
R

*n*

*L*1_{(}_{∂}_{B}_{(}_{x}_{,}_{T}_{−}_{λs}_{))}
*ds*

− *λ*

2

Z *t*

*r*

*L*′(**F***j*(*s*,*z _{ǫ}*(

*s*)))

*v*(

_{ǫ}*s*)2

*2*

_{L}_{(}

_{∂}_{B}_{(}

_{x}_{,}

_{T}_{−}

_{λs}_{))}

*ds*

+

Z *t*

*r*

*L*′(**F***j*(*s*,*zǫ*(*s*)))

(Z

*B*(*x*,*T*−*λs*)

*d*

X

*i*=1
*d*

X

*k*=1

(**a***ik*◦*φm*)

*∂u _{ǫ}*(

*s*)

*∂x _{i}* ,

*∂v _{ǫ}*(

*s*)

*∂x _{k}*
R

*n*)

*ds*+ Z

*t*

*r*

*L*′(**F***j*(*s*,*z _{ǫ}*(

*s*)))n¬

*v*(

_{ǫ}*s*),

_{A}

*mu*(

_{ǫ}*s*) +

*αǫ*(

*s*) +∇

*yFj*(·,

*uǫ*(

*s*))

¶

*L*2_{(}_{B}_{(}_{x}_{,}_{T}_{−}_{λs}_{))}
o

*ds*

+

Z *t*

*r*

*L*′(**F***j*(*s*,*zǫ*(*s*)))

¦

*vǫ*(*s*),*βǫ*(*s*)*dWs*

*L*2_{(}_{B}_{(}_{x}_{,}_{T}_{−}_{λs}_{))}
©
*ds*
+ 1
2
Z *t*
*r*

*L*′(**F***j*(*s*,*zǫ*(*s*)))k*βǫ*(*s*)k2_{L}

2(*U*,*L*2(*B*(*x*,*T*−*λs*))*ds*

+ 1
2
X
*l*
Z *t*
*r*

*L*′′(**F***j*(*s*,*z _{ǫ}*(

*s*)))

*v*(

_{ǫ}*s*),

*βǫ*(

*s*)

*el*

2

*L*2_{(}_{B}_{(}_{x}_{,}_{T}_{−}_{λs}_{))}*ds* (8.5)

for every 0_{≤}*r<t*_{≤}*T* a.s. We may, in fact, find the functions*Fj* satisfying i)-iii) even so that

iv) *Fj*(*w*,_{·})_{→}*F*(*w*,_{·})uniformly on compacts inR*n*, for every*w*∈R*d*,
v) _{∇}* _{y}Fj*(

*w*,

_{·})

_{→ ∇}

*(*

_{y}F*w*,

_{·})uniformly on compacts inR

*n*, for every

*w*

_{∈}R

*d*, vi) and

sup

(

*Fj*(*w*,*y*)
1+_{|}*y*|2 +

_{∇}* _{y}Fj*(

*w*,

*y*)

1+_{|}*y*_{|} :|*w*| ≤*r*,*y*∈R

*n*_{,}_{j}

∈N

)

*<*∞, *r>*0.

Thus

lim_{j}_{→∞}**F***j*(*t*,*z _{ǫ}*(

*t*,

*ω*)) =

**F**(

*t*,

*z*(

_{ǫ}*t*,

*ω*)),

*t*

_{∈}[0,

*T/λ*], sup¦

**F**

*j*(

*t*,

*z*(

_{ǫ}*t*,

*ω*)):

*t*

_{∈}[0,

*T/λ*],

*j*

_{∈}N©

*<*∞,

lim_{j}_{→∞}_{k∇}* _{y}Fj*(

_{·},

*u*(

_{ǫ}*t*,

*ω*))

_{− ∇}

*(*

_{y}F_{·},

*u*(

_{ǫ}*t*,

*ω*))

_{k}

*2*

_{L}_{(}

_{B}_{(}

_{x}_{,}

_{T}_{−}

_{λt}_{))}=0,

*t*∈[0,

*T/λ*],

sup¦_{k∇}* _{y}Fj*(

_{·},

*u*(

_{ǫ}*t*,

*ω*))

_{k}

*2*

_{L}_{(}

_{B}_{(}

_{x}_{,}

_{T}_{−}

_{λt}_{))}:

*t*∈[0,

*T/λ*],

*j*∈N ©

for every*ω*∈Ω. Moreover, by the Gauss theorem,
Z

*B*(*x*,*T*−*λs*)

*d*

X

*i*=1
*d*

X

*k*=1

(**a***ik*◦*φm*)

