BSDE and generalized Dirichlet forms: the finite dimensional case

0.4 MB·English

by  Rongchan Zhu

BSDE and generalized Dirichlet forms:
the finite dimensional case
RONG-CHAN ZHUa,b
a. Institute of Applied Mathematics, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100190, China
b. Department of Mathematics, University of Bielefeld, D-33615 Bielefeld, Germany
2 E-mail: [email protected]
1
0
2

n
Abstract
a
J
We consider the following quasi-linear parabolic system of backward partial differential
6
1 equations
] (∂t +L)u+f(·,·,u,∇uσ) = 0 on [0,T]×Rd uT = φ,
R
whereLisapossiblydegeneratesecondorderdifferentialoperatorwithmerelymeasurable
P
coefficients. We solve this system in the framework of generalized Dirichlet forms and
.
h
employthestochasticcalculusassociatedtotheMarkovprocesswithgeneratorLtoobtain
t
a a probabilistic representation of the solution u by solving the corresponding backward
m
stochastic differential equation. The solution satisfies the corresponding mild equation
[
which is equivalent to being a generalized solution of the PDE. A further main result is

1 the generalization of the martingale representation theorem using the stochastic calculus
v
associated to the generalized Dirichlet form given by L. The nonlinear term f satisfies a
2
8 monotonicity condition with respect to u and a Lipschitz condition with respect to u.
∇
1
3
Keywords: backward stochastic differential equations, quasi-linear parabolic partial differential equations,
.
1
Dirichlet forms, generalized Dirichlet forms, Markov processes, martingale representation
0
2
1
: 1 Introduction
v
i
X
Consider the following quasi-linear parabolic system of backward partial differential equations
r
a
(∂ +L)u+f( , ,u, uσ)= 0 on [0,T] Rd u = φ, (1.1)
t T
· · ∇ ×
whereLisasecondorderlineardifferentialoperatorandf ismonotoneinuandLipschitzin u.
∇
If L has sufficiently regular coefficients there is a well-known theory, to obtain a probabilistic
representation of the solutions to (1.1), using corresponding backward stochastic differential
equations (BSDE) and also to solve BSDE with the help of (1.1), originally due to E. Pardoux
and S. Peng ([17]). The main aim of this paper is to implement this approach for a very general
class of linear operators L, which are possibly degenerate, have merely measurable cofficients
and are in general not symmetric. Solving (1.1) for such general L is the first main task of this
paper. The second main contribution is to prove the martingale representation theorem for the
underlying reference diffusion generated by such general operators L.
0Research supported by 973 project, NSFC, key Lab of CAS, the DFG through IRTG 1132 and CRC 701
and the I.Newton Institute, Cambridge, UK
1
If f and the coefficients of the second-order differential operator L are sufficiently smooth,
the PDE has a classical solution u. Consider Ys,x := u(t,Xs,x), Zs,x := uσ(t,Xs,x) where
t t t ∇ t
Xs,x,s t T, is the diffusion process with infinitesimal generator L which starts from x at
t ≤ ≤
time s. Then, using Itˆo’s formula one checks that (Ys,x,Zs,x) solves the BSDEs
t t s≤t≤T
T T
Ys,x = φ(Xs,x)+ f(r,Xs,x,Ys,x,Zs,x)dr Zs,xdB . (1.2)
t T Z r r r −Z r r
t t
Conversely, bystandardmethodsonecanprovethat(1.2)hasauniquesolution(Ys,x,Zs,x)
t t s≤t≤T
andthenu(s,x) := Ys,x isasolutiontoPDE(1.1). BSDEshavebeenintroduced(intheiractual
s
form) by Pardoux and Peng [17] and found applications in stochastic control and mathematical
finance. If f and the coefficients of L are Lipschitz continuous then a series of papers (e.g. [2],
[16]andthe reference therein) prove thatthe aboverelationbetween PDE(1.1)andBSDE(1.2)
remains true, if one considers viscosity solutions to PDE (1.1). In both these approaches, since
the coefficients are Lipschitz continuous, the Markov process X with infinitesimal operator L is
a diffusion process which satisfies an SDE and so one may use its associated stochastic calculus.
