The stochastic obstacle problem for the harmonic oscillator with damping

Transcription

1 Preprint di Matematica - n. 9 Novembre 2005 The stochastic obstacle problem for the harmonic oscillator with damping Viorel Barbu Giuseppe a Prato SCUOLA NORMALE SUPERIORE PISA

2 The stochastic obstacle problem for the harmonic oscillator with damping Viorel Barbu University Al. I. Cuza, , Iasi, Romania Giuseppe a Prato Scuola Normale Superiore, 5626, Pisa, Italy Abstract. We study the Kolmogorov equation associated with a second order stochastic variational inequality related to the harmonic oscillator Mathematics Subject Classification AMS : 35Q0, 37L40. Key words: stochastic variational inequality, Kolmogorov equation, invariant measure, m dissipative operator. Introduction Consider the second order stochastic differential equation in R, dẋ(t) + (ax(t) + Ẋ(t))dt + β 0(X(t))dt dw (t), X(0) = x, Ẋ(0) = y, x, y R, (.) dx(t) where Ẋ(t) =, a is a positive constant, W (t) is a standard Brownian dt motion in a probability space (Ω, F, P) and β 0 : R 2 R is the multivalued graph, { 0, if r > 0 β 0 (r) = (.2) (, 0], if r = 0. Roughly speaking, (.) describes the dynamics of an harmonic oscillator with damping subject to a stochastic forcing term dw and limited from the left by the obstacle {x = 0}. This is a second order stochastic variational inequality for wich a direct existence theory is missing. In the deterministic case however one can prove Partially supported by the Italian National Project MURST Equazioni di Kolmogorov.

3 the existence of a solution X L (0, ) W, loc (0, ) which satisfies (.) (with 0 replacing W (t)) in the following weak sense (i.e. in the sense of distributions on (0, )) X 0, Ẍ + ax + Ẋ 0, support {Ẍ + ax + Ẋ} {t : X(t) = 0}, d dt ( ( ) ) 2 Ẋ 2 + ax 2 + Ẋ 2 = 0. (See e.g. []). Let us re write equation (.) as a system dx = Y dt dy = (ax + Y + β 0 (X))dt + dw (t) X(0) = x, Y (0) = y. (.3) In order to study (.3) it is natural to introduce the penalized system (as in [0] for a first order equation with reflexion), dx ε = Y ε dt dy ε = (ax ε + Y ε ε X ε )dt + dw (t) X ε (0) = x, Y ε (0) = y. (.4) where X ε = max{ X ε, 0}. The Kolmogorov operator corresponding to (.4) reads as follows, N ε ϕ(x, y) = 2 ϕ yy(x, y) (ax + y ε x )ϕ y (x, y) + yϕ x (x, y) (.5) and by an easy computations (see e.g.[7]) one can see the N ε possesses a unique invariant probability mesure ν ε given by ν ε (dx, dy) = e (ax2 +y 2 +ε (x ) 2 ) e (ax2 +y2 +ε (x )2 ) dx dy. (.6) As ε 0, ν ε ν where ν is concentrated on = (0, ) R and it is given by e (ax2 +y 2 ) ν(dx, dy) = ρ(x, y)dxdy, ρ(x, y) =. (.7) e (ax2 +y2 ) dx dy 2

4 Moreover the Kolmogorov operator N ε converges to the following one Nϕ(x, y) = 2 ϕ yy(x, y) (ax + y)ϕ y (x, y) + yϕ x (x, y), (x, y). (.8) It is natural to require that ν be (infinitesimally) invariant for N, that is that Nϕdν = 0, for any smooth function ϕ. A straighforward integration by parts show that in order to this happen we must require that ϕ fulfill a irichlet homogeneous condition on the boundary of, namely, that ϕ(0, y) = 0, for all y R. Since the operator N is elliptic degenerate, no well posedness theory in space of continuous or L p (with respect to the Lebesgue measure) functions seems to be available for the corresponding irichlet problem. So, we shall try to study this problem in the space L 2 (, ν). We note that in the past few years there was an increasing interest in studying Kolmogorov operators with irregular coefficients in spaces L p with respect to an invariant measure ν, see e.g. [], [2], [3] and references therein. In general, one can show that a Kolmogorov operator, defined in a space of smooth functions is closable in some space L p with respect to ν and its closure is an m dissipative operator and this allows to construct a dynamics. We notice that, due to the strong degeneracy of operator N, we were unable to perform this program in the present case, but we have proved however that a suitable weaker realization (equipped with the boundary condition ϕ(0, y) = 0) of the operator N (still denoted N) is m dissipative in L 2 (, ν), see Theorem 2.4 below. ue to the irichlet boundary condition, the corresponding transition semigroup P t = e tn will be sub-markovian. We shall also prove an ergodicity type result, that is that if ϕ (N) and Nϕ = 0 we have ϕ = 0, see Proposition 3.. Though this is out of the scope of this paper, we notice that with the help of the semigroup P t one can try to construct a Markov process with life time (see e.g. [9, efinition.4]) which is a weak solution of equation (.) using the theory of the irichlet forms, see [8] and [9]. 3