*∂u _{ǫ}*(

*s*)

*∂x _{i}* ,

*∂v _{ǫ}*(

*s*)

*∂x _{k}*

R*n*

+*vǫ*(*s*),A*muǫ*(*s*)

R*n*
*ds*
=
=
*d*
X

*i*=1
*d*

X

*k*=1

Z

*∂B*(*x*,*T*−*λs*)

(**a**_{ik}_{◦}*φm*)

*vǫ*,

*∂uǫ*

*∂xi*

R*n*

*yk*−*xk*

*T*−*λs* *d y*
≤
Z

*∂B*(*x*,*T*−*λs*)

|*vǫ*|

*n*

X

*l*=1

k(**a**_{◦}*φm*)_{∇}*ul _{ǫ}*

_{k}2 1 2

*d y*≤

*λ*Z

*∂B*(*x*,*T*−*λs*)

|*vǫ*|

*n*

X

*l*=1

k(**a**12◦*φm*)∇*ul*

*ǫ*k
2
1
2
*d y*
= *λ*
Z

*∂B*(*x*,*T*−*λs*)

|*vǫ*|

*d*
X

*i*=1
*d*

X

*k*=1

(**a**_{ik}_{◦}*φm*)

_{∂}_{u}

*ǫ*

*∂xi*
,*∂uǫ*

*∂xk*

R*n*
1
2
*d y*
≤ *λ*

2k*vǫ*k
2

*L*2_{(}_{∂}_{B}_{(}_{x}_{,}_{T}_{−}_{λs}_{))}+

*λ*
2
*d*
X

*i*=1
*d*

X

*k*=1

(**a**_{ik}_{◦}*φm*)

_{∂}_{u}

*ǫ*

*∂xi*
,*∂uǫ*

*∂xk*

R*n*

*L*1_{(}_{∂}_{B}_{(}_{x}_{,}_{T}_{−}_{λs}_{))}

(8.6)

so, after applying (8.6) and letting *j*→ ∞in (8.5),

*L*(**F**(*t*,*zǫ*(*t*)))−*L*(**F**(*r*,*zǫ*(*r*)))≤

+

Z *t*

*r*

*L*′(**F**(*s*,*zǫ*(*s*)))

n¬

*vǫ*(*s*),*αǫ*(*s*) +∇*yF*(·,*uǫ*(*s*))

¶

*L*2_{(}_{B}_{(}_{x}_{,}_{T}_{−}_{λs}_{))}
o

*ds*

+

Z *t*

*r*

*L*′(**F**(*s*,*z _{ǫ}*(

*s*)))¦

*v*(

_{ǫ}*s*),

*βǫ*(

*s*)

*dWs*

*L*2_{(}_{B}_{(}_{x}_{,}_{T}_{−}_{λs}_{))}
©
*ds*
+ 1
2
Z *t*
*r*

*L*′(**F**(*s*,*zǫ*(*s*)))k*βǫ*(*s*)k2_{L}

2(*U*,*L*2(*B*(*x*,*T*−*λs*))*ds*

+ 1
2
X
*l*
Z *t*
*r*

*L*′′(**F**(*s*,*zǫ*(*s*)))

*vǫ*(*s*),*βǫ*(*s*)*el*

2

*L*2_{(}_{B}_{(}_{x}_{,}_{T}_{−}_{λs}_{))} *ds* (8.7)

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]).

Now, since

sup
*ǫ>*1k

*ǫ*2(*ǫ*− A*m*)−2_{k}_{L}_{(}* _{L}*2

_{)}

*<*∞ and lim

*ǫ*→∞k*ǫ*

2_{(}_{ǫ}

− A*m*)−2*ϕ*−*ϕ*k*L*2=0, *ϕ*∈*L*2,

there is

lim
*ǫ*→∞

sup
*t*∈[0,*T/λ*]k

*zǫ*(*t*,*ω*)−*z*(*t*,*ω*)k*W*1,2_{⊕}* _{L}*2+ sup

*t*∈[0,*T/λ*]|

**F**(*t*,*zǫ*(*t*,*ω*))−**F**(*t*,*z*(*t*,*ω*))|

=0

sup_{{k}*zǫ*(*t*,*ω*)k*W*1,2_{⊕}* _{L}*2+

**F**(

*t*,

*z*(

_{ǫ}*t*,

*ω*)):

