By Feng-Yu Wang
In this ebook the writer offers a self-contained account of Harnack inequalities and functions 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 is often an basically diverse aspect from the above pointed out classical Harnack inequalities. additionally, the most software within the examine is a brand new coupling strategy (called coupling by way of switch 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
19). For every n ≥ 1, Rn := exp − τn 0 − ξ (s) σ (s)−1 (X(s) −Y (s)), dW (s) |X(s) −Y (s)|ε 1 2 τn 0 ξ (s) σ (s)−1 (X(s) −Y (s)) 2 ds |X(s) −Y (s)|ε is a well-defined probability density such that E Rn log Rn |x − y|2 θ + 2 ≤ 2 θ 2(θ +1) θ ( T 0 EQn h(X(t ∧ τn )) ∨ h(Y (t ∧ τn )) c1 (θ , T ) α +1 2(1−α ) dt) θ (α +1) θ (α +1)+4 θ (α +1) , and for every p > 1, p ERnp−1 p−1 ≤ EQn exp τn c(θ , p)|x − y|2 c1 (θ , T ) θ (α +1)+4 θ (α +1) 0 h(X(t)) ∨ h(Y (t)) α +1 dt 2(1−α ) θ (α +1) p−1 2 , 40 2 Nonlinear Monotone Stochastic Partial Differential Equations where dQn = Rn dP, c(θ , p) := 2(θ +1) θ p+1 θ +2 (p − 1)2 θ .
Next, let μ1 be another invariant probability measure of P. By (2), we have dμ1 = f dμ0 for some probability density function f . e. Let p(x, y) > 0 be the kernel of P with respect to μ0 , and let P∗ (x, dy) = p(y, x)μ0 (dy). Then P∗ g = E g(y)P∗ (·, dy), g ∈ Bb (E), is the adjoint operator of P with respect to μ0 . Since μ0 is P-invariant, we have E gP∗ 1 dμ0 = E Pg dμ0 = E g dμ0 , g ∈ Bb (E). e. e. x ∈ E, the measure P∗ (x, ·) is a probability measure. On the other hand, since μ1 is P-invariant, we have E (P∗ f )g dμ0 = E f Pg dμ0 = E Pg dμ1 = E g d μ1 = E f g dμ0 , g ∈ Bb (E).
43) for some d ∈ (0, 2ε1− α ) implies that m(xei )2 x 2α +1 ≥ c ∑ λiε i≥1 holds for some constant c > 0. 40) holds for ξ (t) = c for some constant c > 0. 1 to γ (t) = 2β (t), q(t) = σ 2 HS , δ (t) = 1, and η (t) = c, for some constant c > 0. 1(1). 2 Let Ψ (t, s) = sα for some α ∈ ( 13 , 1), and let L = Δ be the Dirichlet Laplacian on the open interval (0, π ). 43) 4 +2 α θ , 6αα+1 ) and ε ∈ ( 2(1− holds for d = 1. 42) α ), as required. So taking σi = i are satisfied. 3 hold. 1 Mild Solutions and Finite-Dimensional Approximations ˜ be a larger Hilbert space into Let (H, ·, · , |·|) be a separable Hilbert space, and let H which H is densely and continuously embedded.