In nite dimensional Langevin equations: Uniqueness and rate of convergence for nite dimensional approximations Sigurd Assing Fakultat fur Mathematik, Universitat Bielefeld Postfach 100131, 33501 Bielefeld, Germany e-mail:
[email protected]
Abstract
The paper deals with the in nite dimensional stochastic equation d = ( ) d + d driven by a Wiener process which may also cover stochastic partial di erential equations. We study a certain nite dimensional approximation of ( ) and give a qualitative bound for its rate of convergence to be high enough to ensure the weak uniqueness for solutions of our equation. Examples are given demonstrating the force of the new condition. X
B t; X
t
W
B t; X
KEY WORDS stochastic partial di erential equation, Girsanov theorem, weak uniqueness, martingale problem
Mathematics Subject Classi cation (1991): 60G30, 60H15
1 Introduction One of the most studied stochastic di erential equations is (?)
dX = B (t; X ) dt + dW
which goes back to P. Langevin, C.R. Acad. Sci. Paris 146 (1908), S. 350 . In our equation, B denotes a general measurable drift, for example B (t; X ) = b(Xt ) where
b : E ! E; E locally convex space; while dW is the stochastic It^o-di erential with respect to some Wiener process with a covariance given by the problem that the equation models. In di erence to the deterministic equation without the It^o-di erential dW , for singular drifts B , instaed of a strong solution X , one can often only construct a weak solution, that is a probability measure on some path space over E ; and, the question of weak uniqueness arises: Is this probability measure unique for a given initial distribution? In the case dim E = d where equation (?) is usually assumed to be driven by a d-dimensional Wiener process W , a good answer (cf. Prop.5.3.10 in [10] for example) based on the celebrated Girsanov theorem [4] is the following: For a given initial distribution there is at most one probability measure P on the path space C ([0; 1) ! Rd ) such that the coordinate process X satis es both (1)
P(f
ZT 0
kB (t; X )kRd dt < 1g) = 1; 8 T > 0; 2
The research of the author was supported by the DFG project \Interagierende stochastische Systeme von
hoher Komplexitat".
1
and
Zt
B (s; X ) ds; t 0; is a d-dimensional Wiener process. In the important in nite dimensional case dim E = 1 where the drift B may also cover Xt ? X0 ?
0
di erential operators on function spaces, an analogous result holds true. Indeed, if the driving Wiener process is white in time and has the inner product of a separable Hilbert space H as covariance in space where E 0 H 0 H E densely and continuously, then one just has to replace (1) by ZT (2) P(f kB (t; X )k2H dt < 1g) = 1; 8 T > 0; 0
assuming that an appropriate path space is chosen. Obviously, we can think about (1) as a version of (2) if we understand the inner product of R d as the \covariance in space" of a d-dimensional Wiener process. But in in nite dimensions, the condition (2) is useless for important applications, because B (t; X ) takes values in a space di erent from the Hilbert space H . Let us explain this phenomenon by means of an example which has recently moved again into sight of the research (cf. [14],[8],[12]). To start with, denote the Laplacian with xed boundary conditions (Dirichlet or Neumann) in a bounded open rectangle O R2 by and look at the simple equation dX = ? 21 (1 ? )X dt + dW (3) driven by space-time white noise, that is H = L2 (O). A good choice for E is the space of S generalized functions H ?1 = 2R H where H is the scale of Sobolev spaces generated by (1 ? ) and L2 (O). We know the unique explicit solution of the last equation given by
Xt = S (t)X0 +
Zt 0
where
S (t ? s) dWs ; t 0;
S (t) = expf? 21 t(1 ? )g; t 0: As a consequence, the solution above starting from X0 0 is an H ? -valued process, > 0 arbitrary but xed, such that
kXt kH = kXt kL
O) =
2(
1; t 0; a.s.;
because the semigroup of the Laplacian in 2 dimensions is not smoothing enough to de ne an H -valued stochastic integral with respect to W (cf. [15]). Hence, for b = ? 21 (1 ? ) we also have kB (t; X )kH = kb(Xt )kH = 1; t 0; a.s.; and, (2) is useless or is not satis ed. Fortunately, here we do not need the condition (2) to decide the question of uniqueness for the equation (3). Even more, because of the outstanding properties of its explicit solution above, one can consider perturbations of the equation (3), that is b = ? 21 (1 ? ) + ; getting the result (cf. [7],[13]): For a given initial distribution, there is at most one probability measure P on the corresponding path space such that the coordinate process X satis es both (4)
P(f
ZT 0
k (Xt )kH dt < 1g) = 1; 8 T > 0; 2
2
and
Xt ? X0 ?