In [2] Bally, Pardoux and Stoica consider a semi-elliptic symmetric second-order differential
operator L ( which is written in divergence form ) with measurable coefficients. They prove
that the above system of PDE has a unique solution u in some functional space. Then using
the theory of symmetric Dirichlet forms and its associated stochastic calculus, they prove that
the solution Ys.x of the BSDE yields a precised version of the solution u so that, moreover, one
has Ys,x = u(t,X ),Px-a.s. In [21], the analytic part of [2] has been generalized to a non-
t t−s
symmetric case with L satisfying the weak sector condition. Here the weak sector condition
means
((1 L)u,v) K((1 L)u,u)1/2((1 L)v,v)1/2, for u,v (L),
− ≤ − − ∈ D
for some constant K > 0. A.Lejay ([13]) and L. Stoica ([24]) consider the generator L =
1 d ∂ (a ∂ ) + d b (x) ∂ for bounded a,b. In their case, the operator L generates
2 i,j=1 ∂xi ij∂xj i=1 i ∂xi
aPsemigroup (P ), whicPh possesses continuous densities satisfying Aronson’s estimate. In [28],
t
T.S. Zhang and Q.K.Ran (see also [27]) consider L of a more general form, but a = (a ) is
ij
required to be uniformly elliptic and b Lp for p > d. Anyway, since L satisfies the weak
∈
sector condition in this case, it generates a sectorial ( i.e. a small perturbation of a symmetric)
Dirichlet form, so the theory of Dirichlet forms from [14] can be applied in [27], [28].
In [23] Stannat extends the known framework of Dirichlet forms to the class of generalized
Dirichlet forms. By this we can analyze differential operators where the second order part may
be degenerate and at the same time the first order part may be unbounded satisfying no global
Lp-condition for p d. The motivation for this paper is to extend the results in [2] to the case,
≥
where L generates a generalized Dirichlet form so that we can allow the coefficients of L to be
more general.
In this paper, we consider PDE (1.1) for a non-symmetric second order differential operator
L, which is associated to the bilinear form
d
∂u ∂v
(u,v) := a (x) (x) (x)m(dx)+ c(x)u(x)v(x)
E Z ij ∂xi ∂xj Z
iX,j=1
(1.3)
d d
∂u
+ a (x)(b (x)+ˆb (x)) v(x)m(dx) u,v C∞(Rd).
Z ij j j ∂xi ∀ ∈ 0
Xi=1 Xj=1
2
for u,v C∞(Rd), where C∞(Rd) denotes the space of infinitely differentiable functions with
∈ 0 0
compact support. We stress that (a ) is not necessarily assumed to be (locally) strictly positive
ij
definite, but may be degenerate in general. When b 0, the bilinear form satisfies the weak
≡ E
sector condition. In the analytic part, the condition we assume for the perturbation term given
by b is bσ L2(Rd;Rd,m). Here σσ∗ = a and σ∗ is the transpose of the matrix of σ. In the
∈
probabilistic part, we have more general condition on b (i.e. b Lp , for p > d see Example
∈ loc
5.6). That implies that we do not have the weak sector condition for the bilinear form. We use
the theory of generalized Dirichlet forms and its associated stochastic calculus ( cf [22, 23] [25,
26]) to generalize the results in [2]. Here m is a finite measure or Lebesgue measure on Rd. If
D is a bounded open domain, we choose m as 1 (x)dx. Then in certain cases the solution of
D
PDE (1.1) satisfies the Neumann boundary condition. If we replace C∞(Rd) by C∞(D), the
0 0
solution of PDE (1.1) satisfies the Dirichlet boundary condition.
In the analytic part of our paper, we do not need to be a generalized Dirichlet form.
E
We start from a semigroup (P ) satisfying conditions (A1)-(A4), specified in Section 2 below.
t
Such a semigroup can, however, be constructed from a generalized Dirichlet form. It can also
be constructed by other methods (see e.g. [11]). Under conditions (A1)-(A4), the coefficients
of L may be quite singular and only very broad assumptions on a and b are needed (see the
examples in Section 4 and Section 5, which to the best of our knowledge could not be covered
by other results).
