Take a bounded, open, d-dimensional set and a continuous function f on the boundary. Find a function u, that is continous on the enclosure and is harmonic in R, that converges toward f on the boundary. Brownian motion factors into the problem when fixing a point in R as a start of a Brownian motion process. For all subsequent motions, the resulting induced measure on the boundary is given by
ω → τ(ω) maps the "stopping time". The martingale sequence {sₙ} associated to the increasing sequence of the σ-algebras on the probability space (X, m). The stopping time is an integer-valued function.
Brownian motion is essentially a continuous version of a martingale series. For all (sensible) times t, the σ-algebra generated by all the functions Bₛ, 0 ≤ s ≤ t. Then, sequence {Bₙ} is a martinglae relative to {Aₙ}, and for almost all ω, the path Bₜ(ω) is continuous in t. The maximal inequality immediately leads to the familiar Brownian motion inequality. Two different, well defined "natural stopping times" arise, one is the exit time, and one is the strict exit time.
The stopping time σ can also define a collection A with sets a so that a ∩ {σ(ω) ≤ t} ∈ A for all non-negative t. Suppose a Brownian motion B and stopping time σ, then define another Brownian motion, which is independent of its underlying σ-algebra.
Successive Brownian motion processes are then entirely independent of one another's starting positions. This is the Markov Property. Other forms of this property are embedded in the Borel functions. For a Borel function on the space of all paths:
The Markov property can also be applied to the Dirichlet problem directly. Once again, with two stopping times, defined the stopped process.
For u defined as at the beginning of the section, we can remember that u is harmonic in R, and u(x) → f(y), as x → y. For a regular point y on the boundary of R,
This fixes the upper and lower bounds of the function, since y is regular. s, ϵ > 0 can be used for limiting the function values.
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