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, 2005. 1 / 36
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
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
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
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
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
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
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
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
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
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
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 2 + 2 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
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
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
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 2 + 2 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
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
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
3.1 Preliminaries 3. Levy processes on Hilbert space 3.1 Preliminaries Recall by Levy-Khintchine: λ(ξ) = E ξ, b E + 1 2 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
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
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
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
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 > 0. 22 / 36
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
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
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
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
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)) 2 1 2 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
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
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 4.1-4.3 below for both the Gaussian and Levy case (and subordinations thereof). 29 / 36
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
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
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. 2010 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
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
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
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) = 0. 35 / 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