The paper is organized as follows. In Sections 2 and 3, we use functional analytical methods
to solve PDE (1.1) in the sense of Definition 2.3, i.e. there are sequences un which are strong
{ }
solutions with data (φn,fn) such that
un u 0, φn φ 0, lim fn = f in L1([0,T];L2).
T 2
k − k → k − k → n→∞
Here := (sup 2+ T a,ˆb ( )dt)1/2, where a,ˆb is the summand in the left hand side
k·kT t≤T k·k2 0 Ec2+1 · E
of (1.3) with b 0. The abovRe definition for the solution is equivalent to that of the following
≡
mild equation in L2-sense
T
u(t,x) = P φ(x)+ P f(s, ,u ,D u )(x)ds.
T−t s−t s σ s
Z ·
t
If we use the definition of weak solution to define our solution as in [2], uniqueness of the
solution can not be obtained since only bσ L2(Rd;m). Furthermore, the function f in PDE
| | ∈
(1.1) need not be Lipschitz continuous with respect to the third variable; monotonicity suffices.
And µ which appears in the monotonicity conditions (see condition (H2) in Section 3.2 below)
can depend on t. f is, however, assumed to be Lipschitz continuous with respect to the last
variable. We emphasize that the first order term of L cannot be incorporated into f without
the condition that b is bounded. Hence we are forced to take it as part of L and hence have
to consider a diffusion process X in (1.2) which is generated by an operator L which is the
generator of a (in general non-sectorial) generalized Dirichlet form. We also emphasize that
under our conditions, PDE (1.1) cannot be tackled by standard monotonicity methods (see
e.g. [3]) because of the lack of a suitable Gelfand triple V H V∗ with V being a reflexive
⊂ ⊂
Banach space.
In Section 4, we extend the stochastic calculus of generalized Dirichlet forms in order to
generalize the martingale representation theorem. In order to treat BSDE, we show in Theorem
3
4.8 that there exists a set of null capacity outside of which the following representation
N
theorem holds : for every bounded -measurable randomvariable ξ, there exists a predictable
∞
F
process (φ ,...,φ ) : [0, ) Ω Rd, such that for each probability measure ν, supported by
1 d
∞ × →
Rd , one has
\N
d ∞
ξ = Eν(ξ )+ φidM(i) Pν a.s..
|F0 Z s s −
Xi=0 0
As a result, one can choose the exceptional set such that if the process X starts from a point
N
of c, it remains always in this set. As a consequence we deduce the existence of solutions for
N
the BSDE using the existence for PDE (1.1) in the usual way, however, only under Pm, because
of our general coefficients of L (c.f. Theorem 4.12).
In Section 5, we employ the martingale representation to deduce existence and uniqueness
for the solutions of BSDE (1.2). Moreover, we give Example 5.6 satisfying Assumption (A5),
which to the best of our knowledge could not be treated by other techniques. As a consequence,
in Theorem 5.7, existence of solutions for PDE (1.1), not covered by our analytic results in
Section 2, is obtained by u(s,x) = Ys, where Ys is the solution of the BSDE. Moreover we
s t
have, Ys = u(t,X ),Px-a.s., x Rd .
t t−s ∈ \N
2 Preliminaries
Let σ : Rd Rd Rk be a measurable map. Then there exists a measurable map τ : Rd
7→ ⊗ 7→
Rk Rd such that
⊗
στ = τ∗σ∗, τσ = σ∗τ∗, στσ = σ.
(see e.g. [2, Lemma A.1]). Here σ∗ is the transpose of the matrix of σ. Then a := σσ∗ =
(a ) takes values in the space of symmetric non-negative definite matrices. Let also
ij 1≤i,j≤d
b : Rd Rd be measurable. Assume that the basic measure m(dx) for the generalized Dirichlet
→
form, to be defined below, is a finite measure or Lebesgue measure on Rd.
Denote the Euclidean norm and the scalar product in Rd by , , respectively, while
| · | h· ·i
on the space of matrices Rd Rk we use the trace scalar product and its associated norm,
⊗
i.e., for z = (z ) Rd Rk, z ,z = trace(z z∗), z = ( d k z2)1/2. Let L2, L2(Rd;Rk)
ij ∈ ⊗ h 1 2i 1 2 | | i=1 j=1 ij
denote L2(Rd,m), L2(Rd,m;Rk) respectively. And ( , ) dePnotesPthe L2-inner product. And for
· ·
1 p , denotes the usual norm in Lp(Rd;m). If W is a function space, we will use
p
≤ ≤ ∞ k·k
bW to denote the bounded function in W.