Zt 0
B (s; X ) ds; t 0; is an E -valued Wiener process associated with H .
The example announced above is
(z) = ? : z 3 : ; z 2 H ? ; > 0 constant; that is the third Wick power which models polynomial interaction in the two dimensional quantum eld theory (cf. [5]). From the mathematical point of view, is highly singular since we only know that 2 2 L (H ? ! H ? ; 0 ) and k (z)kL (O) = 1 for 0 -a.e. z 2 H ? where 0 is the Gaussian measure on H ? with the characteristic functional Z expfiH h ; z iH ? g 0 (dz ) = expf? 12 k k2H ? g; 2 H : H ? Now, there are two di erent methods to construct weak solutions of the equation dX = [? 21 (1 ? )X ? : X 3 :] dt + dW: By the rst method, cf. [1], based on Dirichlet form techniques, a full Markov family related to the equation is constructed, while using the second method, cf. [14], based on tightness arguments, only a weak stationary solution starting from the equilibrium X0 ; d = expf? 21 : z 4 : (O)g d0 ; is constructed. But the Markov family also gives a weak stationary solution starting from X0 . Furthermore, in [14] is shown that any weak stationary solution starting from X0 does not satisfy ZT P(f k : Xt3 : k2L (O) dt < 1g) = 1 2
1
2
0
for any T > 0 ; and, because it lacks of another applicable criterion, we do not know whether the solutions starting from X0 constructed by the two di erent methods above are equal or not. In this paper we establish a new condition which ensures, that a class of probability measures solving equation (?) weakly, has at most one element. We also give examples showing that our condition works in situations where the condition (2) resp. (4) is hardly checkable or is even not satis ed. The idea is to re ne the condition (2) as follows. Additionally assume that E is a topological Souslin space and that there exist ek 2 E 0 ; k = 1; 2; :::, separating the points of E such that fek g H 0 H forms an orthonormal basis of H . Then the condition (2) reads
P(f
Z TX 1 0
k=1
E 0 hek ; B (t; X )iE dt < 1g) 2
= 1; 8 T > 0:
Remark: The condition ignores the impact of di erent structures of B corresponding to di erent sequences fek g because the limes of the Fourier series is the same for all possible bases. In order to regard such di erent structures of B , we consider the - elds
FtN = fE0 hek ; Xs iE ; s t; k = 1; :::; N g 3
and investigate the rate of convergence for (5)
2 0 he ; B (t; X )i 3 1 E E 6 . . 66 . N !1 75 FtN C ?! B ( t; X ) A 6 4 E0 heN ; B.(t; X )iE E 0 heN ; B (t; X )iE .
02 0 he ; B (t; X )i E E .. EB @64 .
1
1
.
3 77 77 5
which from a heuristic point of view \measures", how fast the drift B (t; X ) is approximated by N -dimensional functionals of the rst N components of the process X restricted to the time interval [0; t]; t 0, if N goes to in nity. Our condition gives a qualitative bound (see Remark 2a) below) for the rate of convergence to be high enough to ensure the uniqueness for the weak solution P of (?). The rate of convergence for (5) is of course expected to depend on the special choice of the approximating nite dimensional subspaces. So it is not surprising that we easily found an example showing that our condition can be satis ed or not according to the choice of the sequence fek g. Please note that up to now we could neither successfully check our condition nor prove that it is not satis ed in case of the open problem discussed above. But, we may easily reprove all known results about the weak uniqueness for solutions of the equations corresponding to the stochastic quantization in two dimensions (see Remark 5b)).
