[Paper Review] Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations

Robert C. Dalang, Fei Pu|arXiv (Cornell University)|Oct 12, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance29 references3 citations

TL;DR

This paper establishes sharp lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations driven by space-time white noise. Using Malliavin calculus and a novel Gaussian-type upper bound on the two-point transition density, it proves that hitting probabilities in the non-Gaussian case are as sharp as in the Gaussian case, measured via Newtonian capacity, extending prior results on classical stochastic heat equations.

ABSTRACT

We consider a system of $d$ non-linear stochastic fractional heat equations in spatial dimension $1$ driven by multiplicative $d$-dimensional space-time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of $(u(s, y), u (t, x))$. From this result, we deduce optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty[ imes \mathbb{R}\}$ in the non-Gaussian case, in terms of Newtonian capacity, which is as sharp as that in the Gaussian case. This also improves the result in Dalang, Khoshnevisan and Nualart [ extit{Probab. Theory Related Fields} extbf{144} (2009) 371--424] for systems of classical stochastic heat equations. We also establish upper bounds on hitting probabilities of the solution in terms of Hausdorff measure.

Motivation & Objective

• To derive optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations with multiplicative space-time white noise.
• To extend existing results on classical stochastic heat equations (α=2) to the more general fractional case (1<α≤2).
• To establish that hitting probabilities in the non-Gaussian regime are as sharp as in the Gaussian case, using Newtonian capacity as the capacity criterion.
• To improve upon prior results by Dalang, Khoshnevisan, and Nualart (2009) by providing tighter bounds in the non-linear, fractional setting.
• To derive upper bounds on hitting probabilities using Hausdorff measure, providing a geometric characterization of the solution's range.

Proposed method

• Derives a sharp Gaussian-type upper bound on the two-point probability density function of the solution process (u(s,y), u(t,x)) using Malliavin calculus techniques.
• Applies the Malliavin calculus framework to analyze the regularity of the law of the solution and its increments, particularly focusing on the Malliavin derivative.
• Uses the fractional parabolic metric Δα((t,x);(s,y)) = |t−s|^{(α−1)/(2α)} + |x−y|^{(α−1)/2} to characterize Hölder continuity and moment estimates.
• Employs Gronwall’s lemma in conjunction with estimates on Green’s kernel Gα(t,x) to control the growth of Malliavin derivatives and derive moment bounds.
• Establishes moment estimates for the solution and its Malliavin derivative via stochastic integration and Fourier analysis of the Green kernel.
• Applies capacity-theoretic methods, specifically Newtonian capacity and Hausdorff measure, to characterize the hitting probabilities of the solution’s range.

Experimental results

Research questions

• RQ1 What are the optimal lower bounds on the hitting probabilities of non-linear systems of stochastic fractional heat equations with multiplicative space-time white noise?
• RQ2 How do these hitting probabilities compare to those in the Gaussian case, and can they be characterized using Newtonian capacity?
• RQ3 Can the sharpness of hitting probability bounds in the non-Gaussian case be matched to that of the Gaussian case via capacity-theoretic methods?
• RQ4 What role does the fractional order α∈(1,2] play in determining the regularity and hitting behavior of the solution?
• RQ5 How do upper bounds on hitting probabilities relate to the Hausdorff measure of the solution’s range?

Key findings

• The paper establishes a sharp Gaussian-type upper bound on the two-point transition density of the solution process (u(s,y), u(t,x)).
• This density bound leads to optimal lower bounds on hitting probabilities in the non-Gaussian case, matching the sharpness of the Gaussian case in terms of Newtonian capacity.
• The hitting probability of the solution's range over a compact set I×J is bounded below by a positive constant depending on the Newtonian capacity of the set.
• The result improves upon Dalang, Khoshnevisan, and Nualart (2009) by extending their bounds from classical (α=2) to fractional (1<α≤2) stochastic heat equations.
• Upper bounds on hitting probabilities are derived in terms of Hausdorff measure, providing a geometric characterization of the solution’s range.
• The paper proves that the Malliavin derivative of the solution satisfies a Hölder-type moment estimate with exponent β<1−2(α+1)/(p(α−1)), enabling the use of Kolmogorov’s continuity theorem for regularity.