Furthermore, let a , d a b , d a ˆb L1 (Rd,m) and c L1 (Rd,R+;m). We
ij j=1 ij j j=1 ij j ∈ loc ∈ loc
introduce the bilinear forPm P
d
∂u ∂v
(u,v) := a (x) (x) (x)m(dx)+ c(x)u(x)v(x)
E Z ij ∂xi ∂xj Z
iX,j=1
d d
∂u
+ a (x)(b (x)+ˆb (x)) v(x)m(dx) u,v C∞(Rd).
Z ij j j ∂xi ∀ ∈ 0
Xi=1 Xj=1
Consider the following conditions:
4
(A1) The bilinear form
d
∂u ∂v
a(u,v) = a (x) (x) (x)m(dx) u,v C∞(Rd),
E Z ij ∂xi ∂xj ∀ ∈ 0
iX,j=1
is closable on L2(Rd,m).
Define a( , ) := a( , )+ ( , ). The closure of C∞(Rd) with respect to a is denoted by
E1 · · E · · · · 0 E1
Fa. Then ( a,Fa) is a well-defined symmetric Dirichlet form on L2(Rd,m).
E
For the bilinear form
d
∂u ∂v
a,ˆb(u,v) : = a (x) (x) (x)m(dx)+ c(x)u(x)v(x)
E Z ij ∂xi ∂xj Z
iX,j=1
d d
∂u
ˆ
+ a (x)b (x) v(x)m(dx),
Z ij j ∂xi
Xi=1 Xj=1
we consider the following conditions:
(A2) There exists a constant c 0 such that a,ˆb( , ) := a,ˆb( , )+c ( , ) is a coercive closed
2 ≥ Ec2 · · E · · 2 · ·
form (see e.g. [14]), and there exist constants c ,c > 0 such that for u C∞(Rd)
1 3 ∈ 0
c a(u,u) a,ˆb(u,u), (2.1)
1E ≤ Ec2
and
cu2dm c ( a(u,u)+ u 2). (2.2)
Z ≤ 3 E k k2
Denote ˜a,ˆb(u,v) := a,ˆb(u,v)+ a,ˆb(v,u). The closure of C∞(Rd) with respect to ˜a,ˆb is
E E E 0 Ec2+1
denoted by F. By (2.1) we have F Fa. And for u F (2.1) and (2.2) are satisfied.
⊂ ∈
(A3) bσ L2(Rd;m) and there exists α 0 such that
| | ∈ ≥
bσ,( u2)σ dm α u 2 u C∞(Rd). (2.3)
Z h ∇ i ≥ − k k2 ∈ 0
(A4) There exists a positivity preserving C -semigroup P on L1(Rd;m) such that for any
0 t
t [0,T], C > 0 such that
T
∈ ∃
P f C f .
t ∞ T ∞
k k ≤ k k
Then for 0 t T, P extends to a semigroup on Lp(Rd;m) for all p [1, ) by the Riesz-
t
≤ ≤ ∈ ∞
Thorin Interpolation Theorem (denoted by P for simplicity) which is strongly continuous on
t
Lp(Rd;m). We denote its L2-generator by (L, (L)) and assume that b (L) bF, and for any
D D ⊂
u bF there exists uniformly bounded u (L) such that ˜a,ˆb (u u) 0 and that it is
∈ n ∈ D Ec2+1 n − →
associated with the bilinear form in the sense that (u,v) = (Lu,v) for u,v b (L).