5 Theorem 2.4 can be extended to more general equations of the form, dẋ(t) + (MX(t) + Ẋ(t))dt + β(x(t))dt dw (t), (.9) X(0) = x, Ẋ(0) = y, where M is a n n symmetric and positive definite matrix, W (t) is an n dimensional Brownian motion with covariance the identity I and β : R n 2 Rn is the multivalued mapping defined by, β(r,...r n ) = (β (r ),..., β n (r n )) β i (r) = 0 if r > 0, β i (0) = (, 0] for i =,.., m < n, β i (r) = 0 for i = m +,.., n. (.0) We notice that in the case where β = U and U is regular problem (.9) was studied in [4]. However,since in this case the domain of U is the whole space R 2 the elliptic problem is on the whole space and no boundary conditions arise. We end this section giving some notations. We shall use the standard notations for function spaces on R 2 and (0, ) R. In particular where ϕ x = ϕ, ϕ x y = ϕ y H loc() = {ϕ L 2 loc() : ϕ x, ϕ y L 2 loc()}, are taken in the sense of distributions on. By C k () (respectively C k ()) we shall denote the space of all continuously differentiable functions on (respectively on ) up to the order k. We set also C b () = {ϕ C() : sup ϕ(x, y) < + }. (x,y) We shall denote by L 2 (, ν) the space of all Borel functions ϕ: R such that ϕ 2 (x, y)ν(dxdy) <. enote by, the scalar product in R n and by the corresponding Euclidean norm. C 0 () will represent the space of all infinitely differentiable real functions on with compact support and by () the space of distributions on. 4

6 2 The Kolmogorov operator associated to equation (.) For a rigorous definition of N we introduce the spaces and respectively, V 0 = {ψ L 2 (, ν) : ψ y L 2 (, ν)} V = {ψ H loc() : xψ, yψ, ψ x, ψ y L 2 (, ν)}, all derivatives being in the sense of distributions. Both spaces V 0 and V are endowed with their natural inner products and norms, ψ 2 V 0 = (ψ 2 + ψy)ρdxdy, 2 and ψ 2 V = (ψ 2 ( + x 2 + y 2 ) + ψ 2 x + ψ 2 y)ρdxdy. We shall denote by V the dual of V. We note that V L 2 (, ν) algebrically and topologically. Moreover, V is dense in L 2 (, ν). Indeed if ψ H () L 2 (, ν) (which is dense in L 2 ψ (, ν)) we see that ψ ε = V (+ε(x 2 +y 2 )) /2 and ψ ε ψ in V as ε 0. We have therefore V L 2 (, ν) V, algebrically and topologically with dense immersions. We notice that V is not a distributional space however, because V does not include C 0 () as dense subset. Let now introduce the operator N 0 : V 0 V which is defined by ( ) ψ, N 0 ϕ V,V = 2 ϕ y(ψρ) y + (ax + y)ϕ y ψρ + y(ρψ) x ϕ dxdy, (2.) for all ϕ V 0, ψ V (Here, V,V note that N 0 is bounded. is the pairing between V and V ). We 5

7 It is convenient to introduce the approximating elliptic boundary problem, λϕ ε 2 εϕε xx 2 ϕε yy + (ax + y)ϕ ε y H ε (y)ϕ ε x = f(x, y) (2.2) ϕ ε (0, y) = 0, (x, y), where H ε is the following approximation of function y, y, if y ε, H ε (y) = ε, if y > ε, ε, if y < ε. By [6, Proposition 2.2] it follows that for any f C b (), problem (2.2) has a unique classical solution ϕ ε C 2 () C b (). Moreover, by the maximum principle, [6, Proposition A.] we have that ϕ ε L () λ f L (). (2.3) The latter follows by applying the maximum principle to ϕ ε λ f L () and to ϕ ε + λ f L () respectively. In order to apply Proposition A. we note the Lyapunov function Φ(x, y) = x 2 + y 2 satisfies assumption (iv) in [6, Hypothesis.]. We have introduced the approximation H ε (y) of y into equation (2.2) in order to get later sharp estimates for (ϕ ε ) x (0, y). Proposition p2. below is the main ingredient for a rigorous construction of N. Proposition 2. Let f C 0 (), λ > 0 and let ϕ ε be the solution of (2.2). Then there is a subsequence of (ϕ ε ), still denoted (ϕ ε ) such that, (i) ϕ ε ϕ weak in L () and weakly in L 2 (, ν). (ii) ϕ ε y ϕ y weakly in L 2 (, ν). (iii) (λi N 0 )ϕ ε f weakly in V. (iv) ϕ belongs to (N 0 ) and we have (λi N 0 )ϕ = f. 6

