on his 70th birthday
THE KOLMOGOROV OPERATOR ASSOCIATED WITH A STOCHASTIC VARIATIONAL INEQUALITY IN R
nWITH CONVEX POTENTIAL
VIOREL BARBU and GIUSEPPE DA PRATO
One identies the Kolmogorov operator associated with the transition semigroup generated by a multivalued stochastic dierential equation in Rn with convex continuous potential whose gradient is singular on a nowhere dense setΓ. This may be viewed as an elliptic operator on Rn\Γwith Wentzell type conditions onΓ.
AMS 2000 Subject Classication: 60J60, 58E35, 60J25.
Key words: stochastic variational inequality, gradient system, Wentzell condition, transition semigroup.
1. INTRODUCTION
Consider the multivalued stochastic dierential equation (1.1)
( dX(t) +∂U(X(t))dt3dW(t), t≥0, X(0) =x∈Rn,
whereW(t)is ann-dimensional Wiener process on a ltered probability space (Ω,F,{Ft}t≥0,P), whereU :Rn→Rsatises Hypothesis 1.1 below.
Hypothesis 1.1. (i)U is convex and continuous inRn.
(ii) There is a closed set Γ ⊂ Rn such that Rn\Γ is dense in Rn and U ∈C2(Rn\Γ).
(iii) There existC >0and m∈Nsuch that
(1.2) |∇U(x)| ≤C(1 +|x|m), ∀x∈Rn\Γ.
In (1.1)∂U :Rn→2Rn is the subdierential of U, i.e.,
∂U(x) ={y ∈Rn: U(x)−U(u)≤ hy, x−ui, ∀u∈Rn},
REV. ROUMAINE MATH. PURES APPL., 53 (2008), 56, 377388
(h·,·i is the scalar euclidean product of Rn). In particular Hypothesis 1.1 implies that
(1.3) ∂U(x) =∇U(x), ∀x∈Rn\Γ.
Γ must be viewed as the set of singular points ofU and assumption (ii) agrees with the structure of singularities set of continuous convex functions on Rn. Indeed one knows (see [1]) that for such a functionU :Rn→R, the dierential
∇U(x) exists everwhere excepting a set contained in a countable union ofC1 hypersurfaces.
Denition 1.2. By solution of (1.1) on [0, T]we mean a pair of adapted processes (X, η)∈L2W(Ω;C([0, T];Rn))such that
(1.4) X(t) +
Z t 0
η(s)ds=W(t) +x, P-a.s., ∀t∈[0, T], (1.5) η(t)∈∂U(X(t)), P-a.s., a.e.t∈[0, T].
By L2W(Ω;C([0, T];Rn)) we denote the set of all Rn-valued continuous and adapted processes Z inΩsuch that
E
sup
t∈[0,T]
|Z(t)|2
<+∞.
Proposition 1.3. For eachx∈Rn there exists a unique solution (X, η) to equation (1.1). Moreover,
(1.6) E|X(t, x)−x|2 ≤ 1
2 t2[C(1 +|x|m) +n]eCt(1+|x|m), ∀x∈Rn, t∈[0, T].
The above existence result is well known and holds under more general conditions on U (see [6] and [5]). However, we shall prove it in Section 2 via an approximation process which will be used later to prove the essential m-dissipativity of N.
In particular, it follows from (1.6) that the transition semigroup (1.7) Ptϕ(x) =E[ϕ(X(t, x))], x∈Rn, t∈[0, T],
is aC0-semigroup on the space
ϕ∈C(Rn) : sup
x∈Rn
|ϕ(x)|eC(1+|x|m)<+∞
,
and so, it is Feller. But Pt is not aC0-semigroup onCb(Rn)(except whenDU is bounded). However, it is aπ-semigroup onCb(Rn), the space of all bounded and uniformly continuous functions onRn (see [16]).
Let us recall the denition of π-semigroup following [16]. For this we need the notion of π-convergence see [16] (see also [13] where it is called bp- convergence and [14] and [15]).
