[Paper Review] Brownian motion with general drift

Damir Kinzebulatov, Yu. A. Semënov|arXiv (Cornell University)|Oct 18, 2017

Stochastic processes and financial applications3 citations

TL;DR

This paper establishes the existence and uniqueness of a weak solution to the stochastic differential equation $ dX(t) = -b(X(t))dt + \sqrt{2}dW(t) $ for $ d \geq 3 $, when the drift $ b $ belongs to the class of weakly form-bounded vector fields $ \mathbf{F}^{1/2}_{\delta} $, which includes sub-critical and critical classes such as $ L^d + L^\infty $, Kato class, and weak $ L^d $. The key contribution is proving that the associated Feller process has almost surely continuous, finite trajectories, ensuring pathwise regularity under broad drift conditions.

ABSTRACT

We construct and study the weak solution to stochastic differential equation $dX(t)=-b(X(t))dt+\sqrt{2}dW(t)$, $X_0=x$, for every $x \in \mathbb R^d$, $d \geq 3$, with $b$ in the class of weakly form-bounded vector fields, containing, as proper subclasses, a sub-critical class $[L^d+L^\infty]^d$, as well as critical classes such as weak $L^d$ class, Kato class, Campanato-Morrey class, Chang-Wilson-T. Wolff class.

Motivation & Objective

• To extend the existence and uniqueness of weak solutions to stochastic differential equations with drift $ b $ beyond the classical $ L^p $, $ p > d $, or Kato class assumptions.
• To establish pathwise regularity (almost sure continuity and finiteness) of the solution process under a broader class of drifts, including critical and sub-critical vector fields.
• To construct a Feller process via a limiting procedure on regularized drifts $ b_n $, ensuring convergence of the associated semigroups in the strong operator topology on $ C_\infty $.
• To unify and generalize previous results on SDEs with singular drifts by introducing the class $ \mathbf{F}^{1/2}_{\delta} $, which contains $ L^d + L^\infty $, Kato, Campanato-Morrey, and Chang-Wilson-T. Wolff classes.
• To prove that the resulting Markov process has trajectories in $ C([0,\infty[, \mathbb{R}^d) $ under the measure $ \mathbb{P}_x $, ensuring pathwise continuity and non-explosion.

Proposed method

• The authors define the class $ \mathbf{F}^{1/2}_{\delta} $ via a weak form-boundedness condition involving the resolvent $ (\lambda - \Delta)^{-1/4} $, ensuring integrability and regularity of the drift.
• They construct a sequence of smooth, compactly supported approximations $ b_n = \gamma_{\varepsilon_n} * (\mathbf{1}_n b) $ using Friedrichs mollifiers to regularize the drift.
• Using the convergence $ b_n \to b $ in $ L^1_{\text{loc}} $, they prove the strong convergence of the semigroups $ e^{-t\Lambda_{C_\infty}(b_n)} $ to a limit $ e^{-t\Lambda_{C_\infty}(b)} $, forming a Feller semigroup.
• The solution process is constructed as the strong Markov process associated with this Feller semigroup on $ C_\infty(\mathbb{R}^d) $, with paths valued in $ D([0,\infty[, \bar{\mathbb{R}}^d) $.
• The key technical tool is a perturbation estimate involving the resolvent $ (\mu - \Delta)^{-1} $, with bounds derived via operator norm estimates and interpolation techniques.
• They derive a uniform bound on the operator norm $ \| |b_n|^{1/p} \nabla (\mu - \Delta)^{-1} \|_{p \to p} $, showing it is controlled by $ m_d c_p \delta $, which is less than 1 under the condition $ m_d \delta < 4(d-2)/(d-1)^2 $.

Experimental results

Research questions

• RQ1 Does a weak solution exist for the SDE $ dX(t) = -b(X(t))dt + \sqrt{2}dW(t) $ when $ b $ is in the weakly form-bounded class $ \mathbf{F}^{1/2}_{\delta} $, even if $ b $ is not in $ L^p $ for $ p > d $?
• RQ2 Can the solution process be shown to have almost surely continuous and finite trajectories for $ d \geq 3 $, even when $ b $ is singular or critical?
• RQ3 Is the Feller semigroup generated by $ -\Delta + b \cdot \nabla $ well-defined and strongly continuous on $ C_\infty(\mathbb{R}^d) $ for $ b \in \mathbf{F}^{1/2}_{\delta} $?
• RQ4 What is the optimal condition on $ \delta $ such that the perturbation of the Laplacian by $ b \cdot \nabla $ remains a Feller generator?
• RQ5 Can the convergence of regularized semigroups $ e^{-t\Lambda_{C_\infty}(b_n)} $ be established uniformly in $ t \in [0,1] $, ensuring the limit is a Feller semigroup?

Key findings

• The weak solution to the SDE exists and is unique in law for all $ x \in \mathbb{R}^d $, $ d \geq 3 $, when $ b \in \mathbf{F}^{1/2}_{\delta} $ and $ m_d \delta < 4(d-2)/(d-1)^2 $.
• The solution process has almost surely continuous and finite trajectories on $ [0, \infty) $, so $ \mathbb{P}_x $-a.s. $ X \in C([0,\infty[, \mathbb{R}^d) $.
• The semigroup $ e^{-t\Lambda_{C_\infty}(b)} $ is a Feller $ C_0 $-semigroup, positivity-preserving, and $ L^\infty $-contractive, generated by $ -\Delta + b \cdot \nabla $.
• The operator norm $ \| |b_n|^{1/p} \nabla (\mu - \Delta)^{-1} \|_{p \to p} $ is bounded by $ (1+\varepsilon) m_d c_p \delta $, and can be made strictly less than 1 by choosing $ \varepsilon $ small and $ \delta $ small enough.
• The convergence $ e^{-t\Lambda_{C_\infty}(b_n)} \to e^{-t\Lambda_{C_\infty}(b)} $ holds uniformly in $ t \in [0,1] $, ensuring the limit semigroup is strongly continuous on $ C_\infty(\mathbb{R}^d) $.
• The class $ \mathbf{F}^{1/2}_{\delta} $ properly contains critical classes such as the Kato class $ \mathbf{K}^{d+1}_0 $, weak $ L^d $, Campanato-Morrey, and Chang-Wilson-T. Wolff classes.