Potential Theory on Wiener space revisited

Transcription

1 Potential Theory on Wiener space revisited Michael Röckner (University of Bielefeld) Joint work with Aurel Cornea 1 and Lucian Beznea (Rumanian Academy, Bukarest) CRC 701 and BiBoS-Preprint 1 Aurel tragically passed away on Sept. 3, / 36

2 1 1. Motivation 2 2. Brownian Motion on abstract Wiener space 2.1 Preliminaries 2.2 Explicit formulae for compact Lyapunov functions 2.3 Consequences 3 3. Levy processes on Hilbert space 3.1 Preliminaries 3.2 Explicit formulae for compact Lyapunov functions 3.3 Consequences 4 4. Associated Potential Theory 4.1 Reduced functions and polar sets 4.2 Choquet capacities and tightness 4.3 Balayage principle 2 / 36

3 1. Motivation 1. Motivation A stochastic partial differential equation (SPDE) of evolutionary type can often be written as a stochastic differential equation (SDE) on a separable Hilbert (or Banach) space H as follows: where dx (t) = B(t, X (t))dt + σ(t, X (t))dz(t) B : [0, ) H Ω H, σ : [0, ) H Ω L(U, H), U := another separable Hilbert (or Banach) space, L(U, H) := all bounded linear operators from U to H, Z(t), t 0, a U-valued Levy-process on (Ω, F, (F t ), P). 3 / 36

4 1. Motivation 1. Motivation Prominent examples: Stochastic Navier-Stokes equation: dx (t) = ( X (t) P L ((X (t) )X (t)) dt + σ(t, X (t))dz(t) }{{} =B(X (t)) H := {x L 2 (R d, R d ) div x = 0}, d = 2 or d = 3. Stochastic porous media equation: dx (t) = Ψ(X (t))dt + σ(t, X (t))dz(t) H := H 1 0 (O) (= H 1 ), O R d, O open, Ψ : R R increasing. Stochastic quantization SDE: [ ] dx (t) = ( 1)X (t) : X (2N+1) (t) : dt + σ(t, X (t))dz(t), H S (R 2 ), N N. 4 / 36

5 1. Motivation 1. Motivation In finite dimensions, if σ is non-degenerate, the potential theory of the solution process X (t), t 0, is related to the nice potential theory of Z(t), t 0, e.g. if Z(t) := Brownian motion W (t), t 0, and B is not too bad. How compare potential theoretic properties of X (t), t 0, and Z(t), t 0, in infinite dimensions? Though e.g. fine topologies are in general completely different, X (t), t 0, and Z(t), t 0, share joint potential theoretic principles as Markov processes. So, study and try to identify these principles for Z(t), t 0. 5 / 36

6 2. Brownian Motion on abstract Wiener space 2.1 Preliminaries 2.1 Preliminaries (E, H, µ) = abstract Wiener space, i.e. H = separable Hilbert space with inner product, H and corresponding norm H, E = (separable) Banach space with norm E, H E continuously and densely, hence E (H )H E continuously and densely, and thus E, E E H =, H, µ = standard Gauss-measure on E, i.e. µ is the image under embedding H E of a finitely additive measure µ on H with Fourier transform e i ξ,h H µ(dh) = e 1 2 ξ 2 ξ H. Then necessarily H E compact. H 6 / 36

7 2. Brownian Motion on abstract Wiener space 2.1 Preliminaries 2.1 Preliminaries Theorem µ σ-additive on E E is Gross-measurable [Dudley-Feldman-LeCam 1971] : [Gross 1965] Hence for t > 0 also µ t finitely additive measure on H s.th. e i ξ,h H µ t (dh) = e 1 2 t ξ 2 H ξ H, H extends to (σ-additive) measure µ t on E. (So, µ = µ 1 ). 7 / 36