2 Notation and main result Let E be a real locally convex space which is additionally assumed to be a topological Souslin space, i.e. the continuous image of a complete metric space. If E 0 is its topological dual, let h ; i denote the dual pairing between E 0 and E . Suppose that there exist ek 2 E 0 ; k = 1; 2; :::, separating the points of E and introduce the -algebras
AN = fhek ; i; k = 1; :::; N g; N 1; we need for nite dimensional approximations. It is well known in the theory of Souslin locally convex spaces that for the Borel--algebra (6)
_ B(E ) = AN N
holds true (cf. [17]). Let exist a separable Hilbert space H which is densely and continuously embedded into E . By identifying H with its dual H 0 we obtain
E 0 H E densely and continuously. Remark that this property is crucial for the existence of a Wiener process in E (cf. Prop. 7.2.2 in [3]). Furthermore assume that the sequence fek g chosen above forms an orthonormal basis of H . Now introduce the path space to be the set of all E -valued continuous functions on [0; 1) and de ne for t 0 Xt : ! E : ! 7! !(t) as well as _ Ft = fXs; s tg; F = Ft : t0
4
Of course, X = (Xt )t0 is an adapted E -valued stochastic process, the so-called coordinate process, on the measure space ( ; F ) with the ltration F = (Ft )t0 . Consider a B([0; 1)) F =B(E )-measurable function B : [0; 1) ! E particularly satisfying that for every t 0 the mapping ! 7! B (t; !) is Ft =B(E )-measurable; let be a probability measure on (E; B(E )). We say that (; fek g; B ) de nes a martingale problem. De nition 1 A probability measure P on ( ; F ) is said to be a solution of the martingale problem (; fek g; B ) if (i) P X0?1 ; (ii) For all k 1 and all T > 0,
ZT 0
Ejhek ; B (t; X )ij dt < 1 and P(f
(iii) For every ek ; ek 2 fek g, the processes 1
2
Wtki and
:= heki ; Xt i ? heki ; X0 i ?
Zt 0
ZT 0
hek ; B (t; X )i dt < 1g) = 1; 2
heki ; B (s; X )i ds; t 0; i = 1; 2;
Wtk Wtk ? t k ;k ; t 0; 1
2
1
2
are continuous local martingales on the stochastic basis ( ; F ; P; F ). The set of all solutions to (; fek g; B ) is denoted by S(; fek g; B ).
Remark 1 a) The condition (ii) of the de nition above assumes more than is necessary for the
existence of the integrals in De nition 1,(iii). This condition also covers more or less \natural" assumptions related to the uniqueness of the martingale problem (see Section 3). b) For P 2 S(; fek g; B ), the processes W k ; k 2 N , de ned in De nition 1,(iii) are independent one-dimensional standard Wiener processes on ( ; F ; P; F ). c) A weak solution of the equation (?) starting from is a probability measure P on ( ; F ) with P X0?1 such that the process
Wt := Xt ? X0 ?
Zt 0
B (s; X ) ds; t 0;
is well-de ned and presents an E -valued continuous Gaussian process on ( ; F ; P) with covariance Eh ; Wt ih ; Ws i = (t ^ s)( ; )H ; i.e. an E -valued Wiener process associated with H . If P is a weak solution of (?) starting from additionally satisfying De nition 1,(ii), then P 2 S(; fek g; B ), obviously. But the converse is in general not true. Nevertheless, the uniqueness of the martingale problem implies that there is not more than one weak solution of (?) additionally satisfying De nition 1,(ii). d) Each B(E )=B(E ) measurable function b : E ! E naturally de nes a drift function B by B (t; !) = b(!(t)). In what follows we use the small letter b to denote drift functions de ned by this means. 5
As in the introduction, we introduce the -algebras
FtN = fXs? (AN ); s tg; N 1; 1
and remark that for all t 0 (7)
_
Ft = FtN N
follows from (6). We say that a probability measure P on ( ; F ) satis es the condition (CfBek g ) if 8The sequence > (CfBek g )
> > !1 ZTX N < N E(hek ; B (s; X )ijFs ) ? hek ; B (s; X )i ds > k > N > :converges to zero in probability P for every T > 0. 2
0
=1
=1
Remark 2 a) If P 2 S(; fek g; B ) then, from (7) and the properties of the conditional expectation follows, that each summand [E(hek ; B (s; X )ijFsN ) ? hek ; B (s; X )i] converges to zero in L (P), which we have also denoted by (5) above. But, only if the rate of convergence for (5), 1
separately considered for each component, is high enough then the condition (CfBek g ) can be satis ed. b) Often, in concret examples, one can explicitly determine the rate of convergence for the convergence demanded in (CfBek g ) which gives a good quantity for the rate of convergence with respect to (5). The main result of the paper is Theorem 1 There is not more than one solution of the martingale problem (; fek g; B ) which satis es the condition (CfBek g ).
