By Dynkin E.B.

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

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The Laplace functional of (X, P ) can be expressed through P by the formula e− L(f) = f,ν P(dν). 17) LP (f1 , . . , fn) = e− f1 ,ν1 −···− fn ,νn P(dν1, . . , dνn). 1: 54 3. 2. A functional u(f1 , . 18) u(f1i , . . , fni ) → 1 as f1i ↓ 0, . , fni ↓ 0. 4. Constructing a BEM system. We say that a function u from G to B is an N-function if, for every x, u(f)(x) is a real-valued N-function. A, this implies u(f), µ is in the class N for all µ ∈ M(S). The class of P-functions from G to B is defined in a similar way.

5, there exists a sequence of strongly regular open sets Un exhausting S \ Γ. 3, sets Qn = Un ∩ Q are strongly regular. ¯n ⊂ Q ¯ and Q ¯n ∩ Γ ⊂ U ¯n ∩ Γ ⊂ Un+1 ∩ Γ = ∅. Hence Q ¯n ⊂ Q ¯ \ Γ. Note that Q ¯ \ Γ. If K is a compact set disjoint from Γ, then K ⊂ Un for some n. Let x ∈ Q For sufficiently small δ > 0, K = {y : |y − x| ≤ δ} is disjoint from Γ. If xm → x and ¯ n. This xm ∈ Q, then, for sufficiently large m, xm ∈ Un ∩ Q = Qn. 15). The second part holds because Qn ⊂ U c c ¯n , Un+1 Un+1 and d(U ) > 0.

If τ is the limit of τn , then, on the set {τ < ∞}, We have d(η n+1 n n ˜ τ , ητ ) = d(ητ )/2. We conclude that {τ < ∞} ⊂ ητn → ητ and therefore 0 = d(η {ητ ∈ ∂U } ⊂ {τ = τ (U )}. By the definition of the lower semicontinuity, on the set {τ < ∞}, u(ητ ) ≤ lim inf u(ητn ). Therefore, by Fatou’s lemma, Πz 1τ <∞ u(ητ ) ≤ Πz 1τ <∞ lim inf u(ητn ) ≤ lim inf Πz 1τ <∞ u(ητn ) ≤ u(z). 5. Suppose that w is superparabolic in Q and bounded below. Let T be a total subset of ∂Q. 8) lim inf w(z) ≥ 0 as z → z˜, then w ≥ 0 in Q.