8 We notice that since L 2 (, ν) is dense in V so is C 0 (). Before proving Proposition 2. we need a lemma. Lemma 2.2 Let f C b (), λ > 0 and let ϕ ε be the classical solution of (2.2). Then ϕ ε x, ϕ ε y L 2 (, ν) and there exists C > 0 such that (ε(ϕ ε x) 2 + (ϕ ε y) 2 )dν C f 2 L (), ε > 0. (2.4) Moreover, ϕ ε W 2,p ( B n ) C ( B n ), n N, p 2, (2.5) where B n = {(x, y) R 2 : x 2 +y 2 n 2 } and W 2,p is the usual Sobolev space. Proof. Consider the function θ C ([0, )) such that θ, θ = on [0, ], θ = 0 on [2, ) and set ( ) x 2 + y 2 χ n (x, y) = θ, (x, y) R 2. n Multiplying (2.2) by ϕ ε χ n ρ, integrating over and integrating by parts in the second and third integral yields, λ (ϕ ε ) 2 χ n ρdxdy + ε ϕ ε 2 x(ϕ ε χ n ρ) x dxdy + ϕ ε 2 y(ϕ ε χ n ρ) y dxdy + (ax + y)[(ϕ ε ) 2 ] y χ n ρdxdy (2.6) 2 H ε (y)[(ϕ ε ) 2 ] x χ n ρdxdy = fϕ ε χ n ρdxdy. 2 Now, again integrating by parts, we obtain that λ (ϕ ε ) 2 χ n ρdxdy + ε ϕ ε 2 x(ϕ ε χ n ρ) x dxdy + ϕ ε 2 y(ϕ ε χ n ρ) y dxdy (ϕ ε ) 2 [(ax + y)χ n ρ] y dxdy 2 + H ε (y)(ϕ ε ) 2 [χ n ρ] x dxdy = fϕ ε χ n ρdxdy, 2 7 (2.7)

9 which is equivalent to λ (ϕ ε ) 2 χ n ρdxdy + ε 2 (ϕ ε x) 2 χ n ρdxdy + ε [(ϕ ε ) 2 ] x (χ n ρ) x dxdy (ϕ ε y) 2 χ n ρdxdy + [(ϕ ε ) 2 ] y (χ n ρ) y dxdy 4 (ϕ ε ) 2 [(ax + y)χ n ρ] y dxdy H ε (y)(ϕ ε ) 2 [χ n ρ] x dxdy = fϕ ε χ n ρdxdy. Integrating by parts in the third and fifth integral, yields λ (ϕ ε ) 2 χ n ρdxdy + ε 2 (ϕ ε x) 2 χ n ρdxdy ε (ϕ ε ) 2 (χ n ρ) xx dxdy (ϕ ε y) 2 χ n ρdxdy (ϕ ε ) 2 (χ n ρ) yy dxdy 4 (ϕ ε ) 2 [(ax + y)χ n ρ] y dxdy H ε (y)(ϕ ε ) 2 [χ n ρ] x dxdy = fϕ ε χ n ρdxdy, which is equivalent to λ (ϕ ε ) 2 χ n ρdxdy + ( ε(ϕ ε 2 x ) 2 + (ϕ ε y) 2) χ n ρdxdy (ϕ ε ) 2 (ε(χ n ρ) xx + (χ n ρ) yy ) dxdy 4 2 = (ϕ ε ) 2 [((ax + y)χ n ρ) y H ε (y)(χ n ρ) x ] dxdy ( ) /2 ( fϕ ε χ n ρdxdy (ϕ ε ) 2 ρdxdy 8 f 2 ρdxdy) /2, (2.8) (2.9) (2.0)