Remark 3 The theorem only deals with the uniqueness problem, that means, its assumptions
are not sucient for the existence of a weak solution of (?) for which, possibly, further conditions are needed. But, in in nite dimensions many techniques apply to the construction of solutions, each of them having its own framework with special assumptions (also see the next section). Therefore, we decided to consider the uniqueness problem apart from the existence problem using a framework as wide as possible. The theorem will be proven in the last section of the paper; in the following section we demonstrate how the theorem applies.
3 Corollaries and examples First of all we treat the nite dimensional case. Corollary 1 Let E = H = Rd , denote by fek g the sequence fe1 ; :::; ed ; ed ; :::g where (e1 ; :::; ed ) is the canonical basis of Rd , and introduce the corresponding martingale problem (; fek g; B ) as above. Then, for each initial distribution , there is not more than one solution to (; fek g; B ). Proof. Of course, E; H; fek g satisfy all assumptions made at the beginning of Section 2. Emphasize that AN = B(E ) for N d 6
and, thus,
FtN = Ft for N d: As a consequence, if there is a probability measure P 2 S(; fek g; B ) then it satis es the condi-
tion (CfBek g ) proving the corollary by Theorem 1.
Remark 4 The proof above shows that in the nite dimensional case the conditional expectations with respect to the \approximating" -algebras FtN are not needed to decide the weak
uniqueness, and so we do not need the rst part of condition (ii) in De nition 1 ensuring the existence of them either. Hence, the only condition on the drift B is the second part of condition (ii) in De nition 1 which coincides with the well-known condition (1) discussed in the introduction. Now, we give a rst result to what extent the condition (2) resp. (4) is generalized by our theorem in in nite dimensions. Corollary 2 Let A : [0; 1) ! E be a B([0; 1)) F =B(E )-measurable function such that
9 N 8 N N 8 k N 8 t 0 : ! 7! hek ; A(t; !)i is FtN -measurable. 0
0
For an arbitrary but xed initial distribution , consider the subset of solutions
K = fP 2 S(; fek g; B ) : E
ZT 0
k(B ? A)(s; X )kH ds < 1; T > 0g: 2
Then there is not more than one element in K. Proof. First remark that our setting allows to extend the norm in H to E by
kzkH := 2
X k
hek ; zi ; z 2 E; 2
being nite or possibly in nite. Suppose K 6= ;, if K = ; then there is nothing to show. Fix T > 0 and choose P 2 K, arbitrarily. In what follows, we verify the condition (CfBek g ) for P proving the corollary. Obviously, (CfBek g ) is equivalent to (CfBek?gA ) because of our assumption, and so it is sucient to show that lim E N !1
Z TX N 0
k=1
E(hek ; (B ? A)(s; X )ijFsN ) ? hek ; (B ? A)(s; X )i ds = 0
which particularly follows from lim E N !1
Z TX 1 0
k=1
Z TX 1 0
k=1
2
2
E(hek ; (B ? A)(s; X )ijFsN ) ? hek ; (B ? A)(s; X )i ds = 0:
But, we have that for all N
E
2
E(hek ; (B ? A)(s; X )ijFsN ) ? hek ; (B ? A)(s; X )i ds 2
Z TX 1 0
k=1
Ehek ; (B ? A)(s; X )i ds 2
7
= 2E
ZT 0
k(B ? A)(s; X )kH ds < 1 2
since P 2 K. Therefore, applying Lebesgue's theorem, we get lim E N !1
ZTX 1 0
=
k=1
Z TX 1 0
E(hek ; (B ? A)(s; X )ijFsN ) ? hek ; (B ? A)(s; X )i ds
k=1
2
N ) ? he ; (B ? A)(s; X )i 2 ds = 0 lim E E ( h e ; ( B ? A )( s; X ) ijF k k s N !1
which converges to zero by (7). Remark 5 a) If weak uniqueness holds for the solutions of the equation dX = A(t; X ) dt + dW
R
