Space-time harmonic functions yield Brownian local martingales up to exit lifetime

Given: AC, an open $U\subseteq [0,\mathrm{\infty })\times {\mathbb{R}}^d$, a function $f\in C^{1,2}(U)$ with ${\mathrm{\partial }}_{t}f+\frac{1}{2}\mathrm{\Delta }f=0$ on $U$, a standard $d$-dimensional Brownian motion $B$ adapted to a usual filtration with each vector increment independent of the past filtration, its ${\mathcal{F}}_0$-normalized everywhere-continuous representative, and compact sets $K\subseteq U$.

$B$ is a continuous Brownian Ito process. The vector filtration hypothesis implies the scalar standing hypothesis (H) for every coordinate. The normalized $B$ is therefore a continuous Brownian Ito process with drift $0$ and dispersion ${\delta }^{ik}$. The one-block elementary process $1_{(0,T]}$ represents the constant integrand class and has integral $B_t^i-B_0^i=B_t^i$; its localized integral is $B^i$ up to indistinguishability. Continuous Brownian Ito processes $d$-dimensional Brownian motion Brownian motion Elementary predictable Brownian integrands Ito integral of an elementary predictable process Localized Ito integral

Multidimensional Ito formula. For a $C^{1,2}$ function $g$ and a continuous Brownian Ito process $X$, $dg(t,X_t)=({\mathrm{\partial }}_{t}g+{\sum }_i{b}^i{\mathrm{\partial }}_{i}g+\frac{1}{2}{\sum }_{i,j}(\sigma {\sigma }^{\mathsf{T}}{)}^{ij}{\mathrm{\partial }}_i{\mathrm{\partial }}_{j}g)(t,X_t)dt+{\sum }_{i,k}{\mathrm{\partial }}_{i}g(t,X_t){\sigma }_t^{ik}d{B}_t^k$ up to indistinguishability. Multidimensional Ito formula for Brownian-driven processes

Cutoffs on compact subsets and local boundedness. Because $U$ is relatively open, the formula $\tilde{f}(t,x)=3f(-t,x)-2f(-2t,x)$ for small $t<0$ gives a $C^{1,2}$ extension across $t=0$ on a Euclidean-open neighbourhood of each compact $K\subseteq U$: value and time derivative match because $3-2=1$ and $-3+4=1$, and the spatial derivatives match by the same value identity. Choose compact neighbourhoods $K\subseteq \mathrm{int}{K}^{\mathrm{\prime }}$ there. The cutoff lemma gives a continuous compactly supported cutoff equal to $1$ on $K^{\mathrm{\prime }}$; convolving it with a sufficiently small compactly supported mollifier gives $\chi \in C_c^{\mathrm{\infty }}$ equal to $1$ near $K$ and supported in the extension domain. Then $g:=\chi \tilde{f}$, extended by zero, is a global $C^{1,2}$ function and agrees with $f$ and its displayed derivatives near $K$. In particular $\mathrm{\nabla }f$ is bounded there. The spaces $C_c({\mathbb{R}}^n)$ and $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ The mollifier family generated by a unit-mass smooth bump Convolution with a mollifier is smooth, and derivatives pass under the integral sign A compact set inside a bounded open set admits an explicit compactly supported continuous cutoff Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point

Localized-integral interfaces. For a bounded predictable integrand $H$ on $[0,T]$: the integral $\int Hd{B}^k$ is a continuous square-integrable martingale with $E({\int }_0^{T}Hd{B}^k{)}^2=E{\int }_0^T{H}^{2}ds$, the stopping identity identifies stopped integrals with integrals of $H{1}_{[0,\tau ]}$, and a bounded integrand has finite energy. Localized Ito integral Stopping an Ito integral The Ito integral process has a continuous martingale version Ito isometry and linearity in predictable L2 Locally square-integrable predictable Brownian integrands Progressively measurable and predictable processes

Stopping times and exhaustion. Put $S=[0,\mathrm{\infty })\times {\mathbb{R}}^d$. For a relatively open $V\subseteq S$ with nonempty complement $C=S\setminus V$, the distance $q(y)=d(y,C)$ is continuous and zero exactly on $C$. The Lipschitz estimate is supplied by $\mathrm{\mid }d(x,A)-d(y,A)\mathrm{\mid }\le d(x,y)$, so the distance to a fixed nonempty set is $1$-Lipschitz; positivity outside $C$ follows from an open ball disjoint from this closed set. For $Y_s=(s,B_s)$, continuity and compactness of $Y([0,t])$ give $\{{\tau }_V\le t\}=\{{\inf }_{s\in ([0,t]\cap \mathbb{Q})\cup \{t\}}q(Y_s)=0\}\mathrm{.}$ Indeed the continuous distance attains its minimum on the compact path image, and a zero minimum is a hit by time $t$; approximation by rational times gives the same infimum, including $t=0$. The displayed event is ${\mathcal{F}}_t$-measurable. If $V=S$, its exit time is infinity directly. For exhaustion, set $a(y)=\min (1,d(y,S\setminus U))$ when the complement is nonempty, and $a=1$ when $U=S$. Let $L_n=\{(s,x)\in S:s\le n,\mathrm{\mid }x\mathrm{\mid }\le n,a(s,x)\ge 1\mathrm{/}n\}\mathrm{.}$ These sets are closed and bounded, hence compact, contained in $U$, satisfy $L_n\subseteq {\mathrm{int}}_S{L}_{n+1}$, and cover $U$. Discard finitely many initial sets so that the origin belongs to the first interior, and denote the tail by $K_n$. The time caps remain finite. For any such nested exhaustion, every compact path segment before ${\tau }_U$ is covered by finitely many interiors, hence lies in one. Thus ${\tau }_{K_n}\uparrow {\tau }_U$. Moreover ${\tau }_{K_n}<{\tau }_U$: the finite exit point from $\mathrm{int}{K}_n$ lies in $K_n\subseteq U$, and continuity gives a positive interval still in $U$ after this time. Consequently $[0,{\tau }_U)={\bigcup }_n[0,{\tau }_{K_n}]$, so its indicator is predictable by the stopping-indicator generators. Continuous-time stopping times and stopped sigma-algebras Progressively measurable and predictable processes Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous

Finite-energy approximation. Dominated convergence applies on the product of Lebesgue measure on a finite interval and probability when the squared integrand error has the stated integrable majorant. Dominated convergence

AC bookkeeping. Choice is declared for the conditional-expectation and completeness interfaces, and any countable selection of compact cutoffs. The distance exhaustion is explicit. The Axiom of Choice