A sequence (ϕn) ⊂Cb(Rn) is said to be π-convergent to a function ϕ∈ Cb(Rn)(and we shall writeϕn→π ϕ) if for anyx∈Rnwe have lim
n→∞ϕn(x) =ϕ(x) and if supn∈NkϕnkCb(Rn) <+∞.
A subsetΛ of Cb(Rn) is said to be π-dense if for any ϕ∈Cb(Rn) there exists a sequence(ϕn)⊂Λ such thatϕn→π ϕ.
A semigroup Pt of linear bounded operators on Cb(Rn) is called a π- semigroup if
(i) we have
kPtϕkC
b(H)≤ kϕkC
b(H), ϕ∈Cb(H), t≥0;
(ii) ifϕn
→π ϕ thenPtϕn
→π Ptϕ,∀t≥0;
(iii) for allϕ∈Cb(H) and for allx∈H the function [0,+∞)→ R, t→ Rtϕ(x)is continuous.
The main purpose of this paper is to identify the innitesimal generator N :D(N)⊂Cb(Rn)→Cb(Rn) of Pt on Cb(Rn) dened (see [16]) as
(1.8) N ϕ(x) = lim
t→0
1
t (Ptϕ(x)−ϕ(x)), ∀x∈Rn, ∀ϕ∈D(N), whereD(N) is the set ofϕ∈Cb(Rn) such that
(i) there exists the limit
t→0lim 1
t (Ptϕ(x)−ϕ(x)) =N ϕ(x), ∀x∈Rn; (ii) we have
sup
t∈(0,1)
1
t kPtϕ−ϕkCb(Rn)<+∞.
It should be recalled (see [13], [7], [16]) that N, as the generator of a π-semigroup, is m-dissipative in Cb(Rn). However, its domain D(N) is not dense in Cb(Rn) but it is only π-dense. (It is a Yosida operator in the sense of [12].)
To state the main result of the paper, we need a last denition. Let Y be a subspace of D(N) π-dense inCb(Rn) and let N0 be the restriction of N to Y. We say thatN is theπ-closure ofN0 if for any ϕ∈D(N)there exists a sequence {ϕn} ⊂Y such that
ϕn π
→ϕ, N0ϕn π
→N ϕ.
The main result is given by Theorem 1.4 below.
Theorem 1.4. The innitesimal generatorN is the π-closure N0 of the operator N0 dened as follows,
(1.9)
D(N0) =n
ϕ∈C1(Rn)∩C2(Rn\Γ), ∃f ∈Cb(Rn)such that f = 1
2 ∆ϕ− h∇U,∇ϕionRn\Γ o N0ϕ(x) =f(x), ∀x∈Rn.
Notice that ifϕ∈D(N0) we have
(1.10) N0ϕ(x) =
1
2 ∆ϕ(x)− h∇U(x),∇ϕ(x)i, ∀x∈Rn\Γ,
y→x, y∈limRn\Γ
1
2 ∆ϕ(y)− h∇U(y),∇ϕ(y)i
, ∀x∈Γ, Theorem 1.5 below precises the nature ofD(N).
Theorem 1.5. Letϕ∈D(N). Then (i) ϕ∈Cb(Rn)∩Wloc2,p(Rn), ∀p≥2.
(ii) 1
2 ∆ϕ− h∇U,∇ϕi ∈Cb(Rn).
(iii) N ϕ(x) = 1
2 ∆ϕ(x)− h∇U(x),∇ϕ(x)i, ∀x∈Rn\Γ. (iv)N ϕ(x) = lim
y→x,y∈Rn\Γ
1
2 ∆ϕ(y)− h∇U(y),∇ϕ(y)i
, ∀x∈Γ.
Here ∆ϕ is considered in the sense of distributions on Rn\Γ, i.e. in D0(Rn\Γ).