then the set K0 = fP : P is a weak solution of (?) such that E 0T k(B ? A)(s; X )k2H ds < 1, T > 0:g is already known to contain not more than one element. TheR condition on the integral in K0 to be nite can even be replaced by the weaker condition P(f 0T k(B ? A)(s; X )k2H ds < 1g) = 1; T > 0, if the solutions of the equation (?) with drift A are unique in a sense stronger thanR weak uniqueness (cf. [13]). But in in nite dimensions one can often only verify the condition P(f 0T k(B ? A)(s; X )k2H ds < 1g) = 1; T > 0, of which the condition (4) is a special case, for probability measures contained in a subclass of K0 . For a well-known example remind the open problem discussed in the introduction, and, consider the equation (?) with the drift ? 21 (1 ? )1? z ? (1 ? )? (z); z 2 H ?1; driven by an H ?1 -valued Wiener process associated with H ? for some > 0. Choose A = ? 12 (1 ? )1? . Then from the results in [14] follows that K0 and its nonempty subclass fP 2 K0 : X is a stationary process on ( ; F ; P)g coincide. b) Applying Corollary 2, we obtain the same result for the example in the part a) above. Indeed, ? 21 (1 ? )1? presents an easy possibility to x A if fek g is chosen to be the eigenbasis of the corresponding Laplacian. We have already mentioned in the introduction that one feature of our condition (CfBek g ) consists in including the special structure of the drift B with respect to an appropriate sequence fek g, and, the result below is the consequence. Proposition 1 Fix E; H; fek g as in the previous section but additionally choose another sequence fe~k g of the same kind as fek g. If P 2 S(; fek g; B ) \ S(; fe~k g; B ) satis es the condition (CfBek g ) then it does not follow that P also satis es the condition (CfBe~k g ), in general. Moreover, this implication even remains true if P is the only solution to (; fe~k g; B ). Proof. Consider the Laplacian N with Neumann boundary conditions in the S interval (0; 1) of 2 ?1 the real line. Let fek g be its eigenbasis in H = L ((0; 1)) and set E = H = 2R H where H is the scale of Sobolev spaces generated by 1 ? N and L2 ((0; 1)). The space E equipped with its canonical topology becomes a Souslin locally convex space of which its topological dual T 0 1 is E = H = 2R H . It is well-known for fek g to be a subset of E 0 as well as to separate the points of E , and Sobolev's embedding theorem particularly gives
E 0 H E densely and continuously. 8
So, E; H; fek g satisfy all assumptions made at the beginning of Section 2. De ne the function
B : [0; 1) ! E by (8) h ; B (t; !)i = h 00 ; !(t)i; 2 E 0 ; where 00 denotes the second derivative of . Now, in E , consider the equation
Xt =
(9) that is
h ; Xt i =
Zt 0
Zt 0
B (s; X ) ds + Wt; t 0;
h 00 ; Xsi ds + h ; Wt i; t 0; 2 E 0;
driven by an E -valued Wiener process associated with H (see Remark 1). Equation (9) has already been studied in early papers about stochastic partial di erential equations and we know that it has a unique strong solution with outstanding properties (cf. [18] for example). On the one hand, there is exactly one probability measure P on ( ; F ) with P X0?1 = f0g solving the equation (9) weakly, and, on the other hand, P 2 S(f0g ; fek g; B ). But, all elements of S(f0g ; fek g; B ) trivially satisfy the condition (CfBek g ) since
hek ; B (s; X )i = ?k hek ; Xsi; s 0; 8 k 1; where k is the eigenvalue of ?N corresponding to ek . Applying Theorem 1, the only weak solution of the equation (9) is also the only solution to (f g ; fek g; B ). 0
