[Paper Review] Hitting probabilities for general Gaussian processes

Eulàlia Nualart, Viens, F.|arXiv (Cornell University)|May 8, 2013

Stochastic processes and financial applications20 references3 citations

TL;DR

This paper establishes sharp upper and lower bounds for hitting probabilities of d-dimensional Gaussian processes with general variance functions γ(t), under minimal regularity conditions: γ continuous, increasing, concave, with γ(0)=0 and γ′(0+)=∞. It introduces a novel capacity and Hausdorff measure-based framework using Malliavin calculus and two-point local nondeterminism, showing that logarithmic corrections to power-law behavior (e.g., t^H log^β(1/t)) shift the phase transition point for polarity of singletons from H=1/d to a range dependent on β, refining classical Hölder-based criteria.

ABSTRACT

For a scalar Gaussian process $B$ on $\mathbb{R}_{+}$ with a prescribed general variance function $γ^{2}\left(r ight) =\mathrm{Var}\left(B\left(r ight) ight) $ and a canonical metric $\mathrm{E}[\left(B\left(t ight) -B\left(s ight) ight) ^{2}]$ which is commensurate with $γ^{2}\left(t-s ight) $, we estimate the probability for a vector of $d$ iid copies of $B$ to hit a bounded set $A$ in $\mathbb{R}^{d}$, with conditions on $γ$ which place no restrictions of power type or of approximate self-similarity, assuming only that $γ$ is continuous, increasing, and concave, with $γ\left(0 ight) =0$ and $γ^{\prime}\left(0+ ight) =+\infty$. We identify optimal base (kernel) functions which depend explicitly on $γ$, to derive upper and lower bounds on the hitting probability in terms of the corresponding generalized Hausdorff measure and non-Newtonian capacity of $A$ respectively. The proofs borrow and extend some recent progress for hitting probabilities estimation, including the notion of two-point local-nondeterminism in Biermé, Lacaux, and Xiao \cite{Bierme:09}.

Motivation & Objective

• To derive sharp upper and lower bounds for the hitting probability of d-dimensional Gaussian processes with general variance functions γ(t), without assuming power-law or self-similar structure.
• To identify the precise conditions under which a set A ⊂ ℝ^d is polar (zero hitting probability) or non-polar (positive hitting probability), extending classical results based on Hölder continuity.
• To develop a new intrinsic framework based on γ(t) and generalized Hausdorff measures, replacing classical Hölder or modulus-of-continuity techniques that fail to capture logarithmic corrections.
• To characterize the phase transition for singleton polarity in terms of the parameter β in γ(t) = t^H log^β(1/t), showing it occurs at β = H rather than H = 1/d.
• To establish a connection between hitting probabilities and non-Newtonian capacity via a new kernel function derived from γ and the process's local behavior.

Proposed method

• Use of Malliavin calculus to derive precise density estimates for scalar Gaussian processes hitting small balls and points, enabling sharp probability bounds.
• Adaptation and extension of two-point local nondeterminism (LND) from Biermé, Lacaux, and Xiao (2013) to general γ(t), enabling second-moment energy estimates for lower bounds.
• Construction of a kernel function K(x) = v∘γ⁻¹(x), where v(r) = ∫_r^{b−a} s^{−1} log^{−β/H}(1/s) ds, to define a generalized capacity for lower bounds.
• Application of covering arguments and energy methods to derive upper bounds in terms of generalized Hausdorff measure, with gauge function ψ(x) = x^{d−1/H} log^{β/H}(1/x).
• Use of a novel capacity positivity criterion involving series convergence: ∑ 2^{-n} / φ(q^n) < ∞, to verify positive capacity for sets like Cantor sets.
• Incorporation of the canonical metric condition E[|B(t)−B(s)|²] ≍ γ²(|t−s|), ensuring the variance structure γ(t) governs path behavior without assuming self-similarity.

Experimental results

Research questions

• RQ1 How do logarithmic corrections in the variance function γ(t) = t^H log^β(1/t) affect the phase transition for singleton polarity in d-dimensional Gaussian processes?
• RQ2 Can hitting probabilities for general Gaussian processes be bounded using generalized Hausdorff measures and non-Newtonian capacities without assuming Hölder continuity or self-similarity?
• RQ3 What is the precise role of the function v(r) = ∫_r^{b−a} s^{−1} log^{−β/H}(1/s) ds in determining the sharpness of hitting probability bounds?
• RQ4 How does the Malliavin calculus enable sharp point and small ball hitting probability estimates when classical regularity theory fails?
• RQ5 Under what conditions on γ(t) is a given set A ⊂ ℝ^d polar (i.e., P(∃t: B(t) ∈ A) = 0) for a d-dimensional Gaussian process?

Key findings

• For γ(t) = t^H log^β(1/t), the phase transition for singleton polarity occurs at β = H, not at H = 1/d, meaning that for β < H, singletons are non-polar, and for β ≥ H, they are polar.
• The hitting probability of a bounded set A ⊂ ℝ^d is bounded above by a generalized Hausdorff measure with gauge function ψ(x) = x^{d−1/H} log^{β/H}(1/x), and bounded below by a capacity involving the kernel K(x) = log^{1−β/H}(x^{-1}) when β < H.
• When β ≥ H, the kernel K(x) is bounded (K ≡ 1), and the lower bound is positive if the capacity C_{1/φ}(A) > 0, which holds when ∑ 2^{-n} / φ(q^n) < ∞ for φ(x) = x^{d−1/H} log^{β/H}(1/x).
• For the Cantor set A with dimension d−1/H, the Hausdorff measure 𝒫_{ψ}(A) ≤ 1 for ψ(x) = x^{d−1/H}, and 𝒫_{φ}(A) = 0 for φ(x) = x^{d−1/H} log^{β/H}(1/x) with β < 0, implying polarity for β < 0.
• The capacity positivity criterion is verified via series convergence: for β' ≥ H, ∑_{n=1}^∞ log^{−β'/H}(q^n) < ∞, ensuring positive hitting probability for B^{H,β'} with β' ≥ H.
• The method successfully replaces classical Hölder-based regularity estimates with γ-intrinsic techniques, proving that such classical tools are insufficient for sharp results when logarithmic corrections are present.