8 2. Brownian Motion on abstract Wiener space 2.1 Preliminaries 2.1 Preliminaries Semigroup: p t (x, A) := µ t (A x), A B(E), x E, t > 0, acts on bounded Borel functions f : E R P t f (x) := f (y) p t (x, dy) = E E f (x + y) µ t (dy). Theorem [Gross 1967] There exists a Markov process W (t), t 0, on E with continuous sample paths and transition semigroup given by (P t ) t 0 above. W (t), t 0 = Brownian motion on (E, H). 8 / 36

9 2. Brownian Motion on abstract Wiener space 2.2 Explicit formulae for compact Lyapunov functions 2.2 Explicit formulae for compact Lyapunov functions From now on dim H =, hence µ(h) = 0. Consider corresponding resolvent U := (U α ) α>0, where U α f (x) := 0 e αt P t f (x) dt, x E, α > 0. u : E [0, ], Borel measurable, is called 1-excessive if αu α+1 u u and lim αu α+1u = u. α All such E 1 (U ). Note if u := U 1 f, f 0, f Borel, then u excessive. u is called compact Lyapunov function, if u is excessive with compact level sets, i.e. {u α} is compact α > 0. 9 / 36

10 2. Brownian Motion on abstract Wiener space 2.2 Explicit formulae for compact Lyapunov functions 2.2 Explicit formulae for compact Lyapunov functions Consider ONB {e n : n N} of H in E, separating the points of E and for n N define P n : E E by P n z := n E e k, z E e k, z E, k=1 and P n := P n H. Fix x E\H. By [Carmona 1980] we can choose an ONB {e x n : n N} as above such that in addition, E ex n, x E 2 n 2 n N. 10 / 36

11 2. Brownian Motion on abstract Wiener space 2.2 Explicit formulae for compact Lyapunov functions 2.2 Explicit formulae for compact Lyapunov functions It follows by Dudley-Feldman-LeCam Theorem that lim P n z z = 0 in µ-measure, E n and since H E compact can choose subsequence Q n, n N, of P n, n N, such that for Q n := Q n H and all n N Id H Q n L(H,E) 1 2, ({ n }) µ z E : z Q n z > 1 E 2 1 n 2 n 11 / 36

12 2. Brownian Motion on abstract Wiener space 2.2 Explicit formulae for compact Lyapunov functions 2.2 Explicit formulae for compact Lyapunov functions Now define (with Q 0 := 0) ( ) 2 q x (z) := 2 n Q n+1 z Q n z n 2 E en x, z E E n=0 n=1 (wich has compact level sets!) and define 1 2, z E, E x := {q x < } (linear space!). Then improving [Kusuoka 82] we get: 12 / 36

13 2. Brownian Motion on abstract Wiener space 2.2 Explicit formulae for compact Lyapunov functions 2.2 Explicit formulae for compact Lyapunov functions Proposition (i) µ(e x ) = 1, hence µ t (E x ) = µ( te x ) = 1 x / E x. t > 0. Furthermore, (ii) q x (h) 3 h H h H, hence H E x continuously and densely, and thus by (i) in particular µ t (H + x) = 0 t > 0. (iii) z E 2q x (z) z E, in particular, (by Fatou) (E x, q x ) is complete. Furthermore, (E x, q x ) is compactly embedded into (E, E ) and (E x, H, µ) is an abstract Wiener space. 13 / 36

14 2. Brownian Motion on abstract Wiener space 2.2 Explicit formulae for compact Lyapunov functions 2.2 Explicit formulae for compact Lyapunov functions For z E define T z : E E by T z (y) := y + z, y E. Theorem I [Beznea/R. 2010]. Fix x E and define v x 0 := U 1 q 2 x and v x z := v x 0 T z, z E. Then each v x z is a compact Lyapunov function such that E x + z = {v x z < }, and z E P t 1 E x +z = 1 E x +z t > 0 ( Invariance ). 14 / 36