Now involve the space C01 ((0; 1)) of all in nitely many di erentiable functions having compact support in (0; 1). This space equipped with its canonical topology has the space of Schwartz distributions on (0; 1) as its topological dual, which we denote by E~ . Then it holds that C01 ((0; 1)) E 0 H E E~ continuously as well as
C01 ((0; 1)) H E~ densely, and we choose a sequence fe~k g C01 ((0; 1)) separating the points of E~ which also forms an orthonormal basis of H . Remark that such a sequence as a matter of fact exists in our situation. Of course, fe~k g is di erent from fek g because fek g is not included in C01 ((0; 1)). Furthermore, fe~k g also separates the points of E . Again, P 2 S(f0g ; fe~k g; B ) holds true for the measure P on ( ; F ) mentioned above. Even more, P is the only solution to (f0g ; fe~k g; B ). Indeed, if P1 is any solution to (f0g ; fe~k g; B ) then it also solves equation (9) weakly, because fe~k g separates the points of E and the series 1 X k=1
Zt
he~k ; Xt i ? he~k ; B (s; X )i ds e~k = 0
1 * X k=1
e~k ; Xt ?
Z t 0
Xs ds
00+
e~k
converges in E P1 -a.s. to an E -valued Wiener process associated with H as in the proof of Prop.7.2.3 in [3]. Hence, P = P1 by the weak uniqueness for the solutions of (9). But we do not know whether P satis es the condition (CfBe~k g ) or not. ~ H; fe~k g by Denote path space, -algebras resp. coordinate process corresponding to E; ~ ; F~ ; F~t ; F~tN resp. X~ and de ne the function B~ : [0; 1) ~ ! E~ 9
as in (8) only replacing 2 E 0 by 2 C01 ((0; 1)). Clearly, the process X is an F =F~ -measurable mapping X : ! ~ ; and, the image measure P~ := P X ?1 on ( ~ ; F~ ) is a solution to (f0g ; fe~k g; B~ ). Moreover, P satis es the condition (CfBe~k g ) if and only if P~ satis es (CfBe~~k g ). To sum up, using the eigenbasis fek g to the Neumann Laplacian N , we introduced the space E together with an E -valued function B and realized, that there is exactly one solution P to (f0g ; fek g; B ) and that P particularly satis es the condition (CfBek g ). Then, for an appropriate sequence fe~k g di erent from fek g we saw, that the measure P induces a measure P~ on another measure space ( ~ ; F~ ) solving the martingale problem (f0g ; fe~k g; B~ ). Now, we repeat this procedure with respect to the Dirichlet Laplacian D knowing that another eigenbasis appears leading to a space di erent from E and so on. But, the same sequence fe~k g can be chosen and the induced measure on ( ~ ; F~ ), which we denote by Q~ , also solves the martingale problem (f0g ; fe~k g; B~ ). In what follows, we use the symbol Q for the measure of which the image measure is Q~ . Settle that if we speak about Q, then the underlying structure is assumed to be de ned by the eigenbasis to the Dirichlet Laplacian. In this sense, Q is the only solution to (f0g ; fek g; B ), Q particularly satis es (CfBek g ) and Q satis es the condition (CfBe~k g ) if and only if Q~ satis es (CfBe~~k g ). Turn to the measures P~ and Q~ on ( ~ ; F~ ) both solving the martingale problem (f0g ; fe~k g; B~ ). It is well-known in the theory that X~ presents a Gaussian process on ( ~ ; F~ ; P~ ) resp. ( ~ ; F~ ; Q~ ) with variance
Z t Z Z 1
0
resp.
0
1
0
Z t Z Z 1
0
0
0
1
gN (t ? s; x; y) (y) dy gD (t ? s; x; y) (y) dy
2
dxds; t 0; 2 C01 ((0; 1)) ;
2
dxds; t 0; 2 C01 ((0; 1)) ;
where gN resp. gD denotes Green's function of the problem
@ @ @t ? N = 0 resp. @t ? D = 0:
As a consequence, the measures P~ and Q~ are di erent from each other and, applying Theorem 1, one of them cannot satisfy the condition (CfBe~~k g ). Hence, one of the measures P or Q does not satisfy the condition (CfBe~k g ) proving the proposition.