10 by the Hölder inequality. Taking into account that ρ x = 2axρ, ρ y = 2yρ, ρ xx = 2a(2ax 2 ]ρ, ρ yy = 2(2y 2 ]ρ and using (2.3) and the following straightforward inequality (ϕ ε ) 2 (H ε (y)ρχ n ) x dxdy we obtain that λ 0 ( C y e y2 dy + ε /2 y ε /2 (ϕ ε ) 2 χ n dν + 2 y ε /2 e y2 ) dy C, (ε(ϕ ε x) 2 + (ϕ ε y) 2 )χ n dν C f 2 L (), (2.) where C is independent of ε and n. Then letting n we get estimate (2.4) as claimed. Next by (2.2) we see that ψn ε = ϕ ε χ n is the solution to elliptic boundary value problem λψn ε ε 2 (ψε n) xx 2 (ψε n) yy = g n in B 2n, ψ ε n = 0 on ( B 2n ), where g n L 2 ( B 2n ). Then by the Agmon ouglis Nirenberg inequalities we infer that ψ ε n H 2 ( B 2n ) H 0( B 2n ) and by a bootstrap argument we conclude that ψ ε n W 2,p ( B 2n ) for all p 2. Hence ψ ε W 2,p ( B 2n ) for all n as claimed. In particular, this implies that ψ ε C ( B 2n ) for all n and so ψ ε x(0, ) C(R) is well defined. The proof of Lemma is complete. We are now ready to prove Proposition 2.. Proof of Proposition 2.. (i) follows from (2.3) and (ii) from (2.4). Moreover, from (2.4) it follows also that εϕ ε x 0, strongly in L 2 (, ν). (2.2) 9

11 Indeed by (2.4) we have that (εϕ ε x) 2 dν Cε, ε > 0. For further estimates on ϕ ε it is convenient to replace ϕ ε by ϕ ε (x, y) = ϕ ε ( εx, y), (x, y). (2.3) We have λ ϕ ε ϕ 2 ε + (ax + y)( ϕ ε ) y Hε(y) ε ( ϕ ε ) x = f( ε x, y) ϕ ε (0, y) = 0, (x, y). Consider the solution z ε (x) to the equation λz ε 2 z ε ε z ε = d ε z ε (0) = 0, x (0, + ), (2.4) (2.5) where d ε = sup{f( ε x, y) : (x, y) } 0. We note that, since f C 0 () for any m N there exists L m > 0 such that d ε L m ε m. (2.6) The solution of (2.5) is given by ( q x ) z ε (x) = d ε λ e ε + ε 2 +2λ. By (2.4), (2.5) we see that λ( ϕ ε z ε ) 2 ( ϕ ε z ε ) + (ax + y)( ϕ ε z ε ) y Hε(y) ε = f( ε x, y) + d ε + ( Hε(y) ε ε ) (z ε ) x 0, x, y, ( ϕ ε z ε ) x ( ϕ ε z ε )(0, y) = 0, y R. 0

12 Consequently, by (2.6) and the maximum principle (see e.g. [6, Proposition A.]) we see that there exists C > 0 such that ϕ ε (x, y) z ε (x) C ε d εx CLε m x, (x, y). (2.7) Similarly, the same argument applied to function ϕ ε = ϕ ε and z ε solution to (2.5) where d ε is replaced by yields d ε = sup{ f( ε x, y) : (x, y) } 0, ϕ ε (x, y) z ε (x) C ε d εx CLε m x, (x, y), (2.8) where C > 0. By (2.7) and (2.8) it follows that, for some constant C 2 > 0, which implies Finally, by (2.3) we see that ϕ ε (x, y) CLε m x, (x, y), ( ϕ ε ) x (0, y) CLε m, y R. (ϕ ε ) x (0, y) CLε m /2, y R. (2.9) Now we are ready to prove the last part of the proposition. Multiplying both sides of (2.2) by ψρ, where ψ V and integrating over, yields [ λϕ ε ψρ + 2 εϕε x(ψρ) x + ] 2 ϕε y(ψρ) y + (ax + y)ϕ ε yψρ + H ε (y)ϕ ε (ψρ) x dxdy + 2 ε R ϕ ε x(0, y)ψ(0, y)ρ(0, y)dy = fψρdxdy. A little problem arises here however, since we do not know if ϕ ε xxψρ and ϕ ε yyψρ are in L (). In order to avoid this inconvenience we shall first to establish the latter for ψ V with support included in B n (recall that by (2.5), ϕ ε xx, ϕ ε yy L p ( B n ), for all p 2, n N) and after for general ψ V via a standard approximation argument. Concerning the terms where

