ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A singular line law has a smooth positive-time density

Example

Let $λ_{1}$ be one-dimensional Lebesgue measure and define $μ (A) = λ_{1} ({u \in [0, 1] : (u, 0) \in A})$ for every Borel $A \subseteq R^{2}$ . The box measure formula gives $λ_{1} ([0, 1]) = 1$ , so $μ$ is a Borel probability supported on $L = [0, 1] \times {0}$ . Let $g : [0, T] \to (0, \infty)$ be continuously differentiable with $0 < g_{\min} \leq g (t) \leq g_{\max} < \infty$ and set $v (t) = \int_{0}^{t} g (r)^{2} d r$ . For every $t > 0$ , the Gaussian convolution of $μ$ has the density

$p_{t} (x_{1}, x_{2}) = \frac{1}{2 π v (t)} \int_{0}^{1} \exp ⁣ (- \frac{(x_{1} - u)^{2} + x_{2}^{2}}{2 v (t)}) d u .$

The initial law $μ$ is singular with respect to two-dimensional Lebesgue measure, meaning it assigns mass one to a Borel set of planar Lebesgue measure zero, while $p_{t}$ is strictly positive and infinitely differentiable in $(x_{1}, x_{2})$ for each $t > 0$ .

Example references

Hunter, Notes on Partial Differential Equations, Chapter 5, §§5.1.1–5.1.2

Särkkä and Solin, Applied Stochastic Differential Equations, Chapter 5, §§5.1–5.2

Facts & Assumptions

Given: A uniform random variable on $[0, 1]$ , the embedded line-supported law, and the compactly supported scalar-noise Gaussian smoothing setup.

[A1]

The full Axiom of Choice is inherited from the Gaussian smoothing and probability interfaces. The Axiom of Choice

[A2]

Countable Choice is assumed by the box-measure theorem used below; the item's SDE and smoothing interfaces also carry the stronger Full AC assumption. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

For any compactly supported probability law in $R^{2}$ , the positive-time law has the explicit Gaussian-convolution density and is strictly positive. Gaussian-smoothed data law and transition density

[F2]

The convolution density is infinitely differentiable in space for each positive time. Gaussian smoothing density and forward equation

[F3]

The Lebesgue measure of an axis-aligned rectangle in $R^{2}$ is the product of its side lengths. A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included

[F4]

The law of a measurable map is its pushforward probability measure. Law or distribution of a random element

Proof

Given: The example's random vector and the Gaussian smoothing setup.

1.1A2F3F4givenalgebra

The map $u \mapsto (u, 0)$ is continuous and hence Borel measurable; [F4] makes the formula in the statement its pushforward measure, and [A2, F3] give its total mass $λ_{1} ([0, 1]) = 1$ . It assigns mass one to the Borel segment $L$ . For each integer $n \geq 1$ , the segment lies in the rectangle $[0, 1] \times [- 1 / n, 1 / n]$ , whose Lebesgue area is $2 / n$ by [F3]. Therefore its outer measure is at most $2 / n$ for every $n$ , hence its Lebesgue measure is zero. Since $μ (L) = 1$ and Lebesgue measure of $L$ is zero, $μ$ is singular with respect to two-dimensional Lebesgue measure.

2.1A1F1F2step 1.1given∎

Applying the convolution formula [F1] to $μ$ gives $p_{t} (x) = \int_{0}^{1} ϕ_{v (t)} (x - (u, 0)) d u$ . In dimension two, $ϕ_{v (t)} (x - (u, 0)) = (2 π v (t))^{- 1} \exp (- ((x_{1} - u)^{2} + x_{2}^{2}) / (2 v (t)))$ , which is the displayed expression. The integrand is strictly positive for every $u$ , so its integral over a unit interval is strictly positive; [F2] gives infinite spatial differentiability for every $t > 0$ . Thus the initial singularity does not prevent positive-time smoothing.

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Sources

John K. Hunter, Notes on Partial Differential Equations (standard reference, not scraped)
Simo Särkkä and Arno Solin, Applied Stochastic Differential Equations (standard reference, not scraped)