As mentioned earlier, equation (1.1) can be viewed as a stochastic die- rential equation with drift DU and diusion coecient I. It was extensively studied in recent years under dierent conditions on the convex function U and in dierent spaces including Lp(Rn, µ) spaces where µ is the invariant measure dµ= ce−2Uds(see [2], [4], [5], [9], [10], [11]). However, the form of the corresponding Kolmogorov operator is strongly dependent of the domain
D(U) ={x∈Rn: U(x)<+∞}
as well as of U itself. For instance, if U = IK, the indicator function of a convex, closed set K, then the Kolmogorov operator reduces to an elliptic operator on the interior K˚ ofK equipped with Neumann boundary conditions on ∂K (see [4], [5], [9]). If
x→∂D(U)lim U(x) = +∞
then no explicit boundary condition on∂(D(U))arises however (see [10], [11]).
The situation encountered here is dierent. Under Hypothesis 1.1, the Kolmogorov equation reduces to an elliptic equation on O := Rn\ Γ with
Wentzell boundary conditions on the set Γ of singularities of the gradient of U. Indeed, we may view the solution to equation
λϕ−N ϕ=f as that of the elliptic boundary value problem
(1.11)
λϕ(x)−1
2 ∆ϕ(x)+h∇U(x),∇ϕ(x)i=f(x), ∀x∈Rn\Γ :=O,
y→x,y∈limRn\Γ
−1
2 ∆ϕ(y)+h∇U(y),∇ϕ(y)i
=f−λϕ(x), ∀x∈Γ, and this is an elliptc problem with boundary value conditions of Wentzell type on the boundary Γ ofO.
Accordingly, the Kolmogorov equation corresponding to (1.1) can be writ- ten as
(1.12)
(i) ∂u
∂t(t, x) = 1
2 ∆u(t, x)− h∇U(x),∇u(t, x)i int >0, x∈Rn\Γ, (ii) lim
y→x,y∈Rn\Γ
1
2 ∆u(t, y)−h∇U(y),∇u(t, y)i
=∂u
∂t(t, x) inx∈Γ, (iii) u(0, x) =ϕ(x) inRn,
where the boundary condition (ii) is taken in the sense of traces. In the one- dimensional case and for Γ ={0}, (1.12) reduces to
(1.13)
(i) ∂u
∂t(t, x) = 1
2 ux,x(t, x)−U0(x)ux(t, x) int >0, x∈R\{0}, (ii) lim
y→0
1
2 uxx(t, y)−U0(y)ux(t, y)
= ∂u
∂t(t,0) inx= 0, (iii) u(0, x) =ϕ(x) inRn.
Problem (1.13) itself may be viewed as a Wentzell transmission problem in(−∞,+∞).
2. EXISTENCE AND UNIQUENESS FOR PROBLEM (1.1) Proof of Proposition 1.3. We start with the approximating equation (2.1)
( dX(t) +∇U(X(t))dt= dW(t), t≥0, X(0) =x∈Rn.
Here U∈C∞(Rn)∩Lip(Rn) is the regularization U(x) =−n
Z
Rn
U(y)ρ
x−y
dy=
Z
Rn
U(x−y)ρ(y)dy, ∀x∈Rn, of U, whereU is the Moreau-Yosida approximation ofU, i.e.,
U(y) = inf
z∈Rn
U(z) + |y−z|2 2
,
ρ∈C∞(Rn)has support in the ball of center0and radius1, andR
Rnρ(x)dx= 1.
In other words, U is obtained by molling the Moreau-Yosida approximation of U (which is onlyCLip1 whereasU isC∞).
To go further, we need the following result.
Lemma 2.1. There existsC1 >0 and C2 >0 such that (2.2) |U(x)−U(y)| ≤C1(|x|m+|y|m)|x−y|, ∀x, y∈Rn. (2.3) |U(x)−U(x)| ≤C2(1 +|x|2m), ∀x∈Rn, ∀ >0.
Proof. It is enough to prove (2.2) when
{ξx+ (1−ξ)y: ξ∈[0,1]} ∈Rn\Γ.