13 first order derivatives of ϕ ε appear, they are well defined thanks to (2.) and since ψ V. Letting ε tend to 0 we obtain from (i), (ii), (2.4) and (2.9) that [ λϕ ε ψρ + ] 2 ϕε y(ψρ) y + (ax + y)ϕ ε yψρ + y ϕ ε (ψρ) x dxdy fψρdxdy, because [ λϕ ε ψρ + ] 2 ϕε y(ψρ) y + (ax + y)ϕ ε yψρ + H ε (y)ϕ ε (ψρ) x dxdy and by definition of H ε we have (H ε (y) y)ϕ ε (ψρ) x dxdy 2 { y ε /2 } (2.20) fψρdxdy, e a(x2 +y 2) y ( ψ x + 2ax ψ )dxdy ( ) /2 ( ) /2 2 e a(x2 +y 2) ( ψ x 2 + 4a 2 x 2 ψ 2 )dxdy e (ax2 +y 2) y 2 dxdy { y ε /2 } 0, as ε 0. By (2.20) it follows that as ε 0. In other words ψ, (λi N 0 )ϕ ε V,V ψ, f V,V, ψ V, lim (λi N 0)ϕ ε = f weakly in V. ε 0 Since a subsequence of ϕ ε is weakly convergent to ϕ in V 0 and N 0 : V 0 V is bounded, we can conclude that (λi N 0 )ϕ = f as claimed. efinition 2.3 The Kolmogorov operator associated to equation (.) is the closure N = N in L 2 (, ν) of the operator N : (N ) L 2 (, ν) L 2 (, ν) defined by N ϕ = N 0 ϕ, ϕ (N ), (2.2) (N ) = {ϕ V 0 : N 0 ϕ L 2 (, ν)}. 2

14 The above definition makes sense because by (2.) and (2.2) it follows that N is closable. Let in fact {ϕ n } V 0 such that and let ψ C 0 (). Then we have, ϕ n 0, N 0 ϕ n ξ in L 2 (, ν) ψ, ξ V,V = lim ψ, N 0 ϕ n V,V n ( ) = lim n 2 ϕn y(ψρ) y + (ax + y)ϕ n yψρ + y(ψρ) x ϕ n dxdy ( ) = lim ϕ n n 2 (ψρ) yy + [(ax + y)ψρ] y + y(ψρ) x ϕ n dxdy = 0. So, ξ = 0 and we have proved that N is closable. Notice now that (N ) V 0 and that if ϕ (N ) and N ϕ = g L 2 (, ν) then by (2.) and (2.2) we have ( ) 2 ϕ y(ψρ) y + (ax + y)ϕ y ψρ + y(ρψ) x ϕ dxdy = gψρdxdy, ψ V. In particular, for ψ C 0 () this yields (2.22) 2 ϕ yy (ax + y)ϕ y + yϕ x = g in (). (2.23) Now if ϕ is sufficiently regular we see by (2.22), (2.23) via integration by parts that ϕ(0, y) = 0 for all y R. This means that N defined by (2.2) and consequently N is the realization of the elliptic operator (.8) with irichlet homogeneous boundary conditions in the space () (i.e. in the weak sense). Theorem 2.4 The operator N is invariant with respect to the measure ν and m dissipative in L 2 (, ν). Moreover, one has Nϕ ϕ dν ϕ 2 2 ydν, ϕ (N). (2.24) 3

15 Proof. By (2.) with ϕ V 0 and ψ = we see that ( ) N 0 ϕ, V,V = N 0 ϕdν = 2 ϕ yρ y + (ax + y)ϕ y ρ + yρ x ϕ dxdy ( ) ϕ 2 ρ yy + ((ax + y)ρ) y + yρ x dxdy = 0 and therefore N 0ϕdν = 0 for all ϕ V 0. Since (N ) V 0 we infer that N ϕdν = 0, ϕ (N ), and so Nϕdν = 0, ϕ (N), which proves the invariance of ν with respect to ν. It remains to prove dissipativity of ν and (2.24). For this fix λ > 0 and set Γ: = {ϕ (N ) : λϕ N 0 ϕ C 0 ()}. Let ϕ Γ and let ϕ ε be the solution of (2.2). Then by Proposition 2. (iii) we have, ϕ = lim ε 0 ϕ ε, N ϕ = N 0 ϕ = lim ε 0 (λϕ ε f) weakly in L 2 (, ν). (2.25) Moreover, by (2.2) we have (λϕ f)ϕdν = N 0 ϕ ϕ dν lim inf ε 0 lim inf ε 0 = lim inf ε 0 = 2 (λϕ ε f)ϕ ε dν ( 2 εϕε xx + ) 2 ϕε yy (ax + y)ϕ ε y + H ε (y)ϕ ε x dν lim inf ε 0 ( 2 ϕε y(ϕ ε ρ) y + ) 2 εϕε x(ϕ ε ρ) x + (ax + y)ϕ ε yϕ ε ρ + H ε (y)(ϕ ε ρ) x ϕ ε dxdy ( ϕ ε y 2 + ε ϕ ε x 2 )dν, 4 (2.26)

