Fernique's theorem for rotation-invariant measures #

Let μ be a finite measure on a second-countable normed space E such that the product measure μ.prod μ on E × E is invariant by rotation of angle -π/4. Then there exists a constant C > 0 such that the function x ↦ exp (C * ‖x‖ ^ 2) is integrable with respect to μ.

Sketch of the proof #

The main case of the proof is for μ a probability measure such that there exists a positive a : ℝ such that 2⁻¹ < μ {x | ‖x‖ ≤ a} < 1. If μ is a probability measure and a does not exist then we can show that there is a ball with finite radius of measure 1, and the result is true for C = 1 (for example), since x ↦ exp (‖x‖ ^ 2) is almost surely bounded. We then choose such an a.

In order to show the existence of C such that x ↦ exp (C * ‖x‖ ^ 2) is integrable, we prove as intermediate result that for a, c with 2⁻¹ < c ≤ μ {x | ‖x‖ ≤ a}, the integral ∫⁻ x, exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) ∂μ is bounded by a finite quantity (logRatio c is a multiple of log (c / (1 - c))). We can then take C = logRatio c * a⁻¹ ^ 2.

We now turn to the proof of the intermediate result.

First in measure_le_mul_measure_gt_le_of_map_rotation_eq_self we prove that if a measure μ is such that μ.prod μ is invariant by rotation of angle -π/4 then μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2. The rotation invariance is used only through that inequality.

We define a sequence of thresholds t n inductively by t 0 = a and t (n + 1) = √2 * t n + a. They are chosen such that the invariance by rotation gives μ {x | ‖x‖ ≤ a} * μ {x | t (n + 1) < ‖x‖} ≤ μ {x | t n < ‖x‖} ^ 2. Thanks to that inequality we can show that μ {x | t n < ‖x‖} decreases fast with n: for mₐ = μ {x | ‖x‖ ≤ a}, μ {x | t n < ‖x‖} ≤ mₐ * exp (- log (mₐ / (1 - mₐ)) * 2 ^ n).

We cut the space into annuli {x | t n < ‖x‖ ≤ t n + 1} and bound the integral separately on each annulus. On that set the function exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) is bounded by exp (logRatio c * a⁻¹ ^ 2 * t (n + 1) ^ 2), which is in turn less than exp (2⁻¹ * log (c / (1 - c)) * 2 ^ n) (from the definition of the threshold t and logRatio c). The measure of the annulus is bounded by μ {x | t n < ‖x‖}, for which we derived an upper bound above. The function gets exponentially large, but μ {x | t n < ‖x‖} decreases even faster, so the integral is bounded by a quantity of the form exp (- u * 2 ^ n) for u>0. Summing over all annuli (over n) gives a finite value for the integral.

Main statements #

lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self: for μ a probability measure whose product with itself is invariant by rotation and for a, c with 2⁻¹ < c ≤ μ {x | ‖x‖ ≤ a}, the integral ∫⁻ x, exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) ∂μ is bounded by a quantity that does not depend on a.
exists_integrable_exp_sq_of_map_rotation_eq_self: Fernique's theorem for finite measures whose product is invariant by rotation.

References #

[Xavier Fernique, Intégrabilité des vecteurs gaussiens][fernique1970integrabilite]
[Martin Hairer, An introduction to stochastic PDEs][hairer2009introduction]

TODO #

From the intermediate result lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self, we can deduce bounds on all the moments of the measure μ as function of powers of the first moment.