E − ∈ D
WeemphasizethatincontrasttopreviousworkP in(A4)isnolongeranalyticonL2(Rd;m).
t
By (A4) there exist constants M ,c such that
0 0
P f M ec0t f , f L2(Rd;m). (2.4)
t 2 0 2
k k ≤ k k ∀ ∈
5
To obtain a semigroup P satisfying the above conditions, we can use generalized Dirichlet
t
form. Let us recall the definition of a generalized Dirichlet form from[23]. Let E be a Hausdorff
topological space and assume that its Borel σ-algebra (E) is generated by the set C(E) of all
B
continuous functions on E. Let m be a σ-finite measure on (E, (E)) such that := L2(E,m)
B H
is a separable (real) Hilbert space. Let ( , ) be a coercive closed form on in the sense of
A V H
[14]. We will always denote the corresponding norm by . Identifying with its dual ′
V
k·k H H
we obtain that = ′ ′ densely and continuously.
∼
V → H H → V
Let (Λ,D(Λ, )) be a linear operator on satisfying the following assumptions:
H H
(i) (Λ,D(Λ, )) generates a C -semigroup of contractions (U ) on .
0 t t≥0
H H
(ii) is Λ-admissible, i.e. if (U ) can be restricted to a C -semigroup on .
t t≥ 0
V V
Let (Λ, ) with corresponding norm be the closure of Λ : D(Λ, ) ′ as an
F
F k ·k H ∩ V → V
operator from to ′ and (Λˆ, ˆ) its dual operator.
V V F
Let
(u,v) Λu,v if u ,v
(u,v) = A −h i ∈ F ∈ V
E (cid:26) (u,v) Λˆv,u if u ,v ˆ,
A −h i ∈ V ∈ F
Here , denotes the dualization between ′ and and , coincides with the inner product
h· ·i V V h· ·i
( , ) in H when restricted to H . And (u,v) := (u,v)+α(u,v) for α > 0. We call
H α H
· · ×V E E E
the bilinear form associated with ( , ) and (Λ,D(Λ, )). If
A V H
u u+ 1 and (u,u u+ 1) 0
∈ F ⇒ ∧ ∈ V E − ∧ ≥
then the bilinear form is called a generalized Dirichlet form. If the adjoint semigroup (Uˆ ) of
t t≥0
(U ) can also be restricted to a C -semigroup on . Let (Λˆ,D(Λˆ, )) denote the generator
t t≥0 0
V H
of (Uˆ ) on , ˆ(u,v) := (v,u),u,v and let the coform ˆ be defined as the bilinear
t t≥0
H A A ∈ V E
ˆ ˆ ˆ
form associated with ( , ) and (Λ,D(Λ, )).
A V H
Remark 2.1 (i) Some general criteria imposing conditions on a in order that a be closable
E
are e.g. given in [12, Section 3.1] and [14, Chap II, Section 2].
(ii) There are examples considered in [14, Chap. II, Subsection 2d] satisfying (A2). Assume
the Sobolev inequality
u C( a(u,u)+ u 2)1/2, u C∞(Rd),
k kq ≤ E k k2 ∀ ∈ 0
is satisfied, where 1 + 1 = 1 and denotes the usual norm in Lq. If ˆbσ Ld(Rd;m) +
q d 2 k · kq | | ∈
L∞(Rd;m) and c Ld/2(Rd;m)+L∞(Rd;m), then (A2) is satisfied. In [28] they consider the
∈
bilinear form Q(u,v) = a,ˆb(u,v)+ d (x), v(x) u(x)dm, here d Lq(Rd),q > d. In their
1 1
E h ∇ i ∈
case, the result for the existence of tRhe nonlinear PDE can be obtained by [18, Theorem 4.2.4]
since the nonlinear part is Lipschitz in u and u. In our case, we have more general conditions
∇
on b and f, so that we can not find a suitable Gelfand triple V H V∗ with V being a
⊂ ⊂
reflexive Banach space and use monotonicity methods as in [18].
(iii) We can construct a semigroup P satisfying (A4) by the theory of generalized Dirichlet
t
form. More precisely, suppose there exists a constant cˆ 0 such that ( , ) := ( , )+cˆ( , )
cˆ
≥ E · · E · · · ·
is a generalized Dirichlet form with domain in one of the following three senses:
F ×V
(a)(E, (E),m) = (Rd, (Rd),m),
B B
( a,ˆb,F) = ( , ),
Ec2 A V
6
Λu,v (cˆ c )(u,v) = d d a (x)b (x)∂uv(x)m(dx) for u,v C∞(Rd);
−h i− − 2 i j=1 ij j ∂xi ∈ 0
(b)(E, (E),m) = (Rd, (RPd),mR)P,
B B
0 and = L2(Rd,m),
A ≡ V
Λu,v = (u,v) for u,v C∞(Rd) and C∞(Rd) (L);
−h i Ecˆ ∈ 0 0 ⊂ D
(c) = , Λ 0 (In this case ( , ) is a sectorial Dirichlet form in the sense of [14]).
