Ornstein–Uhlenbeck semigroup and Mehler kernel

Wiki card — concept node for exp-families-stability.

(i) Formal statement

On \(\mathbb{R}^d\), the Ornstein–Uhlenbeck (OU) semigroup \((P_t)_{t\ge 0}\) is generated by

\[ L \;=\; \tfrac12 \Delta \;-\; x \cdot \nabla. \tag{OU-L} \]

It is the semigroup of the SDE \(dX_t = -X_t\, dt + dB_t\). Invariant measure: standard Gaussian \(\gamma_1(dx) = (2\pi)^{-d/2} e^{-\|x\|^2/2}\, dx\).

Mehler formula. For \(f : \mathbb{R}^d \to \mathbb{R}\) suitably integrable,

\[ \boxed{\;(P_t f)(x) \;=\; \int_{\mathbb{R}^d} f\bigl(e^{-t} x \;+\; \sqrt{1 - e^{-2t}}\, y\bigr)\, d\gamma_1(y).\;} \tag{Mehler} \]

Spectral decomposition. The Hermite polynomials \(H_\alpha(x) = \prod_i H_{\alpha_i}(x_i)\) (multi-index \(\alpha \in \mathbb{N}^d\)) form an orthogonal basis of \(L^2(\gamma_1)\), and \(L H_\alpha = -|\alpha|\, H_\alpha\) (where \(|\alpha| = \sum\alpha_i\)). Hence \(P_t H_\alpha = e^{-|\alpha| t} H_\alpha\).

Carré-du-champ. \(\Gamma(f, h) = \tfrac12 \langle \nabla f, \nabla h\rangle\) (drift \(-x\cdot\nabla\) is order one and does not contribute). \(CD(1, \infty)\) holds.

(ii) Role in Q1 / Q2 / Q3

• Q1 (invariant reformulation, example 2). OU on \((\mathbb{R}^d, \mathrm{Id}, \gamma_1)\) is the second canonical example of Wheeler's \((M, g, L, \Gamma)\) framework (see invariant-reformulation.md §5). The quadratic-linear class \(\varphi_i = x^\top A_i x + b_i^\top x\) satisfies \((\dagger)\): \(L\varphi_i = -2\,\varphi_i + b_i^\top x + \mathrm{tr}(A_i)\), \(\Gamma(\varphi_i, \varphi_j) = 2 x^\top (A_i A_j)^{\mathrm{sym}} x + (A_ib_j + A_jb_i)^\top x + \tfrac12 b_i^\top b_j\).
• Q2 (modified Riccati). The induced flow (5.6) is \(\dot M = -M^2 - 2M\), \(\dot b = -(I+M)\, b\) — the heat-flow Riccati \(\dot M = -M^2\) with a linear correction coming from the OU drift. Exponential ergodicity to \(\gamma_1\) corresponds to \(M \to 0\).
• Q3. The Cole–Hopf identity on \((M, g, L_{\mathrm{OU}})\) produces the same Hamilton–Jacobi structure, with additional linear terms reflecting the drift.

(iii) References

• Nelson (1973) — The Free Markoff Field. doi:10.1016/0022-1236(73)90025-6.
• Bakry, Gentil, Ledoux (2014) — Analysis and Geometry of Markov Diffusion Operators, §2.7 (OU, Mehler kernel, Hermite decomposition).
• Bakry, Émery (1985) — Diffusions hypercontractives (OU as motivating example for \(CD(K, \infty)\)).

(iv) Worked miniature — Mehler acting on \(\varphi(x) = x^2\)

Take \(d = 1\), \(f(x) = x^2\). Apply Mehler: \[ (P_t f)(x) \;=\; \int_\mathbb{R} (e^{-t} x + \sqrt{1 - e^{-2t}}\, y)^2\, d\gamma_1(y) \;=\; e^{-2t} x^2 \;+\; (1 - e^{-2t}), \] using \(\int y\,d\gamma_1 = 0\), \(\int y^2\, d\gamma_1 = 1\).

Read-off. \(P_t\) maps \(x^2 \mapsto e^{-2t} x^2 + (1 - e^{-2t})\): a linear map on \(\mathrm{Span}\{x^2, 1\}\). This is (i) stability of the quadratic class under \(P_t\), and (ii) ergodicity (\(P_t x^2 \to 1 = \mathbb{E}_{\gamma_1}[x^2]\) as \(t \to \infty\)).

Comparison with heat flow. On \((\mathbb{R}, \tfrac12\Delta)\), the heat flow gives \(P_t^{\mathrm{heat}} x^2 = x^2 + t\) (no contraction, no invariant measure). The OU semigroup has the same quadratic stability but adds the confining contraction \(e^{-2t}\). Both are in the "stable class" of the problem — the \((\dagger)\) condition selects identically in both cases.

Hermite interpretation. \(x^2 = H_2(x) + 1\) in the probabilists' Hermite basis (with \(H_2(x) = x^2 - 1\)). Then \(P_t (H_2 + 1) = e^{-2t} H_2 + 1 = e^{-2t} (x^2 - 1) + 1 = e^{-2t} x^2 + (1 - e^{-2t})\). The quadratic class = \(\mathrm{Span}\{H_0, H_1, H_2\}\) = degree-\(\le 2\) Hermite space, spectrally stable under \(L\).