Then by the mean value theorem we have
|U(x)−U(y)| ≤ sup
ξ∈[0,1]
|∇(ξx+ (1−ξ)y)|
and (2.2) follows from Hypotesis 1.1(iii).
Let us prove (2.3). It is obvious that U(x)−U(x) ≥ 0 for all x ∈Rn. Moreover,
U(x)−U(x) =U(x)−U(x) +|x−x|2
2 ,
wherex= (I+∂U)−1(x)(see e.g. [3]). Thenx−x ∈∂U(x), so that
|x−x| ≤C(1 +|x|m).
Therefore,
U(x)−U(x)≤C(1 +|x|m) +C2(1 +|x|m)2, which yields (2.3).
We now continue the proof of Proposition 1.3 by noting that problem (2.1) has a unique solution X=X(t, x) with
X ∈L2W(Ω;C([0, T];Rn)) for all T >0. Next, we proceed in two steps.
Step 1. Proof of (1.6). Using Itô's formula we nd
d|X(t, x)−x|2 = 2EhX(t, x)−x,dX(t, x)i+ndt,
from which, integrating and taking expectation, E|X(t, x)−x|2+ 2E
Z t 0
h∇U(X(s, x)), X(s, x)−xids+nt.
Taking into account the monotonicity of U we have E|X(t, x)−x|2 ≤2E
Z t 0
h∇U(x), X(s, x)−xids+nt
≤C(1 +|x|m) Z t
0
E|X(s, x)−x|2ds+ [C(1 +|x|m) +n]t.
So, by a standard comparison result, E|X(t, x)−x|2 ≤
Z t 0
eC(t−s)(1+|x|m)[(C(1 +|x|m) +n]sds, which yields (1.6).
Step 2. Conclusion. By Itô's formula we see that 1
2 d
dt|X(t, x)−Xλ(t, x)|2
+h∇U(X(t, x))− ∇Uλ(Xλ(t, x)), X(t, x)−Xλ(t, x)i= 0, P-a.s.. for all , λ >0. Since U and Uλ are convex and dierentiable, we have
−h∇Uλ(Xλ(t, x)), X(t, x)−Xλ(t, x)i ≥Uλ(Xλ(t, x))−Uλ(X(t, x)) h∇U(X(t, x)), X(t, x)−Xλ(t, x)i ≥U(X(t, x))−U(Xλ(t, x)).
For any t≥0,P-a.s., this yields 1
2 |X(t, x)−Xλ(t, x)|2 (2.4)
≤ Z t
0
(U(Xλ(s, x))−U(X(s, x)) +Uλ(X(s, x))−Uλ(Xλ(s, x)))ds.
Note also that
|U(x)−U(x)| ≤ Z
Rn
|(U(x−y)−U(x))|ρ(y)dy (2.5)
≤ Z
Rn
|(U(x−y)−U(x−y))|ρ(y)dy+ Z
Rn
|U(x−y)−U(x))|ρ(y)dy.
Now, using Lemma 2.1 we deduce by (2.4) that there exists a constantC3 >0 such that
(2.6) 1
2|X(t, x)−Xλ(t, x)|2≤C Z t
0
((1+|X(s, x)|)2m+λ(1+|Xλ(t, x)|)2m)ds.
We want to take expectation of both sides of (2.6). For this we have to estimate E|X(t, x)|2m. By Itô's formula we see that
E|X(t, x)|2m+ 2mE Z t
0
h∇U(X(s, x)), X(s, x)i|X(s, x)|2m−2ds
=|x|2m+m(n+ 2m−2)E Z t
0
|X(s, x)|m−2ds, ∀t≥0.
Using the monotonicity of U, we obtain E|X(t, x)|2m+ 2mE
Z t 0
h∇U(0), X(s, x)i|X(s, x)|2m−2ds
≤ |x|2m+m(n+ 2m−2)E Z t
0
|X(2s, x)|m−2ds, ∀t≥0.