16 because, as it can be checked by straightforward integrations by parts, one has ( ) 2 ϕε yϕ ε ρ y + (ax + y)ϕ ε yϕ ε ρ + y(ϕ ε ρ) x ϕ ε dxdy = 0 and as seen earlier in the proof of Proposition 2. (H ε (y) y)ϕ ε (ϕ ε ρ) x dxdy = 0. lim ε 0 Then by Proposition 2. (ii) and (2.26) we infer that N 0 ϕ ϕ dν + ϕ y 2 dν 0, ϕ Γ. 2 Since C0 () is dense in L 2 (, ν) and by previous inequality N 0 is dissipative on Γ we infer that Γ is a core for N. This implies that Nϕ ϕ dν + ϕ y 2 dν 0, ϕ (N). (2.27) 2 as claimed. In particular it follows that N is dissipative. On the other hand by Proposition 2. we know that the range of λi N is dense in L 2 (, ν) because C 0 () R(λI N 0 ) L 2 (, ν) for all λ > 0, i.e. N is m dissipative by the Lumer Phillips theorem. Remark 2.5 It is obvious that Theorem 2.4 extends to domains of the form = (b, ) R and so to equation (.) with β 0 (r) = 0 for r > b, β 0 (r) = (, 0] where b R. Remark 2.6 An open problem is whether or not one has equality in (2.4), i.e. the carré du champs formula holds for Kolmogorov operator N. Anyway this happens for all ϕ (N) which are sufficiently smooth but is not clear that there is a core of this form. 3 Ergodicity of ν Proposition 3. Assume that ϕ (N) and Nϕ = 0. ϕ = 0. Then we have 5

17 Proof. By (2.24) we see that ϕ y = 0 and therefore ϕ (which belongs to V 0 since (N) V 0 ) is constant with respect to y, i.e. ϕ(x, y) = ϕ(x), x > 0, y R. On the other hand, by the definition of N, there exists a sequence {ϕ n } (N ) such that lim n ϕn = ϕ, lim N ϕ n = Nϕ = 0, n in L 2 (, ν). Since N ϕ n = N 0 ϕ n, we have by definition of N 0 that for any ψ V, 0 = ψ, N 0 ϕ V,V ( ) = lim n 2 (ϕn ) y (ψρ) y + (ax + y)(ϕ n ) y ψρ + y(ρψ) x ϕ n dxdy. For ψ C0 () this means that (recall that ϕ n ϕ in L 2 (, ν)) 2 ϕ yy + (ax + y)ϕ y (yϕ) x = 0 in () and therefore yϕ x = 0 in (). Hence yϕ(x)ψ (y)ψ 2(x)dxdy = 0, ψ C0 (R), ψ 2 C0 ((0, )). This yields and therefore 0 R yψ (y)dy ϕ(x)ψ 2(x)dx = 0, 0 ϕ(x)ψ 2(x)dx = 0 ψ 2 C 0 ((0, )), i.e. ϕ = C is constant. Now we show that ϕ(0) = 0. By definition of N = N we see that for all ψ V (because Nϕ = 0) 0 = lim n ( 2 (ϕn ) y (ψρ) y + (ax + y)(ϕ n ) y ψρ + y(ρψ) x ϕ n ( ) = lim n 2 ϕn (ψρ) yy + ϕ n [(ax + y)ψρ] y y(ρψ) x ϕ n dxdy ( ) = C 2 (ψρ) yy + [(ax + y)ψρ] y y(ρψ) x dxdy 6 ) dxdy

