Harnack Inequalities and Applications for Stochastic Equations Shun- ([email protected]) International Graduate College “Stochastics and Real World Models” Bielefeld University

2.2 Preliminary: Null Controllable Hence we get the following Harnack inequality 1. Introduction β α α E β α Consider the system (PT f) (x) ≤ [ PρT ] Ptf (y).  1/2 1.1 Stochastic Equations  dxt = Axt dt + R ut dt, H (1) We consider Harnack inequalities and their ap-  x0 = y − x ∈ . 4.2 Girsanov’s Theorem for Levy´ Pro- plications for the following stochastic equations We say a system of the form (1) is Null Control- cess (SEs). lable in time T if for each initial data, there exist Let (Zt)0≤t≤T be aL evy´ process with charac- Single-valued SEs: dXt = AXt dt + F (t, Xt) dt + dLt. 2 u ∈ L ([0,T ], H) such that xT = 0. teristic triple (b, R, ν) under probability measure R T ∗ P S(O)DE⇐ with irregular drift Define QT = 0 StRSt dt. Then (Ω, (Ft)0≤t≤T , ). Under some new probability fBM Q W P Gauss H 1/2 H measure = ρ (T ) , process BM Null Controllable ⇔ ST ( ) ⊂ QT ( ) SE Z t (linear) −1/2 Lévy Let Γ = Q S . We have a representation for Ze(t) := Z(t) − ψ(s) ds, 0 ≤ t ≤ T T T T 0 SPDE the Minimal Energy of driving the initial states is also a (b, R, ν)–Levy´ process on (Ω, , ( ) ), y − x 0 T F Ft 0≤t≤T nPerturbation of to in time : 1/2 semi- where we will take ψ(t) = R ut , 0 ≤ t ≤ T , and ⇐ (Z T ) linear 2 2 2 Gauss OU process |ΓT (x − y)| = inf |us| ds: u ∈ L . ut is the null control of the system (1). 0 ˜ 4.3 Levy´ Driven OU Process Multi-valued SEs: dXt ∈ AXt dt + BXt dt + σtdWt. y Consider Yt driven by Zt on (Ω, Ft, P): Rd MSDEs (on )& MSEEs (on Gelfand triple) 3. Gauss Case y We just show the main technical by establish- Y0 = y, dY y = AY y dt + dZ . ing Harnack inequalities for OU processes. 3.1 Gauss OU Process and Semigroup t t t y We define P f(y) = EPf(Y ). We also consider 1.2 Transition Semigroup Mild solution of the Langevin’s Equation: t t x Q Q W P Let H be a real separable Hilbert space with inner Xt driven by Zet on (Ω, Ft, ).( = ρ ) 1/2 H h·, ·i | · | X dXt = AXt dt + R dWt,X0 = x ∈ . x product and norm . Let be a solution X0 = x, process to some stochastic equation. Assume that Qt, 0 ≤ t ≤ ∞, is of trace class. x x dXt = AXt dt + dZet, For every t ∈ [0, ∞), Xt(·), Xt : (Ω, Ft, P) → Then = AXx dt + dZ −R1/2u dt. (H, B(H)) is a r.v.. Denote the marginal distribu- t t t x Z t x tion of Xt by µ : 1/2 EQ t Xt = Stx + St−sR dWs, t ≥ 0. We see Ptf(x) = f(Xt ). By noting that Zet is x x −1 0 µ := P ◦ (X ) . a drift transformation of Zt and note that xT = 0, t t We know Then we define the transition semigroup by we get Xt ∼ N(Stx,Qt). x y Z Z Xt = Yt −xt ⇒ XT = YT x P E x x Ptf(x) = f(Xt ) d = Pf(Xt )= f(z) dµt Therefore the Gaussian Ornstein-Uhlenbeck H Ω H Z t x Semigroup is given by Yt := Stx + St−s dZs for every bounded measurable function f on H. 0 Z x Ptf(x) = f(Stx + z) N(0,Qt)(dz). 1.3 Harnack Inequalities H The Harnack inequality we are interested in which 3.2 Harnack Inequality for Gaussian Z t Z t is first introduced by Wang in 1997 [2], is of the x x 1/2 y Xt := Stx + St−s dZes = Yt − St−sR us ds = Yt − xt following form OU Semigroup 0 0 α α Assume y (Ptf) (x) ≤ C(t, α, x, y)Ptf (y), H + H • Qt is of trace class; for every x, y ∈ , f ∈ Cb ( ), α > 1, where 1/2 H H o Z t t • St( ) ⊂ Qt ( ). y T C(t, α, x, y) is a constant independent of f. Yt := Sty + St−s dZs 0 Applications of Harnack inequalities including By using the Cameron-Martin formula for Gaus- Regularizing Property, Contractivity, Heat Kernel sian measures, we have Estimates, Bound of the Norm of Heat Kernel etc.. β ! (P f)α(x) ≤ exp |Γ (x − y)|2 · (P f α(y)) , (2) t 2 t t 2. Main Technique 4.4 Harnack Inequality for Levy´ Driven + H H OU Process for every t > 0, f ∈ Cb ( ), x, y ∈ and α, β > 1 satisfying 1 + 1 = 1. 2.1 How to get Harnack inequalities? α β By the procedure introduced in subsection 4.1, for Levy´ OU semigroup Pt, we have The main technique we use is applying Holder’s¨ 3.3 Application: Strong Feller Prop- Z t ! inequality after a measure transformation. There erty α β 2 α (Ptf) (x) ≤ exp |us| ds · (Ptf (y)) , 2 0 are two levels of measure transformation which For Gauss OU semigroup Pt. The following state- are used in [3] (Relative density method) and [1] ments are equivalent to each other. 1 1 + H ∀t > 0, α, β > 1 satisfying α + β = 1, f ∈ Cb ( ), (Coupling argument+Girsanov’s transformation). 1/2 • St(H) ⊂ Qt (H); x, y ∈ H. • Harnack inequality holds; Taking infimum over all null control u, we still can Coupling get inequality with the same form as (2). Relative Density + • Pt is strongly Feller. Girsanov Trans. References 4. Levy´ Case H¨older’s Ineq. [1] M. Arnaudon, A. Thalmaier, and F.-Y. Wang, 4.1 Transformation of Measures on Ω Harnack inequality and heat kernel estimates + x Q y P on manifolds with curvature unbounded below, Let (Xt , ) and (Yt , ) be two processes on Bull. Sci. Math. 130 (2006), no. 3, 223–233. Measure Trans. (Ω, FT ). Assume three conditions E x E y [2] F.-Y. Wang, Logarithmic Sobolev inequali- 1. Ptf(x) = Qf(Xt ), Ptf(y) = Pf(Yt ); ties on noncompact Riemannian manifolds, 2. Q = ρT P; y Trans. of µt Trans. of P x Probab. Theory Related Fields 109 (1997), 3. XT = YT . on (H, (H)) on (Ω, F ) B Then no. 3, 417–424. P E x E y E y [3]M.R ockner¨ and F.-Y. Wang, Harnack and func- T f(x)= Qf(XT )= Qf(YT )= PρT f(YT ) β 1/β α y 1/α tional inequalities for generalized Mehler semi- We use the image measure transformation for ≤ (EPρ ) (EPf (Y )) T T groups, J. Funct. Anal. 203 (2003), no. 1, 237– Gauss case; and use the measure transformation β 1/β α 1/α = (EPρ ) (P f (y)) . on Ω for Levy´ case. T T 261.