Quantum Potential Theory by Philippe Biane, Luc Bouten, Fabio Cipriani, Norio Konno,

By Philippe Biane, Luc Bouten, Fabio Cipriani, Norio Konno, Quanhua Xu, Uwe Franz, Michael Schuermann

This quantity comprises the revised and accomplished notes of lectures given on the institution "Quantum capability concept: constitution and functions to Physics," held on the Alfried-Krupp-Wissenschaftskolleg in Greifswald from February 26 to March 10, 2007.

Quantum power idea reviews noncommutative (or quantum) analogs of classical power thought. those lectures offer an creation to this conception, targeting probabilistic power thought and it quantum analogs, i.e. quantum Markov procedures and semigroups, quantum random walks, Dirichlet varieties on C* and von Neumann algebras, and boundary idea. functions to quantum physics, particularly the filtering challenge in quantum optics, also are presented.

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I) If 0 ≤ a ≤ t we have ∞ IE us dMs Ft = IE [G(Mb − Ma )|Ft ] 0 = GIE [(Mb − Ma )|Ft ] 38 N. Privault = GIE [(Mb − Mt )|Ft ] + GIE [(Mt − Ma )|Ft ] = G(Mt − Ma ) ∞ = 1[0,t] (s)us dMs . 0 ii) If 0 ≤ t ≤ a we have for all bounded Ft -measurable random variable F : ∞ IE F us dMs = IE [F G(Mb − Ma )] = 0, 0 hence ∞ IE ∞ us dMs Ft = IE [G(Mb − Ma )|Ft ] = 0 = 0 1[0,t] (s)us dMs . 0 This statement is extended by linearity and density, since from the continuity of the conditional expectation on L2 we have: ∞ t us dMs − IE IE 0 2 us dMs Ft 0 ∞ t uns dMs − IE = lim IE n→∞ 0 0 ∞ = lim IE n→∞ IE n→∞ n→∞ 2 us dMs Ft 0 2 (uns − us )dMs 0 ∞ = lim IE n→∞ ∞ uns dMs − 0 ∞ 2 us dMs Ft 0 ∞ ≤ lim IE IE ∞ uns dMs − 0 ≤ lim IE 2 us dMs Ft |uns − us |2 ds 0 = 0.

17) 46 N. Privault The Green kernel is defined as ∞ gD (x, y) := pD t (x, y)dt, 0 and the associated Green potential is ∞ GD µ(x) = gD (x, y)µ(dy), 0 with in particular τ∂D GD f (x) := IE f (Bt )dt 0 when µ(dx) = f (x)dx has density f with respect to the Lebesgue measure. s. hence GRn = G. 22. The function (x, y) → gD (x, y) is symmetric and continuous on D2 , and x → gD (x, y) is harmonic on D\{y}, y ∈ D. Proof. For all bounded domains A in Rn , the function GD 1A defined as τ∂D GD 1A (x) = Rn gD (x, y)1A (y)dy = IEx 1A (Bt )dt 0 ¯ and the property extends to gD by has the mean value property in D\A, linear combinations and an limiting argument.

Ft = Fs , t ∈ R+ . s>t Recall that a process (Mt )t∈R+ in L1 (Ω) is called an Ft -martingale if IE[Mt |Fs ] = Ms , 0 ≤ s ≤ t. For example, if (Xt )t∈[0,T ] is a (non homogeneous) Markov process with semigroup (Ps,t )0≤s≤t≤T satisfying Ps,t f (Xs ) = IE[f (Xt ) | Xs ] = IE[f (Xt ) | Fs ], 0 ≤ s ≤ t ≤ T, on Cb2 (Rn ) functions, with Ps,t ◦ Pt,u = Ps,u , 0 ≤ s ≤ t ≤ u ≤ T, then (Pt,T f (Xt ))t∈[0,T ] is an Ft -martingale: IE[Pt,T f (Xt ) | Fs ] = IE[IE[f (XT ) | Ft ] | Fs ] = IE[f (XT ) | Fs ] 0 ≤ s ≤ t ≤ T.

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