BV functions in a Gelfand triple and the stochastic reflection problem on a convex set

Transcription

1 BV functions in a Gelfand triple and the stochastic reflection problem on a convex set Xiangchan Zhu Joint work with Prof. Michael Röckner and Rongchan Zhu Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 1 set / 14

2 Outline Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 2 set / 14

3 Reflection problem Consider the following stochastic differential inclusion in the Hilbert space H: dx(t) + (AX(t) + X(0) = x Γ. N Γ (X(t)) )dt dw (t), }{{} Reflection term (1.1) Here A : D(A) H H is a positively definite self-adjoint operator, Γ is a closed convex set, N Γ (x) = {z H : z, y x 0 y Γ}, is the normal cone to Γ at x and W (t) is a cylindrical Wiener process in H. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 3 set / 14

4 BV functions in finite dimensions Recall that: a function u is called a BV function on R n if and only if one of the following is satisfied: Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 4 set / 14

5 BV functions in finite dimensions Recall that: a function u is called a BV function on R n if and only if one of the following is satisfied: There exist real finite measures µ j, j = 1,..., n on R n such that: ud j φdx = φdµ j, φ Cc 1 (R n ), R n R n Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 4 set / 14

6 BV functions in finite dimensions Recall that: a function u is called a BV function on R n if and only if one of the following is satisfied: There exist real finite measures µ j, j = 1,..., n on R n such that: ud j φdx = φdµ j, φ Cc 1 (R n ), R n R n V (u) := sup{ udivφdx : φ [Cc 1 (R n )] n, φ R n }{{} 1} <. test vector fields Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 4 set / 14

7 BV functions in finite dimensions Recall that: a function u is called a BV function on R n if and only if one of the following is satisfied: There exist real finite measures µ j, j = 1,..., n on R n such that: ud j φdx = φdµ j, φ Cc 1 (R n ), R n R n (Riesz-Markov representation theorem) V (u) := sup{ udivφdx : φ [Cc 1 (R n )] n, φ R n }{{} 1} <. test vector fields Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 4 set / 14

8 BV functions in infinite dimensions Below µ will denote the Gaussian measure in H with mean 0 and covariance operator Q := 1 2 A 1. For l H, a function u is called a (directional) BV l function on H if and only if one of the following is satisfied: Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 5 set / 14

9 BV functions in infinite dimensions Below µ will denote the Gaussian measure in H with mean 0 and covariance operator Q := 1 2 A 1. For l H, a function u is called a (directional) BV l function on H if and only if one of the following is satisfied: There exist a real finite measure µ l on H such that: l, Dv(z) u(z)dµ(z) = v(z)µ l (dz) H H Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 5 set / 14

10 BV functions in infinite dimensions Below µ will denote the Gaussian measure in H with mean 0 and covariance operator Q := 1 2 A 1. For l H, a function u is called a (directional) BV l function on H if and only if one of the following is satisfied: There exist a real finite measure µ l on H such that: l, Dv(z) u(z)dµ(z) = v(z)µ l (dz) H H V l (u) := sup{ u(z) l, Dv(z) dµ(z) : v Cb 1 (H), v 1} <. H Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 5 set / 14

11 BV functions in infinite dimensions Below µ will denote the Gaussian measure in H with mean 0 and covariance operator Q := 1 2 A 1. For l H, a function u is called a (directional) BV l function on H if and only if one of the following is satisfied: There exist a real finite measure µ l on H such that: l, Dv(z) u(z)dµ(z) = v(z)µ l (dz) H H (?) V l (u) := sup{ u(z) l, Dv(z) dµ(z) : v Cb 1 (H), v 1} <. H Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 5 set / 14

12 BV functions in infinite dimensions Extend to infinite dimensions: Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 6 set / 14

13 BV functions in infinite dimensions Extend to infinite dimensions: Problem Riesz-Markov representation theorem cannot be used since lack of local compactness. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 6 set / 14

14 BV functions in infinite dimensions Extend to infinite dimensions: Problem Riesz-Markov representation theorem cannot be used since lack of local compactness. If we take BV l as the definition of BV functions in infinite dimensions, this does not give rise to vector-valued measures representing their total weak derivatives or gradients. iangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 6 set / 14