Finally, after some standard manipulations involving Gronwall's lemma we end up with
E|X(t, x)|2m≤C4(1 +|x|2m), ∀t∈[0, T], x∈Rn. Taking expectation into (2.6) and substituting the latter, we get
(2.7) E|X(t, x)−Xλ(t, x)|2≤C4(+λ)(1+|x|2m), ∀t∈[0, T], x∈Rn, , λ >0.
Hence there is X=X(t, x) such that
(2.8) lim
→0E|X(t, x)−X(t, x)|2 = 0, ∀t >0, x∈Rn, uniformly on each [0, T]×BRwhereBR is the ball inRn of radius R.
In particular, it follows thatX∈C([0,+∞);Rn) and is pathway contin- uous.
Moreover, by (1.2) we see that
|∇U(x)| ≤C(1 +|x|m), ∀x∈Rn, therefore,
|∇U(X(t, x))| ≤C(1 +|X(t, x)|m) a.e. in(0, T)×Rn×Ω.
Thus on a subsequence we have
∇U(X(t, x))→η weak star inL∞((0, T)×Ω).
Since by (2.8) we also have
X(t, x)→X(t, x) a.e. in(0, T)×Ω, by a standard argument we infer that (details are omitted)
η∈∂U(X) a.e. in(0, T)×Ω, and so, (1.4)(1.5) follow as claimed.
The uniqueness ofX is immediate.
3. PROOF OF THEOREMS 1.4 AND 1.5
Proof of Theorem 1.4. Consider the transition semigroup Pt in Cb(Rn) associated with equation (2.1), namely,
Ptϕ(x) =E[ϕ(X(t, x))], t≥0, x∈Rn, ϕ∈Cb(Rn).
We notice also that by (2.7), (2.8) we have
(3.1) lim
→0Ptϕ(x) =Ptϕ(x), t≥0, x∈Rn, ϕ∈Cb(Rn), uniformly on bounded sets of Rn.
Lemma 3.1. We have
(3.2) |X(t, x)| ≤1, ∀t≥0, x∈Rn.
Lemma 3.2. Let λ > 0 and f ∈ Cb2(Rn). Then there exists a unique ϕ ∈Cb2(Rn) such that
(3.3) λϕ−1
2 ∆ϕ+h∇U,∇ϕi=f.
Moreover, ϕ is given by
(3.4) ϕ(x) =E
Z ∞ 0
e−λtf(X(t, x))dt, x∈Rn, and
(3.5) kϕkCb(Rn)≤ 1
λ kfkCb(Rn), and
(3.6) k∇ϕkCb(Rn)≤ 1
λ k∇fkCb(Rn). Proof. It is classical that setting
u(t, x) =Ptf(x), t≥0, x∈Rn, and using Itô's formula, we have
d
dtu(t, x) = 1
2 ∆u(t, x)− h∇U(x),∇u(t, x)i, t≥0, x∈Rn. Now, setting
ϕ(x) =E Z ∞
0
e−λtu(t, x)dt,
we can easily check that ϕ enjoys the required properties.
Lemma 3.3. Let λ > 0 and f ∈ Cb2(Rn) and let ϕ ∈ Cb2(Rn) be the solution to (3.3). Then there exists ϕ∈D(N0) such that
(3.7) lim
→0ϕ(x) =ϕ(x), x∈Rn. Moreover, ϕis given by
(3.8) ϕ(x) =E Z ∞
0
e−λtf(X(t, x))dt= (λ−N)−1f(x), x∈Rn. Proof. Letα∈C0∞(BR) be such thatα= 1 inBR/2.By (3.3) we have (3.9)
λ(αϕ)−1
2∆(αϕ) =αf+1
2ϕ∆α+∇α·∇ϕ−αh∇U,∇ϕi inBR, αϕ = 0 on ∂BR.