15 2. Brownian Motion on abstract Wiener space 2.2 Explicit formulae for compact Lyapunov functions 2.2 Explicit formulae for compact Lyapunov functions Proof. Note P t (E l, ) 2 E (y) = ( E l, y E + E l, z E ) 2 µ t (dz) E l, y 2 E l E. E Hence P t qx 2 qx. 2 Also: P t qx 2 2qx qx(y) 2 µ t (dy). Hence 2q 2 x + const. v x 0 = E } {{ } < by Fernique, since (E x, H, µ) abstract Wiener space! U 1 qx 2 = }{{} 1 excessive! 0 e t P t qx 2 dt q }{{} x. 2 qx 2 But q 2 x has compact level sets in E, since E x E compact, hence so has v x 0 and then also so has v x z. Invariance is elementary by construction. 15 / 36

16 2. Brownian Motion on abstract Wiener space 2.3 Consequences 2.3 Consequences Let x, z E. Since vz x is a compact Lyapunov function in E, we have E x + z is invariant for the Brownian motion on E. In particular, we have Goodman slicing for each x E E x + αx, α R, are uncountable disjoint sets, each invariant for the Brownian motion on E. Or more generally, E = (E x + z), where τ contains exactly one representative of each z τ equivalence class, where z 1 z 2 : z 1 z 2 E x. The topology on E is locally, i.e. on each {v0 x α} (compact!), equivalent to a Ray topology (i.e. generated by excessive functions). So, many theorems from Potential Theory applicable. 16 / 36

17 3.1 Preliminaries 3. Levy processes on Hilbert space 3.1 Preliminaries H = separable Hilbert space with inner product, H and corresponding norm H, E = (separable) Hilbert space with inner product, E and corresponding norm E, H E continuously and densely by Hilbert-Schmidt map, hence E (H )H E continuously and densely and thus E, E E H =, H, λ : H C continuous, negative definitive, λ(0) = 0, ν t = probability measure on E, which is image under embedding of a finitely additive measure ν t on H with Fourier transform e i ξ,h H ν t (dh) = e tλ(ξ) ξ H. H ν t exists by Bochner-Minlos Theorem. 17 / 36

18 3.1 Preliminaries 3. Levy processes on Hilbert space 3.1 Preliminaries Recall by Levy-Khintchine: λ(ξ) = E ξ, b E E ξ, Rξ E E ( e i E ξ,z E 1 i E ξ, z E 1 + z 2 E ) M(dz) ξ E, where b E, R : E E such that its composition R i E with Riesz isomorphism i E : E E is (linear) non-negative definite symmetric trace class operator on E and M a Levy measure on E, i.e. a nonnegative measure on E such that M({0}) = 0 and (1 z 2 E )M(dz) <. E 18 / 36

19 3.1 Preliminaries 3. Levy processes on Hilbert space 3.1 Preliminaries Semigroup: p t (x, A) := ν t (A x), A B(E), x E, t > 0, acts on bounded Borel functions f : E R P t f (x) := f (y) p t (x, dy) = E E f (x + y) ν t (dy). Theorem (e.g. [Fuhrman/R. 2000]) There exists a Markov process Z(t), t 0, on E with cadlag paths and transition semigroup given by (P t ) t 0 above. (Z(t), t 0) = Levy process on (E, H) 19 / 36

20 3.1 Preliminaries 3. Levy processes on Hilbert space 3.1 Preliminaries From now on assume: (H) (i) ν t (H) = 0 t > 0, ( dim H = ) (ii) C > 0 such that E E ξ, z 2 E ν t (dz) C(1 + t 2 ) ξ 2 H t > 0, ξ E. 20 / 36

21 3.1 Preliminaries 3. Levy processes on Hilbert space 3.1 Preliminaries Examples: (1) b = 0, R = i H ih i 1 E, where i H : H E, and M is such that E ξ, z 2 E M(dz) < ξ E. Then (H) holds. E (2) O R d, open, bounded; H := L 2 (O) and λ(h) := (1 e ih(x) ) }{{} dx, h H. Lebesgue measure O E = Sobolev space of sufficiently high (w.r.t. to d) order. Then Z(t), t 0, Poisson Process on (E, H) and (H) holds. 21 / 36