18 We choose now ψ depending only on x. Then we deduce 0 = C y(ρψ) x dxdy = Cψ(0) ye ay2 dy. Since ψ(0) can be chosen arbitrary, since ψ is arbitrary in V, we infer that C = 0 as claimed. R 4 The semigroup P t Let N be the operator defined by Theorem 2.4 and let P t be the semigroup generated by N in L 2 (, ν). We recall that P t is called sub-markovian if P t and P t ϕ 0 if ϕ 0. Proposition 4. The semigroup P t is sub markovian. Proof. In order to prove that P t is sub Markovian it suffices to show that for any λ > 0 we have by the Kato Trotter theorem: 0 (λi N) (), λ 2 0 (λi N) f 0 for all f 0. We set N ε ϕ = εϕ 2 xx + ϕ 2 yy (ax + y)ϕ y + H ε (y)ϕ x, (N ε ) = {ϕ V : εϕ 2 xx + ϕ 2 yy L 2 (, ν), ϕ(0, y) = 0, y R}, (4.) By Proposition 2., we know that lim (λi N ε) f = (λi N) f, f C0 (), λ > 0 (4.2) ε 0 in the weak topology of L 2 (, ν). On the other hand by the maximum principle applied to equation (2.2) we have that (λi N ε ) f 0 on if f 0, f C0 () and by (4.2) we infer that (λi N) f 0 for all f C0 (), f 0. Since (λi N) is continuous in L 2 (, ν) and C0 () is dense in L 2 (, ν), we have that (λi N) f 0 for all f L 2 (, ν), f 0 a.e.. Next we prove that (λi N) () λ, λ > 0, a.e. on 7

19 or equivalently that (I λn) (), λ > 0, a.e. on. For f C0 () such that f on we set ϕ ε = (I λn ε ) f, i.e. ϕ ε λ 2 εϕε xx λ 2 ϕε yy + λ(ax + y)ϕ ε y λh ε (y)ϕ ε x = f(x, y) ϕ ε (0, y) = 0, (x, y). Equivalently (ϕ ε ) λ 2 ε(ϕε ) xx λ 2 (ϕε ) yy +λ(ax + y)(ϕ ε ) y λh ε (y)(ϕ ε ) x 0 (ϕ ε )(0, y) 0, (x, y). By the maximum principle we infer that ϕ ε on. Then by (4.2) we see (I λn) (f), f C 0 (), f. (In fact the inequality ϕ ε a.e in is preserved in the weak convergence on L 2 (, ν)) Since (I λn) is continuous in L 2 (, ν), we have by density of the embedding C 0 () L 2 (, ν), that as claimed. (I λn) (f), f L 2 (, ν), f, Remark 4.2 The solution u to equation i.e. u t = Nu, u(0) = ϕ, u t = u 2 yy (ax + y)u y + yu x in (0,, u(t, 0, y) = 0 for t 0, y R u(0, x, y) = ϕ(x, y) in, (4.3) is related to the optimal stopping time problem on for the linear stochastic equation, { dx = Y dt (4.4) dy = (ax + Y )dt + dw (t). 8

20 More precisely we have u(t, x, y) = E [ ϕ(x T t x,y (τ), Y T t x,y (τ)) ], (x, y), 0 < t < T, (4.5) where τ = τ t x,y is the first exit time from of solution (X, Y ) of (4.4) with initial conditions X(T t) = x, Y (T t) = y. It should be mentioned that if x > 0 and y R then on the time interval (0, τ) the solution (X, Y ) of (4.4) is a solution to variational equation (.3) as well. 5 The n equation (.9) The previous results extend mutatis mutandis to the n dimensional system (.9), where M is a n n symmetric and positive definite matrix, W (t) is an n dimensional Brownian motion with covariance the identity I and β : R n 2 Rn is the multivalued mapping defined by (.0). We take here = {x = (x,..., x n ) R n, y R n : x i > b, i =,..., n}, where b R is fixed. Formally, the corresponding Kolmogorov operator is given by Nϕ(x, y) = 2 yϕ(x, y) Mx + y, ϕ y (x, y) + y, ϕ x (x, y), (5.) where ϕ = 0 on. efine the operator N 0 : V 0 V by ( ) ψ, N 0 ϕ V,V = 2 ϕ y, (ψρ) y + Mx + y, ϕ y ψρ + y, (ρψ) x ϕ dxdy, for all ϕ V 0, ψ V (Here ϕ x = x ϕ, ϕ y = y ϕ). We consider on the probability measure ν(dx, dy) = ρ(x, y)dxdy, ρ(x, y) = (5.2) e Mx,x y 2 e Mx,x y 2 dx dy. (5.3) efinition 5. The Kolmogorov operator associated to equation (.9) is the closure N = N in L 2 (, ν) of the operator N : (N ) L 2 (, ν) L 2 (, ν) defined by N ϕ = N 0 ϕ ϕ (N ), (N ) = {ϕ V 0 : N 0 ϕ L 2 (, ν)}. (5.4) 9

