Classical Potential Theory and Its Probabilistic Counterpart by Joseph L. Doob

By Joseph L. Doob

From the experiences: "Here is a momumental paintings by means of Doob, one of many masters, during which half 1 develops the capability conception linked to Laplace's equation and the warmth equation, and half 2 develops these elements (martingales and Brownian movement) of stochastic method idea that are heavily concerning half 1". --G.E.H. Reuter briefly publication stories (1985)

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Differentiable Superharmonic Functions Theorem. If D is an open subset of R' and if u E V ')(D), then u is superharmonic if and only if Au 5 0. g 0 when u is superharmonic. Conversely if Au :r. 2) for sufficiently small b. 2) becomes the superharmonic function 1 inequality for u when c -' 0. 2) is defined and infinitely differentiable on the set { a D: I - 3DI > S}. According to Section 4 this function is superharmonic and majorized by u, and according to Section 6(f), for each point of D b < 1 - aDl.

Z 0, according to what we have proved, and u = A n) represents u as the limit of an increasing sequence of positive bounded harmonic functions. (b4) (bl) If u is bounded positive and harmonic on B, condition (b2) is satisfied, so (b I) is satisfied. Moreover, if u, is an increasing sequence of positive bounded harmonic functions on B with limit u, the sequence f,4 15. The Fatoi. Boundary Limit Theorem 31 is an increasing sequence (up to 1N_, null sets) with limit some function f, and the equation u = PI(B, f ) becomes in the limit u = PI(B, f ).

Exactly the same proof, in which oo is the only boundary 0 when N > 2. 1 the Green point, shows that function Go will be defined for every Greenian set D. The property 0 remains true and in fact is very nearly a defining property of GD. 2.

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