A Harmonic-Polynomial Escape Sequence for an Ornstein–Uhlenbeck Transmission Problem Across Hilbert–Schmidt Ellipsoids

We give a constructive high-dimensional escape sequence for the equation (∆ − x ·∇)u = u associated with the symmetric Ornstein–Uhlenbeck operator in Gaussian space. Let (a_i)_i≥1 be a positive square summable sequence and let B_n = {x ∈ Rⁿ : ∑ⁿ_i=1 a_i² x_i² < 1}. We construct functions u_n that are continuous on Rⁿ, smooth on both sides of ∂B_n, solve the positive spectral equation away from ∂B_n, and have finite Gaussian H¹ energy. The construction uses a single real harmonic polynomial, Re(x1 +ix2)^m_n with m_n = ⌊n^1/8⌋, multiplied by the finite-energy Tricomi branch of the separated radial Ornstein–Uhlenbeck equation and then extended into B_n by the weighted Dirichlet principle. The exterior energy has a lower bound of order (2π)^n/2n^−1/2(2m_n/e)^m_n, whereas the interior minimizing energy is bounded by (2π)^n/2n^CC_a^m_n. Hence the ratio of total Gaussian H¹ energy to the energy inside B_n tends to infinity. The proof is written with all non-standard notation defined explicitly, and two examples, including an ℓ¹-small sequence with ∑_i a_i < 1, are included as checks of the hypotheses.