Results on Gaussian semigroups and Ehrhard's inequality: integration by parts, smoothing, the gradient bound, the monotone quantity and the theorem, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Ehrhard's inequality for convex sets (Ehrhard) — Theorem A. For closed convex K, L in R^n with Gaussian measures in (0,1) and 0 <= lambda <= 1, Phi^{-1}(gamma_n(lambda K + (1-lambda) L)) >= lambda Phi^{-1}(gamma_n(K)) + (1-lambda) Phi^{-1}(gamma_n(L)). Proved in the last lesson.
• Gaussian integration by parts — Lemma 1.1. For a C^1 function psi of exponential growth, the integral of w_i psi against gamma_C equals sum_j C_ij times the integral of d_j psi.
• Smoothing and the heat equation — Proposition 1.2. For bounded measurable F and t > 0, Q^C_t F is smooth with derivatives bounded by c t^(-|alpha|/2) sup|F|, its gradient is t^(-1/2) times the integral of F(z + sqrt t w) C^{-1} w, and d/dt Q^C_t F = L_C Q^C_t F.
• A moving function under the semigroup — Proposition 1.5. d/du Q^C_{b-u} H_u = Q^C_{b-u}(d_u H_u - L_C H_u) for families of C^2_b functions with continuous bounded derivatives in (u, z).
• Gradient bound for the heat semigroup — Proposition 1.7. For measurable f with values in [0,1] and t > 0, |grad Q_t f| <= t^(-1/2) I(Q_t f) with I = phi o Phi^{-1}; equality for half-spaces.
• A quantity monotone along the heat flow (after Neeman-Paouris) — Theorem 2.1. If the smoothed functions Q_eps f(x), Q_eps g(y), Q_{sigma^2 eps} h(lambda x + mu y) satisfy lambda Phi^{-1} + mu Phi^{-1} - sigma Phi^{-1} <= -eps_1 everywhere, the same combination of the Gaussian integrals of f, g and h(sigma .) is negative.
• Perturbation of positive semidefinite matrices (Neeman-Paouris) — Lemma 2.4. If A_0 >= 0, B_0 is symmetric, B_0 >= delta_0 on ker A_0 and A_0 >= delta_0 on its orthogonal complement, then A_0 + eta B_0 >= 0 for 0 <= eta <= delta_0^2/(||B_0||^2 + delta_0 ||B_0||).
• Functional form for uniformly continuous functions — Proposition 3.1. If uniformly continuous f, g, h with values in [Phi(-N), Phi(N)] satisfy lambda Phi^{-1}(f(x)) + mu Phi^{-1}(g(y)) - sigma Phi^{-1}(h(lambda x + mu y)) <= -eps_3 for all x, y, the same combination of the Gaussian integrals of f, g and h(sigma .) is negative.
• Boundaries of convex sets — Lemma 3.2. A convex set with an interior point has a Lebesgue-null boundary and the same interior as its closure; a convex set without interior points lies in an affine hyperplane.
• Concavity along translates — Corollary 3.4. For a closed convex K and a vector v with gamma_n(K + uv) in (0,1) for all real u, the function u -> Phi^{-1}(gamma_n(K + uv)) is concave.

Reading order

Read the lessons in order. The first lesson collects the facts on Gaussian heat semigroups and can be read on its own; the second proves the monotonicity theorem; the third passes to convex sets.