By (3.6) the right hand side of (3.9) is bounded in BR and so, by the Agmon- Douglis-Nirenberg estimates, we have
(3.10) |αϕ|W2,p(BR)≤Cp, ∀ >0, ∀p≥2,
and therefore, sinceRis arbitrary,{ϕ}is bounded inWloc2,p(Rn)for allp, which implies that it is bounded in C1(BR) for every R > 0. Hence there exists a subsequence{ϕk} convergent to a functionϕ∈T2,p
loc(Rn)⊂C1(Rn).
Next, we shall takeα∈C0∞(Rn\Γ). Dierentiating (3.9) and taking into account that f ∈C1(Rn),U∈C2(Rn\Γ)and
|D2U(x)| ≤Cδ(1 +|x|m), ∀x∈Γδ, < δ, whereΓδ={x:dist(x,Γ)≥δ},we conclude as above that
|αϕ|W3,p(Rn\Γ) ≤C, ∀ >0, p≥2,
and therefore αϕ ∈ W3,p(Rn\Γ). Since α ∈ C0∞(Rn) is arbitrary, we infer that {ϕ} is bounded in T
p≥2Wloc3,p(Rn\Γ) ⊂ C2(Rn\Γ) and therefore, by the Sobolev embedding theorem, it is compact in C2(K) on each compact K ⊂Rn\Γ.Hence
ϕ∈C2(Rn\Γ)∩C1(Rn)∩Cb(Rn) and
(3.11)
→0lim∇ϕ =∇ϕ uniformly on compacts ofRn
→0limϕ=ϕ inC2(BR\Γ), ∀R >0.
Moreover, ϕ∈ D(N0) and is a solution to equation λϕ−N0ϕ = f. Indeed, letting →0 into (3.8) or (3.9) we see by (3.11) that
(3.12) λϕ(x)− 1
2 ∆ϕ(x) +h∇U(x),∇ϕ(x)i=f(x), ∀x∈Rn\Γ,
and by (3.12) that
y→x, y∈Rlim n\Γ
1
2 ∆ϕ(y)− h∇U(y),∇ϕ(y)i
=λϕ(x)−f(x), ∀x∈Γ.
This completes the proof of Theorem1.4.
Proof of Theorem 1.5. We know that
ϕ= (λ−N)−1f →π ϕ= (λ−N0)−1f
for f ∈Cb(Rn)and estimate (3.10) extends for such aϕ. Hence (ii) holds and so ϕ ∈ C1(Rn)∩Wloc2,2(Rn) for all p ≥ 2, as claimed. It is also obvious that (1.11) holds in D0(Rn\Γ), i.e. in the sense of distributions. This completes the proof.
4. REMARKS AND FURTHER RESULTS
1) The above theorem extend to stochastic dierential equations of the form
(4.1)
( dX(t) + (∂U(X(t)) +F(X(t)))dt3dW(t) X(0) =x,
whereU satises Hypothesis 1.1 andF :Rn→Rn is aC1 function such that (4.2) hF x−F y, x−yi ≥ −γ|x−y|2, ∀x, y∈Rn.
The proof is exactly the same.
2) The transition semigroupPtis strongly Feller. Indeed by the Bismut- Elworthy formula for any h∈Rn we have
|hDPtϕ(x), hi| ≤ 1 t E
ϕ(X(t, x) Z t
0
hDxX(s, x),dW(s)i
≤C|h|, because DxX(t, x) :=η(s, x), as solution to the equation
(4.3)
( dη0 +D2U(X(t, x))η= 00 η(0, x) =h,
satises the estimate |η,x| ≤ |h|,∀t≥0.Hence
|Ptϕ(x)−Ptϕ(y)| ≤ C
√
t |x−y|, x, y∈Rn, which implies that Pt is strong Feller (see e.g. [8]).
Acknowledgements. The authors were partialy supported by a CEx-05 Grant of the Romanian Ministry of Education and Research, and by the Italian National Project MURST Equazioni di Kolmogorov, respectively.
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Received 5 December 2007 Al.I. Cuza University
700506 Ia³i, Romania and
Scuola Normale Superiore 52126 Pisa, Italy