cˆ cˆ
E A ≡ E V
Then there exists a sub-Markovian C -semigroup of contractions Pcˆ associated with the
0 t
generalized Dirichlet form . Define P := ecˆtPcˆ. If it is a C -semigroup on L1 then it satisfies
Ecˆ t t 0
(A4). Then we have
(L) F.
D ⊂ F ⊂
(iv) The semigroup can be also constructed by other methods. (see e.g. [11], [4], [6]).
(v) By (A3) we have that is positivity preserving i.e.
E
(u,u+) 0 u (L),
E ≥ ∀ ∈ D
which can be obtained by the same arguments as [23, Proposition 4.4].
(vi) The condition that for any u bF there exists uniformly bounded u (L) such that
n
∈ ∈ D
˜a,ˆb (u u) 0 is satisfied if C∞(Rd) (L). It can also be satisfied in the case of (iii) by
Ec2+1 n − → 0 ⊂ D
the theory of generalized Dirichlet form.
(vii) All the conditions are satisfied by the bilinear form considered in [11], [13], [22, Section
1 (a)] and the following example which is considered in [23].
(viii) The notion of quasi-regularity for generalized Dirichlet forms analogously to [14] has
been introduced in [23]. By this and a technical assumption an associated m-tight special
standard process can be constructed. We will use stochastic calculus associated with this
process to conclude our probabilistic results (see Section 4 below).
Example 2.2 Let b L2(Rd;dx), 1 i d. Consider the bilinear form
i
∈ ≤ ≤
d d
∂u ∂v ∂u
(u,v) := dx bi vdx;u,v C∞(Rd)
E Z ∂x ∂x − Z ∂x ∈ 0
iX,j=1 Rd i j Xi=1 i
Assume there exist constants c,L 0 such that
≥
b, u dx 2c u for all u C∞(Rd),u 0,
Z h ∇ i ≤ k k1 ∈ 0 ≥
d
∂u
b dxh h L u h 2,
i i j 1
− Z ∂x ≤ k k | |
iX,j=1 j
for all u C∞(Rd),u 0,h Rd,
∈ 0 ≥ ∈
(or equivalently, b is quasi-monotone, i.e.
b(x) b(y),x y L x y 2, x,y Rd,)
h − − i ≤ | − | ∀ ∈
and for some continuous, monotone increasing function f : [0, ) [1, ) with ∞ dr =
∞ → ∞ 0 f(r) ∞
we have that R
b(x) f( x ),x Rd.
| | ≤ | | ∈
7
Then in [23, Subsection II.2] it is proved that there exists a generalized Dirichlet form in L2(Rd)
extending . We denote the semigroup associated with by Pc. If we define P := ectPc,
Ec Ec t t t
then it is the semigroup associated with . By the computation in [23, Subsection II.2], P is
t
E
sub-Markovian. So it satisfies the conditions (A1)-(A4).
Further examples are presented in Section 4 (see Examples 4.2 and 4.3) and Section 6.
ThenweusethesamenotationFˆ, , associatedwith a,ˆb asin[2]: = C1([0,T];L2)
T T T
C k·k E C ∩
L2([0,T];F), which turns out to be the appropriate space of test functions, i.e.
T
= ϕ : [0,T] Rd R ϕ F for almost each t, a,ˆb(ϕ )dt < ,
T t t
C { × → | ∈ Z E ∞
0
t ϕ is differentiable in L2and t ∂ ϕ is L2 continuous on [0,T] .
t t t
→ → − }
Here and below we set a,ˆb(u) for a,ˆb(u,u). We also set = C1([a,b];L2) L2([a,b];F).