Remark 6 Consider the drift B in the one case above where (CfBek g ) is not satis ed. Then B
can also be given by the function (see Remark 1,d))
b(z) =
1 X k=1
~
?k hek ; ziek ; z 2 E;
but, with respect to fe~k g such a \nice" structure fails. Manifestly, the rate of convergence for (5) with respect to fe~k g is too low for (CfBe~k g ) to be satis ed. Emphasize that the martingale problem (f0g ; fe~k g; B ) is well-posed since it has only one solution. So, one cannot say that fe~k g would be badly chosen.
10
In the following example we discuss a situation where (remind Remark 5a)) the condition
P(fR T k(B ? A)(s; X )kH ds < 1g) = 1 is never satis ed, while the condition (CfBek g ) applies 2
0
easily.
Example Introduce N ; fek g; H ; E; H; fk g as in the proof of Proposition 1 but, in what
follows, we abbreviate the symbol N by because Neumann boundary conditions are only considered. Let g be the continuous real function de ned by ( pjxj : jxj 1 g(x) = 1 : jxj 1 which is not lipschitz continuous. We use g to construct the drift function ~b(z ) = z + X g(hek ; z i)ek ; z 2 E: 1
k=1
The corresponding equation (?) starting from zero has a pathwise unique strong solution. Indeed, choose a probability measure P on ( ; F ) admitting a sequence of independent one-dimensional Wiener processes (W k ). Then for each k 1 exists a pathwise unique solution to the onedimensional stochastic equation
ytk =
P yk e ) and the process ( 1 t k t k
Zt 0
(?k ysk + g(ysk )) ds + Wtk ; t 0;
is the only strong solution of the corresponding equation (?) P 1 starting from zero driven by the E -valued Wiener process Wt = k=1 Wtk ek ; t 0, associated with H . Of course, we can easily handle the equation (?) with drift ~b since it leads to a separate system of one-dimensional stochastic equations. A simple idea to couple the components of a drift is presented by 1 X b(z) = z + g(hek+1 ; zi)ek ; z 2 E: =1
0
k=1
We will see that, although, the structure of b is in some sense close to the structure of ~b, the equation (?) corresponding to b is far o to be handled easily. At rst, in order to construct a solution, various techniques fail: Picard iteration does not work because g is not lipschitz continuous; Girsanov's method is not known to be applicable since the expectation of the density process cannot suciently be estimated; construction via comparison theorem as in [6] does not work in this situation either. However, a possible construction of a weak solution is the following one, where all solutions constructed below are supposed to start from zero. For N 1 de ne N X bN (z) = z + g(hek+1 ; zi)ek ; z 2 E; k=1
and let Q denote the unique probability measure on ( ; F ) weakly solving the equation dX = X dt + dW: Then we have for all T 0 ZT X N 1 EQ expf 2 k g(hek+1 ; zi)ek k2H dtg < 1 0 k=1 11
since g is bounded, that is Novikov's condition is satis ed. Applying Girsanov's theorem, there exists a probability measure PN on ( ; F ) weakly solving the equation (?) corresponding to bN which is absolutely continuous with respect to Q. Fix k 1 and de ne W k as in De nition 1,(iii). Then the process hek ; X i satis es the stochastic di erential inequality
Zt
hek ; Xt i hek ; Xs i ? k hek ; Xr i dr + (t ? s) + Wtk ? Wsk ; s t; PN -a.s.; s
resp.