15 BV functions in infinite dimensions iangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 7 set / 14

16 BV functions in infinite dimensions A) Hilbert Space [Ambrosio,Da Prato,Pallara,2009] Define V (ρ) := sup DµG(z)ρ(z)µ(dz) G H 1 H Definition A function ρ on H is called a BV function in the Hilbert space H, if V (ρ) is finite. iangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 7 set / 14

17 BV functions in infinite dimensions A) Hilbert Space [Ambrosio,Da Prato,Pallara,2009] Define V (ρ) := sup DµG(z)ρ(z)µ(dz) G H 1 H B) Gelfand triple Let (H 1, H, H1 ) be a Gelfand triple. Define V (ρ) := sup DµG(z)ρ(z)µ(dz) G H1 1 H Definition A function ρ on H is called a BV function in the Hilbert space H, if V (ρ) is finite. Definition A function ρ on H is called a BV function in the Gelfand triple (H 1, H, H 1 )(denoted ρ BV (H, H 1 )), if V (ρ) is finite. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 7 set / 14

18 BV functions in infinite dimensions A) Hilbert Space [Ambrosio,Da Prato,Pallara,2009] Define V (ρ) := sup DµG(z)ρ(z)µ(dz) G H 1 H B) Gelfand triple Let (H 1, H, H1 ) be a Gelfand triple. Define V (ρ) := sup DµG(z)ρ(z)µ(dz) G H1 1 H Definition A function ρ on H is called a BV function in the Hilbert space H, if V (ρ) is finite. Definition A function ρ on H is called a BV function in the Gelfand triple (H 1, H, H 1 )(denoted ρ BV (H, H 1 )), if V (ρ) is finite. There exists ρ BV (H, H 1 ) which is not a BV function in H! Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 7 set / 14

19 BV functions in infinite dimensions Theorem 1.1 Suppose ρ BV (H, H 1 ), then there exist a positive finite measure dρ on H and a map σ ρ : H H1 such that test vector fields G : H H 1 Dµ(G(z))ρ(z)µ(dz) = H H H 1 G(z), σ ρ (z) H 1 dρ (dz). (1.2) Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 8 set / 14

20 BV functions in infinite dimensions Theorem 1.1 Suppose ρ BV (H, H 1 ), then there exist a positive finite measure dρ on H and a map σ ρ : H H1 such that test vector fields G : H H 1 Dµ(G(z))ρ(z)µ(dz) = H H H 1 G(z), σ ρ (z) H 1 dρ (dz). (1.2) Conversely, if Eq.(1.2) holds for ρ and for some positive finite measure dρ and a map σ ρ with the stated properties, then ρ BV (H, H 1 ). Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 8 set / 14

21 Reflected OU processes on regular convex sets Consider the situation when ρ = I Γ, the indicator of a regular convex set Γ = {x H : g(x) 1} for some regular function g : H R (e.g.{x H, x H 1}). Then I Γ BV (H, H). Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 9 set / 14

22 Reflected OU processes on regular convex sets Consider the situation when ρ = I Γ, the indicator of a regular convex set Γ = {x H : g(x) 1} for some regular function g : H R (e.g.{x H, x H 1}). Then I Γ BV (H, H). We have E ρ (u, v) = 1 2 H Du, Dv ρ(z)µ(dz) is a quasi-regular local Dirichlet form on L 2 (H; ρ µ). Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 9 set / 14

23 Reflected OU processes on regular convex sets Consider the situation when ρ = I Γ, the indicator of a regular convex set Γ = {x H : g(x) 1} for some regular function g : H R (e.g.{x H, x H 1}). Then I Γ BV (H, H). We have E ρ (u, v) = 1 2 H Du, Dv ρ(z)µ(dz) is a quasi-regular local Dirichlet form on L 2 (H; ρ µ). By [Ma, Röckner,92], there exists a diffusion process M ρ = (Ω, M, {M t }, X t, P z ) associated with this Dirichlet form. M ρ will be called reflected OU (Ornslein-Uhlenbek)-process. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex 9 set / 14