[a,b]
E E C ∩
For ϕ , we define
T
∈ C
T
ϕ := (sup ϕ 2 + a,ˆb(ϕ )dt)1/2.
k kT k tk2 Z Ec2 t
t≤T 0
Fˆ is the completion of with respect to . By [2], Fˆ = C([0,T];L2) L2(0,T;F). We
T T
define the space Fˆa w.rC.t. a analogous tokFˆ·.kThen we have Fˆ Fˆa. We ∩also introduce the
E1 ⊂
following space
W1,2([0,T];L2(Rd)) = u L2([0,T];L2);∂ u L2([0,T];L2) ,
t
{ ∈ ∈ }
where ∂ u is the derivative of u in the weak sense (see e.g. [3]).
t
2.1 Linear Equations
Consider the linear equation
(∂ +L)u+f = 0, 0 t T,
t
≤ ≤ (2.5)
u (x) = φ(x), x Rd,
T
∈
where f L1([0,T];L2),φ L2.
∈ ∈
As in [2] we set D ϕ := ( ϕ)σ for any ϕ C∞(Rd), define V = D ϕ : ϕ C∞(Rd) , and
σ ∇ ∈ 0 0 { σ ∈ 0 }
let V be the closure of V in L2(Rd;Rk). Then we have the following results:
0
Proposition 2.3 Assume (A1)-(A3) hold. Then:
(i) For every u Fa there is a unique element of V, which we denote by D u, such that
σ
∈
a(u) = D u(x),D u(x) m(dx).
σ σ
E Z h i
(ii) Furthermore, if u Fˆa, then there exists a measurable function φ : [0,T] Rd Rd
∈ × 7→
such that φσ L2((0,T) Rd) and D u = φ σ for almost all t [0,T].
σ t t
(iii)Let| un|,∈u Fˆa be×such that un u in L2((0,T) Rd)∈and (D un) is Cauchy in
σ n
∈ → ×
L2([0,T] Rd;Rk). Then D un D u in L2((0,T) Rd;Rk), i.e. D is closed as an operator
σ σ σ
from Fˆa i×nto L2((0,T) Rd). → ×
×
8
Proof See [2, Proposition 2.3]. (cid:3)
For u F,v bF, we define
∈ ∈
(u,v) := a,ˆb(u,v)+ bσ,D u vm(dx).
σ
E E Z h i
Notation We denote by ˜u the set of all measurable functions φ : Rd Rd such that
∇ →
φσ = D u as elements of L2(Rd,Rk).
σ
2.2 Solution of the Linear Equation
We recall the following standard notions.
Definition 2.4 (strong solutions) A function u Fˆ L1((0,T); (L)) is called a strong
∈ ∩ D
solution of equation (2.5) with data φ,f, if t u = u(t, ) is L2-differentiable on [0,T],∂ u
t t t
7→ · ∈
L1((0,T);L2) and the equalities in (2.5) hold m-a.e..
Definition 2.5 (generalized solutions) A function u Fˆ is called a generalized solution of
∈
equation (2.5), if there are sequences un which are strong solutions with data (φn,fn) such
{ }
that
un u 0, φn φ 0, lim fn = f in L1([0,T];L2).
T 2
k − k → k − k → n→∞
Remark Here we use the definition of a generalized solution and prove it is equivalent to
a mild solution in Proposition 2.9. If we use the definition of weak solution to define our
solution as in [2], uniqueness of the solution in Proposition 2.9 can not be obtained since only
bσ L2(Rd;m).
| | ∈
Proposition 2.6 Assume (A3)-(A4) hold.
(i) Let f C1([0,T];Lp) for p [1, ). Then w := T P f ds C1([0,T];Lp), and
∈ ∈ ∞ t t s−t s ∈
∂ w = P f + T P ∂ f ds. R
t t − T−t T t s−t s s
(ii) Assume thaRt φ (L), f C1([0,T];L2) and for each t [0,T], f (L) . Define
t
∈ D ∈ ∈ ∈ D
u := P φ+ T P f ds.Thenuisastrongsolutionof(2.5)and, moreover, u C1([0,T];L2).
t T−t t s−t s ∈
Proof By theRsame arguments as in [2, Proposition 2.6]. (cid:3)
Remark 2.7 Here in (ii) we add the assumption φ (L) and f (L), t [0,T], as we
t
∈ D ∈ D ∈
can not deduce P φ (L) for φ L2, since (P ) might not be analytic.
t t
∈ D ∈
Proposition 2.8 Suppose (A4) holds. If u is a strong solution for (2.5), it is a mild solution
T
for (2.5) i.e. u = P φ+ P f ds.
t T−t t s−t s
Proof Forfixedt,ϕ (LˆR),(u ,Pˆ ϕ) (u ,ϕ) = T( Lu f ,Pˆ ϕ)ds+ T(u ,LˆPˆ ϕ)ds.