21 We have Theorem 5.2 The operator N is invariant with respect to the measure ν and m dissipative in L 2 (, ν). Moreover, one has Nϕ ϕ dν ϕ y 2 dν, ϕ (N). (5.5) 2 Proof. The proof is exactly the same as that of Theorem 2.4 so, it will be sketched only. The main step is the proof of density of R(λI N) in L 2 (, ν). We fix f C0 () and consider the equation λϕ ε ε 2 xϕ ε 2 yϕ ε + Mx + y, ϕ ε y H ε (y), ϕ ε x = f(x, y) (5.6) ϕ ε (z, y) = 0, (x, y), z = (b,..., b). This has a unique classical solution ϕ ε C 2 () C b () (again by [6]) for which the following estimates hold ( ϕ ε y 2 + ε ϕ ε x 2) dν C, ϕ ε Cb () C ε > 0, ϕ ε x(b, y) Cε y R n. Then we infer as in the proof of Proposition 2. that ϕ ε ϕ, weak in L (), ϕ ε y ϕ y, weakly in L 2 (, ν), where (λi N 0 )ϕ = f. Then we may conclude the proof as in the previous case. Example 5.3 (The elastic string with discontinuous distribution of masses) Consider an elastic string [0, L] damped at z = 0, z = L. At the points z i = ih, i =,..., N, h = L act gravitational forces with mass m. If N x i = x i (t) is the displacement of the string at z i, the dynamic is described by the system, see [2, pp ] ( xi x i mẍ i = a + x ) i x i+ x h 2 h 2 i, i =,..., N (5.7) x 0 = x N = b 0 b, x i (0) = x 0 i, x i (0) = x i, i =,..., N. 20

22 If the motion is limited from below by a rigid obstacle, i.e. x i b, i =,..., N and subject to a random force dw (t) = (dw (t),..., dw N (t)) then the dynamics (5.7) is replaced by the stochastic variational inequality, m dx i = a x 0 = x N = b 0 b, ( xi x i + x i x i+ h 2 h 2 ) dt x i dt β(x i )dt + dw i (t), x i (0) = x 0 i, x i (0) = x i, i =,..., N, where β 0 is the multivalued graph defined by (.2). Equivalently dẋ(t) + (MX(t) + Ẋ(t))dt + β(x(t))dt dw (t), X(0) = x, Ẋ(0) = y, where β(x) = col {β 0 (x i )} N i=, X = col {x i} N i= and (take m = ) M = a h (5.8) (5.9) Thus Theorem 5.2 is applicable in the present situation. Though in the deterministic case (W (t) = 0) equation (5.9) was studied in [], a direct study of it in the stochastic case seems to be open. We mention that formally (5.8) may be wiewed as a finite difference approximation for the stochastic wave equation dẋ(t) + ( ξx(t) + Ẋ(t))dt + β(x(t))dt dw (t), (5.0) X(0, ξ) = x(ξ), Ẋ(0, ξ) = y(ξ), ξ [0, L]. Acknowledgements This work was written during the visit of authors to University of Bielefeld. We are indebted to M. Röckner for kind hospitality and fruitfull discussions on subject of this work. Also the authors are indebted to E. Priola which pointed out his joint paper [6]. 2

23 References [] V.I. Bogachev, N.V. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial iff. Equations, 26, no. 2, 200. [2] R. S. Crawford, Waves, Berkeley Physics Course, 968. [3] G. a Prato, Kolmogorov equations for stochastic PEs, Birkhäuser, 2004 [4] G. a Prato and A. Lunardi, Maximal dissipativity of a class of elliptic degenerate operators in weighted L 2 spaces, CS, to appear. [5] G. a Prato and J. Zabczyk, Ergodicity for infinite dimensional systems, London Mathematical Society Lecture Notes, 229, Cambridge University Press, 996. [6] S. Fornaro, G. Metafune and E. Priola, Gradient estimates for irichlet problems in unbounded domains, J. ifferential Equations, 205, , [7] M. Freidlin, Some remarks on the Smoluchowski-Kramers approximation, J. Statist. Phys. 7, no. 3-4, , [8] M. Fukushima, irichlet forms and symmetric Markov processes, North Holland, 980. [9] Z. M. Ma and M. Röckner, Introduction to the theory of (non symmetric) irichlet forms, Springer-Verlag, 992. [0]. Nualart and E. Pardoux, White noise driven quasilinear SPEs with reflection, Prob. Theory and Rel. Fields 93, 77-89, 992. North Holland, Amsterdam, 980. [] M. Schatzman, A class of nonlinear differential equations of second order in time, Nonlinear Analysis TMA, , 978. [2] W. Stannat, (Nonsymmetric) irichlet operators on L : existence, uniqueness and associated Markov processes, Ann. Scuola Norm. Sup. Pisa, Ser. IV, 28, 99-40,

24 Stampato in proprio NOVEMBRE 2005 Scuola Normale Superiore, Piazza dei Cavalieri, 7 - PISA