22 3. Levy processes on Hilbert space 3.2 Explicit formulae for compact Lyapunov functions 3.2 Explicit formulae for compact Lyapunov functions Analogous to Gaussian case consider corresponding resolvent U := (U α ) α>0, where U α f (x) := 0 e αt P t f (x) dt, x E, α > 0. and recall: u : E [0, ], Borel measurable, is called 1-excessive if αu α+1 u u and lim αu α+1u = u. α All such E 1 (U ). Note if u := U 1 f, f 0, f Borel, then u excessive. u is called compact Lyapunov function, if u is excessive with compact level sets, i.e. {u α} is compact α > / 36

23 3. Levy processes on Hilbert space 3.2 Explicit formulae for compact Lyapunov functions 3.2 Explicit formulae for compact Lyapunov functions Since H E Hilbert-Schmidt, can choose ONB {e n : n N} of H which diagonalizes this embedding, that is: λ n (0, ), n N, such that λ n < and ē n := λ 1 2 n e n, n N, n=1 form an ONB of E. In fact {e n : n N} E, separating the points of E. For n N (as before) define P n : E E by n P n z := E e k, z E e k, z E, k=1 and P n := P n H. Then n N, z E n P n z = ē k, z E ē k. (orthogonal projection in E!) In particular, k=1 lim P n z z = 0 z E. E n 23 / 36

24 3. Levy processes on Hilbert space 3.2 Explicit formulae for compact Lyapunov functions 3.2 Explicit formulae for compact Lyapunov functions Fix x E\H. By [Carmona 1980] (as before) we can choose another ONB {e x n : n N} of H in E, separating the points of E such that E ex n, x E 2 n n N. Now pick α n > 0, α n, such that α n λ n <, n=1 and define ( ) 2 q x (z) := α n λ n E e n, z 2 E + 2 n 2 E en x, z E n=1 n=1 (which has compact level sets!) and define E x := {q x < } (linear space!). 1 2, 24 / 36

25 3. Levy processes on Hilbert space 3.2 Explicit formulae for compact Lyapunov functions 3.2 Explicit formulae for compact Lyapunov functions Then, as in Gaussian case (even better), have: Proposition (i) q x L 2 (E, ν t ), in particular ν t (E x ) = 1 x / E x. t > 0. Furthermore, (ii) H E x continuously and thus by (i) ν t (H + x) = 0 t > 0. (iii) z E q x (z) z E, in particular, (by Fatou) (E x, q x ) is complete. Furthermore, (E x, q x ) is compactly embedded into (E, E ). 25 / 36

26 3. Levy processes on Hilbert space 3.2 Explicit formulae for compact Lyapunov functions 3.2 Explicit formulae for compact Lyapunov functions Theorem II [Beznea/R. 2010]. Fix x E and define v x 0 := U 1 q 2 x and v x z := v x 0 T z, z E. Then each v x z is a compact Lyapunov function such that E x + z = {v x z < }, and z E P t 1 E x +z = 1 E x +z t > 0 ( Invariance ). 26 / 36

27 3. Levy processes on Hilbert space 3.2 Explicit formulae for compact Lyapunov functions 3.2 Explicit formulae for compact Lyapunov functions Proof. By (H)(i) q 2 x(z) ν t (dz) C(1 + t 2 ). Hence P t q 2 x(y) 2q 2 x(y) + 2 C(1 + t 2 ) y E. Furthermore, since q x seminorm we have q 2 x(y + z) (q x (y) q x (z)) q2 x(y) q 2 x(z) y, z E. Therefore, P t q 2 x(y) 1 2 q2 x(y) C(1 + t 2 ). Rest, as in the Gaussian case. 27 / 36

28 3.3 Consequences 3. Levy processes on Hilbert space 3.3 Consequences Let x, z E. Since vz x is a compact Lyapunov function in E, we have E x + z is invariant for the Levy process on E. In particular, we have Goodman slicing for each x E E x + αx, α R, are uncountable disjoint sets, each invariant for the Levy process on E. Or more generally, E = (E x + z), where τ contains exactly one representative of each z τ equivalence class, where z 1 z 2 : z 1 z 2 E x. The topology on E is locally, i.e. on each {v0 x α} (compact!), equivalent to a Ray topology (i.e. generated by excessive functions). So, many theorems from Potential Theory applicable. 28 / 36

