Chapter ownership and proof scope
Primary proof chapter: Chapter 28
Related tools and applications: Chapter 17 · Chapter 23 · Chapter 38 (not yet written)
- Written proof or calculation (limited scope)
The Brownian/Gaussian-kernel and heat-equation calculations are retained.
- Mathematical intuition
The connection from random paths to deterministic evolution motivates, but does not replace, general operator theory.
- Written proof or calculation (limited scope)
Chapter 28 and its three companions now provide positive contraction generation, boundary domains, and conservative locally compact Polish Feller paths and strong Markov. Author review and human mathematical final review remain pending; PDE developments remain in Chapter 38.
Tools needed by the individual arguments
Lebesgue integration, conditional distributions, the Markov property, operator semigroups, and domains of generators.
These states describe the existing material; they do not certify a complete theorem treatment or final mathematical review.
Advanced Reading: From Random Motion to Semigroups and Generators
4.9.3 A Unification: Random Paths and Deterministic Evolution
We now have two entirely different descriptions.
The first is the random path:
t\longmapsto B_t(\omega).Each \omega corresponds to a different random trajectory.
The second is the probability density:
(t,x)\longmapsto p_t(x).This function is governed by the deterministic heat equation.
For a sufficiently regular function f, define
P_tf(x)
=
\mathbb E[f(x+B_t)].Since Brownian motion has independent, stationary increments,
P_{t+s}f=P_t(P_sf).Thus (P_t)_{t\ge0} forms a Markov semigroup.
Its infinitesimal generator satisfies
\lim_{h\downarrow0}
\frac{P_hf(x)-f(x)}{h}
=
\frac12f''(x),for example, for f\in C_c^\infty(\mathbb R).
Therefore
\boxed{
L=\frac12\frac{d^2}{dx^2}.
}The corresponding evolution equation is precisely
\frac{\partial u}{\partial t}=Lu.We obtain an exceptionally important unifying relationship:
\boxed{
\text{stochastic processes}
\longleftrightarrow
\text{Markov semigroups}
\longleftrightarrow
\text{partial differential equations}.
}This is more than a symbolic correspondence.
It means that random motion can explain partial differential equations, while partial differential equations can be used to study the probability distributions of stochastic processes.
In stochastic calculus, the Feynman–Kac formula, and stochastic differential equations, we will repeatedly encounter the importance of this connection.