*ǫ >*0,

*t*∈[0,

*T/λ*]}

*<*∞

9

## 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.** *Letµbe a finite spectral measure on*R*d, let m*_{∈}N*, letν* *be a Borel probability measure*
*supported in a ball in*_{H}*, let fi*,*gi* :R*d*_{×}R*n* _{→}R*n*×*n* *for i*_{∈ {}0, . . . ,*d*_{}}*, fd*+1,*gd*+1:R*d*_{×}R*n*_{→}R*n*

*be measurable functions such that*

sup¦_{|}*fi*(*w*,*y*)_{|}+_{|}*gi*(*w*,*y*)_{|}:_{|}*w*_{| ≤}*r*, *y*_{∈}R*n*,*i*_{∈ {}0, . . . ,*d*_{}}©*<*∞

sup

¨

|*fd*+1(*w*,*y*)_{|}+_{|}*gd*+1(*w*,*y*)_{|}

1+_{|}*y*_{|} :|*w*| ≤*r*, *y*∈R

*n*

«

*<*∞,

sup

¨

|*fi*(*w*,*y*_{1})_{−}*fi*(*w*,*y*_{2})_{|}

|*y*_{1}_{−}*y*_{2}_{|} :|*w*| ≤*r*, *y*16= *y*2∈R

*n*_{,}_{i}

∈ {0, . . . ,*d*+1_{}}

«

*<*∞

sup

¨

|*gi*(*w*,*y*1)−*gi*(*w*,*y*2)|

|*y*1−*y*2|

:_{|}*w*_{| ≤}*r*, *y*_{1}_{6}= *y*_{2}_{∈}R*n*,*i*_{∈ {}0, . . . ,*d*+1_{}}

«

*<*∞

*hold for every r* *>* 0*. Then there exists a completely filtered stochastic basis* (Ω,F,(_{F}* _{t}*),P)

*with*

*a spatially homogeneous*(

_{F}

*)*

_{t}*-Wiener process W with the spectral measure*

*µ*

*and an*(

_{F}

*)*

_{t}*-adapted*

*process z with continuous paths in*

_{H}

*which is a solution of the equation*

*u _{t t}*=

_{A}

*mu*+

**1**

_{B}*mf*(·,*u*,*ut*,∇*u*) +**1***Bmg*(·,*u*,*ut*,∇*u*)*W*˙ (9.1)

*in the sense of Section 4 with the notation (4.1), (4.1),ν* *is the law of z*(0)*and Bmis the open centered*

*ball in*R*d* *with radius m.*

The proof of Lemma 9.1 will be carried out in a sequence of lemmas. For, let us introduce the
mappings *f _{k}*:

_{H → H}and

*g*:

_{k}_{H → L}

_{2}(

*Hµ*,H)defined by

*f _{k}*(

*u*,

*v*) =

**1**

_{B}*m*

0

*f*0(_{·},*u*)(*ζk*∗*v*) +

P*d*
*i*=1*f*

*i*_{(}

·,*u*)(*ζk*∗*uxi*) + *f*

*d*+1_{(}

·,*u*)

,

*g _{k}*(

*u*,

*v*) =

**1**

_{B}*m*

0

*g*0(_{·},*u*)(*ζk*∗*v*) +

P*d*
*i*=1*g*

*i*_{(}

·,*u*)(*ζk*∗*uxi*) +*g*

*d*+1_{(}

·,*u*)

*ξ*, *ξ*∈*H _{µ}*.

**Lemma 9.2.** *For every k* _{∈} N*, there exists a completely filtered stochastic basis* (Ω*k*,_{F}*k*,P*k*) *with a*
*spatially homogeneous*(_{F}* _{t}k*)

*-Wiener process Wk*

*with spectral measureµand an*(

_{F}

*)*

_{t}k*-adapted process*

*zk*= (

*uk*,

*vk*)

*with*H

*-continuous paths such thatν*

*is the law of zk*(0)

*under*P

*k*

*and*

*zk*(*t*) =*S _{t}mzk*(0) +

Z *t*

0

*Sm _{t}*

_{−}

*(*

_{s}fk*zk*(

*s*))

*ds*+

Z *t*

0

*S _{t}m*

_{−}

*(*

_{s}gk*zk*(

*s*))

*dWsk*,

*t*≥0.