Zt
hek ; Xt i hek ; Xs i ? k hek ; Xr i dr + Wtk ? Wsk ; s t; PN -a.s. s Using [2], hek ; X i can be estimated by the solution to the stochastic di erential equation dok = (?k ok + 1) dt + dW k ; ok = 0; 0
resp.
duk = ?k uk dt + dW k ; uk0 = 0;
we explicitly know:
Zt
that is (10)
0
e?k (t?s) dW k s
hek ; Xt i
Zt 0
e?k (t?s) ds +
8 1 Zt > < + j e?k t?s dWsk j jhek ; Xt ij > k : t + jWtk j (
)
0
Zt 0
e?k (t?s) dWsk ;
if k 6= 0
if k = 0 for all t 0 PN -a.s. As a consequence, we obtain for every T 0 1 X
(11)
sup EPN kXt k2H = sup (1 + k ) EPN hek ; Xt i2 t2[0;T ] k=1 t2[0;T ] ! 1 X 1 2 C T + (1 + ) ; 2 R; 8 N 1; k=2
k
k
which immediately implies (applay Aldous criterion in [9] for example) that the sequence (PN ) is tight. Let P denote the weak limit of some subsequence of (PN ). Then, applying (11) again, P is easily shown to be a weak solution of the equation (?) corresponding to b with P(fX0 = 0g) = 1. Of course, we are interested in the question whether P is the only solution or not. The answer has proved to be an easy application of our main theorem, and, we do not know another method to decide this question. At rst we notice that if P is an arbitrary weak solution of the equation (?) corresponding to b starting from zero, then (10) also holds true for all t 0 P-a.s. simply copying the proof of (10) with respect to PN . On the one hand, from (10) follows for every T 0 that 8 1 1 if 6= 0 > < + 2 2 k 2 sup Ehek ; Xt i2 > k 2k t2[0;T ] : 2T 2 + 2T if k = 0 and, hence (see Remark 1c)), P is a solution to the martingale problem (f0g ; fek g; b). On the other hand, additionally using the Burkholder-Davis-Gundy inequality, we get for k 2 r1 1 sup Ejhek ; Xt ij + c1 2 (1 + c1 ) p1 k t0 k k 12
why P even satis es the condition (Cfbek g ). Indeed,
E = E
2 2 (12)
Z TX N
Z T k 0
ZT
ZT 0
2
=1
E(g(heN ; Xt i)jFtN ) ? g(heN ; Xt i) dt
0
0
E(hek ; b(Xt )ijFtN ) ? hek ; b(Xt )i dt +1
+1
2
Eg(heN ; Xt i) dt 2
+1
EjheN ; Xt ij dt +1
2(1 + c )T p 1
N +1
1
N !1
?! 0; 8 T > 0:
Now, Theorem 1 applies showing that the sequence (PN ) of probability measures on ( ; F ) constructed above weakly converges to the only weak solution of the corresponding equation (?) starting from zero. Emphasize that the same is true if we replace by the d-dimensional Laplacian with Neumann or Dirichlet boundary conditions in a bounded cube O Rd ; d 1. The proof is exactly the same. Only remark that, in the case d = 1, it is sucient to apply (11) for = 0 while, in the case d > 1, an appropriate < 0 has to be chosen. In a last step, return to the condition (13)
ZT X 1
P f
0
k
k=1
!
g(hek+1 ; Xt i)ek kH dt < 1g = 1; 8 T > 0; 2
being an alternative (see Remark 5a)) to our condition (Cfbek g ) used above for the weak uniqueness result. Consider again the Laplacian in di erent dimensions d and x an arbitrary weak solution P of the equation (?) corresponding to b with P(fX0 = 0g) = 1. For d = 1 we are not able to verify (13) but we cannot exclude that (13) is satis ed either. Even so, already in the case d = 2, (13) is not satis ed, which we will show by leading the opposite to a contradiction. Before doing this, remember the following properties of the sequence fk g in the case d = 2: 1 X k=2
?k 1 = 1 and
1 X k=2
?k 1? < 1; 8 > 0:
Now, assume that the condition (13) would be satis ed for the arbitrary solution P above. Then, for a xed time T > 0, we have
1 > which implies (14)
Z TX 1
k Z TX 1 0
g(hek+1 ; Xt i)2 dt
=1
k Z TX 1 0
=1
k=1
0
jhek ; Xt ij1fjhek +1
hek ; Xt i 1fjhek +1
Z TX 1 0
k=1
+1
2
+1
;Xt ij