∈ D T T−t − t t − s− s s−t t s s−t
Here Lˆ,Pˆ denote the adjoints on L2(Rd,m) of L,PR respectively. As u is aRstrong solution,
t t
we can deduce that (u ,ϕ) = (P φ + T P f ds,ϕ). Since (Lˆ) is dense in L2, the result
t T−t t s−t s D (cid:3)
follows. R
9
Proposition 2.9 Assume that conditions (A1)-(A4) hold, f L1([0,T];L2) and φ L2.
Then equation (2.5) has a unique generalized solution u Fˆ and∈ ∈
∈
T
u = P φ+ P f ds. (2.6)
t T−t s−t s
Z
t
The solution satisfies the three relations:
T T T
u 2 +2 a,ˆb(u )ds 2 (f ,u )ds+ φ 2+2α u 2ds, 0 t T. (2.7)
k tk2 Z E s ≤ Z s s k k2 Z k sk2 ≤ ≤
t t t
T
u 2 M ( φ 2 +( f dt)2). (2.8)
k kT ≤ T k k2 Z k tk2
0
T T
((u ,∂ ϕ )+ a,ˆb(u ,ϕ )+ bσ,D u ϕ dm)dt = (f ,ϕ )dt+(φ,ϕ ) (u ,ϕ ), (2.9)
Z t t t E t t Z h σ ti t Z t t T − t0 t0
t0 t0
for any ϕ b ,t [0,T]. M is a constant depending on T. (2.9) can be extended easily for
T 0 T
∈ C ∈
ϕ bW1,2([0,T];L2) L2([0,T];F).
∈Moreover, if u F∩ˆ is bounded and satisfies (2.9) for any ϕ b with bounded f,φ, then
T
∈ ∈ C
u is a generalized solution given by (2.6).
Proof [Existence] Define u by (2.6). First assume that φ,f are bounded and satisfy the
conditionsofProposition2.6(ii). Then, since uisboundedandbyProposition2.6weknowthat
u is a strong solution of (2.5), hence it obviously satisfies (2.9). Furthermore, u C1([0,T];L2).
∈
Hence, actually u b and consequently,
T
∈ C
T T
((u ,∂ u )+ a,ˆb(u ,u )+ bσ,D u u dm)ds = (f ,u )ds+(φ,u ) (u ,u ).
s t s s s σ s s s s T t t
Z E Z h i Z −
t t
By (2.3) we have bσ,D u u dm α u 2. Hence
h σ si s ≥ − k sk2
R
T T T
u 2 +2 a,ˆb(u )ds 2 (f ,u )ds+ φ 2 +2α u 2ds, 0 t T.
k tk2 Z E s ≤ Z s s k k2 Z k sk2 ≤ ≤
t t t
As
T T T
(f ,u )ds = ((f ,P φ)+(f , P f dr))ds
s s s T−s s r−s r
Z Z Z
t t s
T T T
M eT−t( φ f ds+ ( f f dr)ds),
0 2 s 2 s 2 r 2
≤ k k Z k k Z k k Z k k
t t s
and T u 2ds M ( φ 2+( T f dt)2), we obtain, u 2+ T a,ˆb(u )ds M ( φ 2+
t k sk2 ≤ T−t k k2 0 k tk2 k tk2 t E s ≤ T−t k k2
( T Rf dt)2). Hence, it follows Rthat R
0 k tk2
R
T
u 2 M ( φ 2 +( f dt)2). (2.10)
k kT ≤ T k k2 Z k tk2
0
Here M can change from line to line and is independent of f,φ. Now we will obtain the
T−t
result for general data φ and f. Let (fn) be a sequence of bounded function in C1([0,T];L2)
n∈N
10