The Brownian differential generator

Assume the Axiom of Choice and fix a finite integer $d\ge 1$. For a function $f\in C^2({\mathbb{R}}^d)$, where $C^2({\mathbb{R}}^d)$ means that all partial derivatives of order at most two exist and are continuous, the Brownian differential operator, also called the Brownian generator or the Ito differential operator, is $Lf:=\frac{1}{2}\mathrm{\Delta }f=\frac{1}{2}{\sum }_{k=1}^d{\mathrm{\partial }}_{x_k}^{2}f\mathrm{.}$ For a space-time function $f\in C^{1,2}([0,\mathrm{\infty })\times {\mathbb{R}}^d)$ (continuous together with its first time and first and second space derivatives, using the right time derivative at zero), one writes $Lf(t,x):=\frac{1}{2}{\mathrm{\Delta }}_{x}f(t,x)$, the Laplacian being taken in the space variable only.

The following conventions are part of the definition and fix what the symbol does and does not assert.

1. Coefficient convention in Ito notation. Whenever an Ito formula for $f(t,X_t)$ is valid with drift $b$ and dispersion matrix $\sigma$, its displayed second-order expression is $\frac{1}{2}{\sum }_{i,j}(\sigma {\sigma }^{\mathsf{T}}{)}^{ij}{\mathrm{\partial }}_i{\mathrm{\partial }}_{j}f$. If $\sigma {\sigma }^{\mathsf{T}}=I_d$, this expression equals $Lf$, since the off-diagonal coefficients vanish and each diagonal coefficient is one. The complete drift expression is then ${\mathrm{\partial }}_{t}f+{\sum }_i{b}^i{\mathrm{\partial }}_{i}f+Lf$, evaluated at $(t,X_t)$. This is an algebraic identification of the coefficients, not an assertion that an Ito formula holds under AC alone. Applying Multidimensional Ito formula for Brownian-driven processes requires its stochastic hypotheses, including (H) of Elementary predictable Brownian integrands, and compatible versions and integrability conventions. The differential expression itself is defined independently of that application. For a constant covariance-rate matrix $\mathrm{\Sigma }$, the corresponding second-order expression is $\frac{1}{2}{\sum }_{i,j}{\mathrm{\Sigma }}^{ij}{\mathrm{\partial }}_i{\mathrm{\partial }}_{j}f$; a single Gaussian random variable does not by itself specify a stochastic generator.
2. $L$ acts on $C^2$ functions, and that is all that is defined here. The definition assigns to each $f\in C^2({\mathbb{R}}^d)$ the continuous function $Lf$, and for $f\in C^{1,2}$ the space-time function $Lf(t,x)$. It makes no assertion about semigroups: it does not claim that every $C^2$ function lies in the infinitesimal-generator domain of the heat semigroup on $C_0({\mathbb{R}}^d)$, and it does not define a closed operator there. If semigroup-generator language is wanted, the actual domain must be stated and the assertion that $C_c^{\mathrm{\infty }}$ is a core must be proved separately; neither statement is used or asserted on this page.
3. Relation to the heat equation. A $C^{1,2}$ function satisfies ${\mathrm{\partial }}_{t}f+Lf=0$ on an open set exactly when it is space-time harmonic there in the sense of Space-time harmonic functions yield Brownian local martingales up to exit lifetime. This names the differential equation only; a martingale consequence requires a separate stochastic theorem. The later Dynkin formula uses this notation for its compensator. In dimension one $Lf=\frac{1}{2}{f}^{\mathrm{\prime }\mathrm{\prime }}$, the second-order expression appearing in One-dimensional Ito formula.
4. Constant and scaling conventions. $L$ is linear, $Lf=0$ for affine functions, and $L(cf)=cLf$; the operator is determined by the second derivatives only and is invariant under adding affine functions to $f$. All derivatives are ordinary partial derivatives; no weak or distributional interpretation is used, so every application of $L$ in this development verifies that the function is twice continuously differentiable where the operator is applied.

No choice principle is used in the definition itself: $L$ is an explicit differential expression applied to given functions. The Axiom of Choice is declared because the theorems that use $L$ on this page invoke the conditional-expectation and $L^2$ interfaces, and the inherited countable-choice obligations of those interfaces are declared as dependencies of this item.