Diffusions, superdiffusions and PDEs by Dynkin E.B. PDF

By Dynkin E.B.

This publication describes the relationships among linear and semilinear differential equations and the corresponding Markov procedures of diffusions and superdiffusions. Parabolic equations and branching go out Markov structures are tested, as are elliptic equations and diffusions. Appendices hide simple proof approximately Markov tactics, martingales, and elliptic differential equations. status difficulties are defined within the epilogue.

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Example text

Then, for every r, x, Xt = 1t<τ u(ηt) is a supermartingale on [r, ∞) relative to F[r, t] and Πr,x . Proof. Note that σ = τ ∧ t is the first exit time from Q ∩ S s} ∈ F[r, s]. s. 4), Xt dΠr,x = A,s<τr Πηs Xt dΠr,x ≤ A,s<τr u(ηs) dΠr,x . s. and therefore A Xt dΠr,x ≤ A Xs dΠr,x .

A. s.. B. For every µ ∈ M and every bounded measurable function f on M × M, Pµ f(XQ1 , XQ2 ) = Pµ F (XQ1 ) where F (ν) = Pν f(ν, XQ2 ). C. s. D. s. Proof. A. B is true form a linear space closed under the bounded convergence. D, this space contains all functions f1 (ν1)f2 (ν2). 1 in the Appendix A), it contains all bounded measurable functions. C, we consider F (ν) = Pν { ϕ1, ν ≤ ϕ2 , XQ2 }. 5) Pµ { ϕ1 , XQ1 ≤ ϕ2, XQ2 } = Pµ F (XQ1 ). Let ν be the restriction of ν to Qc2. 5). C to ϕ1 = ϕ2 = 1Γ .

2) aij = u(fi + fj ) is a P-matrix. 2) is an N-matrix for every n ≥ 2 and all f1 , . . , fn ∈ G. A. 3) u(f) ≥ 0 and u(f)2 ≤ u(2f)u(0). 4) u(f) ≤ u(0). Proof. 3) holds because a 1 × 1-matrix u(f/2 + f/2) is a P-matrix. The second inequality is true because the determinant of a 2 × 2 P-matrix u(2f) u(f) u(f) u(0) is positive. 3), u(f) = 0 if u(0) = 0. If u(0) > 0, then v(f) = u(f)/u(0) satisfies the condition v(f)2 ≤ v(2f) which implies that, for every n, n v(f)2 ≤ v(2n f). n If v is bounded, then the sequence v(f)2 is bounded and therefore v(f) ≤ 1.

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Diffusions, superdiffusions and PDEs by Dynkin E.B.

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Categories: Differential Equations