[Paper Review] Zero noise limit for multidimensional SDEs driven by a pointy gradient

François Delarue, Mario Maurelli|arXiv (Cornell University)|Sep 18, 2019

Stochastic processes and financial applications22 references4 citations

TL;DR

This paper establishes the zero noise limit for multidimensional SDEs driven by a pointy gradient—where the drift arises from a C¹,¹ potential singular at the origin—showing that as noise intensity vanishes, the solution almost surely exits the origin along paths maximizing the potential. The result extends Bafico and Baldi’s one-dimensional findings, proving selection of extremal solutions via probabilistic exit behavior.

ABSTRACT

The purpose of the article is to address the limiting behavior of the solutions of stochastic differential equations driven by a pointy $d$-dimensional gradient as the intensity of the underlying Brownian motion tends to $0$. By pointy gradient, we here mean that the drift derives from a potential that is ${\mathcal C}^{1,1}$ on any compact subset that does not contain the origin. As a matter of fact, the corresponding deterministic version of the differential equation may have an infinite number of solutions when initialized from $0_{{\mathbb R}^d}$, in which case the limit theorem proved in the paper reads as a selection theorem of the solutions to the zero noise system. Generally speaking, our result says that, under suitable conditions, the probability that the particle leaves the origin by going through regions of higher potential tends to $1$ as the intensity of the noise tends to $0$. In particular, our result extends the earlier one due to Bafico and Baldi for the zero noise limit of one dimensional stochastic differential equations.

Motivation & Objective

• To analyze the limiting behavior of solutions to multidimensional SDEs with a singular drift derived from a C¹,¹ potential that is non-Lipschitz at the origin.
• To resolve the Peano phenomenon in higher dimensions, where deterministic systems may have infinitely many solutions starting from the origin.
• To establish a selection principle for solutions in the zero noise limit, favoring paths that exit the origin through regions of higher potential.
• To extend Bafico and Baldi’s one-dimensional zero noise limit result to the multidimensional setting with a pointy gradient.

Proposed method

• The analysis relies on a probabilistic approach to the zero noise limit, focusing on exit behavior from neighborhoods of the origin as the noise intensity ε → 0.
• The authors use the fact that the drift is derived from a potential V that is C¹,¹ away from the origin, with a singularity at 0, and satisfies a growth condition on |∇V|/V^{p/(p+1)} for some p > 0.
• A key technical tool is the construction of a transition point (t,x) depending on ε, where noise dominates before time t and the drift dominates after, with x being a typical position at time t.
• The proof leverages the regularity of V^{1/(p+1)} and uses a directional derivative argument to show that the boundary of {g > 0} (where g = |∇V|) has zero Lebesgue measure.
• A porosity argument is applied to show that the set {g = 0} has zero measure, which supports the almost sure exit along high-potential directions.
• The main result is derived by showing that the probability of exiting through regions of higher potential tends to 1 as ε → 0.

Experimental results

Research questions

• RQ1 How does the solution of a multidimensional SDE with a pointy gradient behave in the zero noise limit, especially when the deterministic system has multiple solutions from the origin?
• RQ2 Can the zero noise limit be used to select a unique solution among infinitely many possible solutions in the Peano phenomenon setting?
• RQ3 What determines the preferred exit direction from the origin in the zero noise limit—specifically, does the potential gradient influence the selection?
• RQ4 To what extent does the one-dimensional result of Bafico and Baldi extend to higher dimensions with a singular drift?
• RQ5 How does the geometry of the potential's gradient influence the exit behavior of the stochastic process as noise vanishes?

Key findings

• As the noise intensity ε → 0, the probability that the solution exits the origin through a region of higher potential tends to 1.
• The zero noise limit selects solutions that follow paths maximizing the potential gradient, effectively acting as a selection principle among infinitely many deterministic solutions.
• The boundary of the set {g > 0}, where g = |∇V|, has zero Lebesgue measure, implying that the set of directions with non-zero gradient is dense in measure.
• The result generalizes Bafico and Baldi’s one-dimensional zero noise limit, where the limiting measure concentrates on extremal solutions weighted by their exit speed.
• The exit behavior is governed by the local regularity of the potential: V is C¹,¹ on compact sets not containing the origin, and the ratio |∇V|/V^{p/(p+1)} is bounded away from zero in neighborhoods of the origin.
• Numerical simulations with ε = 0.001 show that simulated paths closely follow the expected exit directions, providing empirical support for the theoretical prediction.