24 Reflected OU processes on regular convex sets Theorem 1.2 Under P z there exists an M t - cylindrical Wiener process W z, such that the sample paths of the associated reflected OU-process M ρ with ρ = I Γ satisfy the following: for l D(A) l, X t X 0 = t 0 l, dw z s t 0 l, σ ρ (X s )dl di Γ s t 0 Al, X s ds t 0 where σ ρ = n Γ := Dg Dg is the interior normal to Γ and di Γ = µ Γ, where µ Γ is the surface measure on Γ induced by µ. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex10 set / 14

25 Existence and uniqueness of solutions Definition A pair of continuous H R-valued and F t -adapted processes (X t, L t ), t [0, T ], is called a (strong) solution of (1.1) if (i) X t Γ for all t [0, T ] P a.s.; (ii) L is an increasing process satisfying I Γ (X s )dl s = dl s and for any l D(A) we have l, X t x = t 0 l, dw s t 0 l, n Γ(X s )dl s t 0 Al, X s ds. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex11 set / 14

26 Existence and uniqueness of solutions Definition A pair of continuous H R-valued and F t -adapted processes (X t, L t ), t [0, T ], is called a (strong) solution of (1.1) if (i) X t Γ for all t [0, T ] P a.s.; (ii) L is an increasing process satisfying I Γ (X s )dl s = dl s and for any l D(A) we have l, X t x = t 0 l, dw s t 0 l, n Γ(X s )dl s t 0 Al, X s ds. Proposition If Γ is a regular convex set, then (1.1) is pathwise unique. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex11 set / 14

27 Existence and uniqueness of solutions Definition A pair of continuous H R-valued and F t -adapted processes (X t, L t ), t [0, T ], is called a (strong) solution of (1.1) if (i) X t Γ for all t [0, T ] P a.s.; (ii) L is an increasing process satisfying I Γ (X s )dl s = dl s and for any l D(A) we have l, X t x = t 0 l, dw s t 0 l, n Γ(X s )dl s t 0 Al, X s ds. Proposition If Γ is a regular convex set, then (1.1) is pathwise unique. Theorem 1.3 If Γ is a regular convex set, then (1.1) has a unique strong solution. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex11 set / 14

28 A special case Example Consider the case where H := L 2 (0, 1), ρ = I Kα and A = 1 2 d 2 dr 2 with Dirichlet boundary conditions on (0,1). Here K α := {f H f α}, α > 0 ( a convex set with empty interior). Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex12 set / 14

29 A special case Example Consider the case where H := L 2 (0, 1), ρ = I Kα and A = 1 2 d 2 dr 2 with Dirichlet boundary conditions on (0,1). Here K α := {f H f α}, α > 0 ( a convex set with empty interior). Then for some H 1, we have I Kα BV (H, H 1 ), but not in BV (H, H). Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex12 set / 14

30 Extensions We can extend all these results to non-symmetric Dirichlet forms obtained by a first order perturbation of the above Dirichlet form. { dx(t) + (AX(t) + B(X) + NΓ (X(t)))dt dw (t), X(0) = x. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex13 set / 14

31 Extensions We analogously define BV functions in a Gelfand triple replacing the Gaussian measure with a differentiable measure ν. In particular, if ν is the p(φ) 2 quantum field, we get a martingale solution of the stochastic quantization problem with reflection in infinite volume: dx(t) + ( AX(t)+ : p(x) : }{{} +N Γ (X(t)))dt dw (t), logarithmic derivative of ν X(0) = x. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex14 set / 14

32 Extensions We analogously define BV functions in a Gelfand triple replacing the Gaussian measure with a differentiable measure ν. In particular, if ν is the p(φ) 2 quantum field, we get a martingale solution of the stochastic quantization problem with reflection in infinite volume: dx(t) + ( AX(t)+ : p(x) : }{{} +N Γ (X(t)))dt dw (t), logarithmic derivative of ν X(0) = x. In particular, if ν = e x 4 µ, we get the existence and uniqueness of (probabilistically) strong solutions to the following: { dx(t) + (AX(t) + 4 Xt 2 X t + N U (X(t)))dt dw (t), X(0) = x. Xiangchan Zhu ( Joint work with Prof. Michael BVRöckner functionsand in arongchan Gelfand triple Zhu ) and the stochastic reflection problem on a convex14 set / 14