29 4. Associated Potential Theory 4. Associated Potential Theory Mainly, because topology is locally Ray by above construction of compact Lyapunov functions, we have the following results in Subsections below for both the Gaussian and Levy case (and subordinations thereof). 29 / 36

30 4. Associated Potential Theory 4.1 Reduced functions and polar sets 4.1 Reduced functions and polar sets Let A E and v E 1 (U ). Then define R A 1 v := inf{u E 1 (U ) : u v on A} reduced function of v on A and ˆR A 1 v := lim α αu 1+α(R A 1 v) balayage of v on A. Then by Hunt, if A B(E), and X (t), t 0, Brownian motion or Levy process, R A 1 v(x) = E x [e D A v(x (D A )); D A < ] and ˆR A 1 v(x) = E x [e T A v(x (T A )); T A < ], where D A := inf{t 0 X (t) A}, T A := inf{t > 0 X (t) A}. 30 / 36

31 4. Associated Potential Theory 4.1 Reduced functions and polar sets 4.1 Reduced functions and polar sets N B(E) is called polar if ˆR N 1 1 E = 0 and λ-polar for a σ-finite (nonnegative) measure λ on E, if ˆR N 1 1 E = 0 λ a.e.. In particular, by Hunt N polar P x [T N = ] = 1 x E N λ-polar P x [T N = ] = 1 for λ a.e. x E. 31 / 36

32 4. Associated Potential Theory 4.1 Reduced functions and polar sets 4.1 Reduced functions and polar sets Theorem III [Carmona 1980 in Gaussian case, Beznea/R in Levy case] H is polar. Proof. Let x E\H. Then, since H E x, 1 E 1 E x on H. But P t 1 E x = 1 E x, hence U 1 1 E x = 1 E x, so 1 E x E 1 (U ). 32 / 36

33 4. Associated Potential Theory 4.1 Reduced functions and polar sets 4.1 Reduced functions and polar sets Therefore, in particular, since x E x, Altogether, Hence, since R H 1 1 E = U 1+α (1 H ) = R H 1 1 E 1 E x, R H 1 1 E (x) = 0. { 1 on H 0 on E\H } = 1 H. e (1+α)t P t 1 H dt = 0 because ν t (H + x) = 0 x E, ˆR 1 H 1 E = lim αu 1+α R H α 1 1 }{{ E = 0. } 1 H 33 / 36

34 4. Associated Potential Theory 4.2 Choquet capacities and tightness 4.2 Choquet capacities and tightness Let λ be a finite (nonnegative) measure on E. Define cap λ : 2 E [0, ] by cap λ (A) := inf R1 G 1 E dλ : A G, G open, A E. E 34 / 36

35 4. Associated Potential Theory 4.2 Choquet capacities and tightness 4.2 Choquet capacities and tightness Theorem IV [Beznea, R. 2010] (i) cap λ is Choquet capacity, which is tight, i.e. compact increasing K n E, n N, such that lim cap n λ(e\k n ) = 0. (ii) Let A B(E), then cap λ (A) = E R A 1 1 E dλ. (iii) Let N B(E). Then: N λ-polar and λ(n) = 0 cap λ (N) = / 36

36 4.3 Balayage principle 4. Associated Potential Theory 4.3 Balayage principle Theorem V [Beznea/R. 2010] (i) ( Balayage principle ) Let A B(E) and λ σ-finite (nonnegative) measure on E. Define λ A (B) := ˆR 1 A 1 B dλ, B B(E). Then λ A is a (nonnegative) measure on E which is supported by the fine closure of A. Furthermore, as measures λ A U 1 λu 1 on B(E) and λ A U 1 = λu 1 on B(A). (ii) Consider the Gaussian case. Let O E, O open, x O. Then the harmonic measure µ O x, defined by µ O x (A) := E x [1 A (X (T E\O )), T E\O < ], is supported by the fine boundary f O of O. 36 / 36