*Moreover, for every p*∈[2,_{∞})*, there exists a constant K _{p}*(1

_{,}

*)=*

_{l}*K*(

_{l}_{,}1

*)*

_{m}_{,}

_{g}_{,}

_{f}_{,}

_{p}_{,}

_{ν}_{,}

_{c}_{,}

_{a}*such that*

E*k* sup
*s*∈[0,*l*]k

*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*

E*k*k*vk*(*t*)_{k}2*q*

*Cγ*_{([}_{0,}_{l}_{]}_{;}* _{W}*−1,2

_{)}≤

*K*(2)

*q*,*l*, *k*,*l*∈N. (9.3)

*Proof.* The mappings *f _{k}* :

_{H → H}and

*g*:

_{k}_{H → L}

_{2}(

*H*,

_{µ}_{H}) are Lipschitz on bounded sets and have at most linear growth hence there exists a completely filtered stochastic basis(Ω

*k*,

_{F}

*k*,P

*k*)with a spatially homogeneous(

_{F}

*k*

*t*)-Wiener process *Wk* with spectral measure *µ* and an (F*tk*)-adapted
process*zk*with_{H}-continuous paths such that*ν* is the law of*zk*(0)underP*k*and

*zk*(*t*) =*S _{t}mzk*(0) +

Z *t*

0

*Sm _{t}*

_{−}

*(*

_{s}fk*zk*(

*s*))

*ds*+

Z *t*

0

*Sm _{t}*

_{−}

*(*

_{s}gk*zk*(

*s*))

*dWsk*,

*t*≥0

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

〈G*mz*,*z*〉_{Dom}_{(}_{I}

−A*m*_{)}1_{2}_{⊕}* _{L}*2≤

1
2k*z*k

2

Dom(*I*−A*m*_{)}12_{⊕}* _{L}*2,

*z*∈DomG

*m*

hence the square norm of the local solution*zk*2

Dom(*I*−A*m*_{)}1_{2}_{⊕}* _{L}*2 cannot explode in finite time and

so *zk* is a global solution in the sense of (4.3) by the Chojnowska-Michalik theorem (see [9] or
Theorem 12 in[33]).

By Proposition 8.1 applied on *T* *>* 0, *x* = 0, *l*(*r*) = log(1+*rp*) for *p* _{∈} [2,_{∞}), *F*(*y*) =_{|}*y*_{|}2*/*2,
*λ*0=sup*w*∈R*d*k**a**(*φm*(*w*))k

1

2, with the notation**F*** _{T}* =

**F**

_{m}_{,}

_{λ}0,0,*T* and

**F**_{∞}(*u*,*v*) =1

2

Z

R*d*

( * _{d}*
X

*i*=1
*d*

X

*l*=1

(**a***il*◦*φm*)

¬
*uxi*,*uxl*

¶

R*n*+|*v*|
2
R*n*+|*u*|

2
R*n*

)
*d y*,

there is

*l*(**F*** _{T}*(

*t*,

*zk*(

*t*)))

_{≤}

*l*(

**F**

*(0,*

_{T}*zk*(0))) +

*M*,

_{T}*k*(

*t*)− 1

2〈*MT*,*k*〉(*t*)

+ **c**

2

2

Z *t*

0

*p*(*p*_{−}1)**F*** _{T}p*−2(

*s*,

*zk*(

*s*))

1+**F***p _{T}*(

*s*,

*zk*

_{(}

_{s}_{))}k

*v*

*k*

_{(}

_{s}_{)}

k2* _{L}*2

_{(}

_{B}*T*−λ0*s*)k*g*(·,*u*

*k*_{(}_{s}_{)}_{,}_{v}k_{(}_{s}_{)}_{,}

∇*uk*(*s*))_{k}2* _{L}*2

_{(}

_{B}_{m∩}_{B}*T*−λ0*s*)*ds*

+ **c**

2

2

Z *t*

0

*l*′(**F*** _{T}*(

*s*,

*zk*(

*s*)))

_{k}

*g*(

_{·},

*uk*(

*s*),

*vk*(

*s*),

_{∇}

*uk*(

*s*))

_{k}2

*2*

_{L}_{(}

_{B}_{m∩}_{B}*T*−λ0*s*)*ds*

+

Z *t*

0

*l*′(**F*** _{T}*(

*s*,

*zk*(

*s*)))

_{k}

*vk*(

*s*)

_{k}

*2*

_{L}_{(}

_{B}*T*−λ0*s*)k*u*

*k*_{(}_{s}_{)}

k*L*2_{(}_{B}

*T*−λ0*s*)*ds*

+

Z *t*

0

*l*′(**F*** _{T}*(

*s*,

*zk*(

*s*)))

_{k}

*vk*(

*s*)

_{k}

*2*

_{L}_{(}

_{B}*T−λ*0*s*)k*f*(·,*u*

*k*_{(}_{s}_{)}_{,}_{v}k_{(}_{s}_{)}_{,}

∇*uk*(*s*))_{k}* _{L}*2

_{(}

_{B}_{m∩}_{B}*T
*