By Feng-Yu Wang
ISBN-10: 1461479339
ISBN-13: 9781461479338
ISBN-10: 1461479347
ISBN-13: 9781461479345
In this publication the writer provides a self-contained account of Harnack inequalities and purposes for the semigroup of ideas to stochastic partial and not on time differential equations. because the semigroup refers to Fokker-Planck equations on infinite-dimensional areas, the Harnack inequalities the writer investigates are dimension-free. this can be an basically diverse element from the above pointed out classical Harnack inequalities. in addition, the most software within the research is a brand new coupling technique (called coupling through swap of measures) instead of the standard greatest precept within the present literature.
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Additional resources for Harnack Inequalities for Stochastic Partial Differential Equations
Example text
4]), it remains to find t0 ∈ (0, T ] such that sup E|J(Z)(t) − J(Y )(t)|2 ≤ t∈[0,t0 ] 1 sup E|Z(t) −Y (t)|2 , Z,Y ∈ J . 2), we have sup E|J(Z)(t) − J(Y (t))|2 ≤ 2{sφb (s) + φσ (s)} sup E|Z(t) −Y (t)|2 . 8) holds. 5). 6) imply that the solution has a continuous version. 1) for X(0) = x. We aim to investigate Harnack inequalities for the associated semigroup (Pt )t∈[0,T ] : Pt f (x) = E f (X x (t)), x ∈ H, f ∈ Bb (H). 3) (A, D(A)) has a discrete spectrum, so that there exists an orthonormal basis {en , n ≥ 1} ⊂ D(A) of H such that −Aen = λn en , n ≥ 1, where λn ≥ 0, n ≥ 0 are all eigenvalues of −A including multiplicities.
7) that sup E|X(s)|2 ≤ 3E|X(0)|2 + 6{t φb (t) + φσ (t)} sup E|X(s)|2 s∈[0,t] s∈[0,t] t +6 t sup |S(s)b(r, 0)|2 + sup r∈[0,T ] 0 r∈[0,T ] S(s)σ (r, 0) 2 HS ds. 2 HS ds. Taking t0 ∈ (0, T ] such that 6{t0 φb (t0 ) + φσ (t0 )} ≤ 12 , we obtain sup E|X(s)|2 s∈[0,t0 ] t0 ≤ 6E|X(0)|2 + 12 t0 sup |S(s)b(r, 0)|2 + sup r∈[0,T ] 0 r∈[0,T ] S(s)σ (r, 0) Therefore, letting h(n) = sups∈[0,T ∧(nt0 )] E|X(s)|2 and repeating the argument for the equation starting from time T ∧ ((n − 1)t0 ), we obtain t0 h(n)≤6h(n−1)+12 0 t0 sup |S(s)b(r, 0)|2 + sup r∈[0,T ] r∈[0,T ] S(s)σ (r, 0) 2 HS ds, n≥1.
2) with respect to a complete filtered probability space (Ω , F , {Ft }t≥0 , P). 1) Hemicontinuity. For every t ≥ 0 and v1 , v2 , v ∈ V, R s →V∗ b(t, v1 + sv2 ), v V is continuous. 2) Monotonicity. For every v1 , v2 ∈ V,t ≥ 0, 2V∗ b(t, v1 ) − b(t, v2 ), v1 − v2 V+ σ (t, v1 ) − σ (t, v2 ) 2 HS ≤ K(t)|v1 − v2 |2 . -Y. 3) Coercivity. For every t ≥ 0, v ∈ V, 2V∗ b(t, v), v V+ σ (t, v) 2 HS ≤ φ (t) + K(t)|v|2 − ψ (t) v α +1 V . 4) Growth. For every u, v ∈ V,t ≥ 0, α V+ |V∗ b(t, v), u V | ≤ φ (t) + K(t){ v u α +1 + |u|2 + |v|2 }.
Harnack Inequalities for Stochastic Partial Differential Equations by Feng-Yu Wang
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