[Paper Review] Nonlinear Fokker-Planck equations for Probability Measures on Path Space and Path-Distribution Dependent SDEs

Xing Huang, Michael Röckner+1 more|arXiv (Cornell University)|Sep 2, 2017

Stochastic processes and financial applications17 references8 citations

TL;DR

This paper establishes existence, uniqueness, and regularity of solutions to nonlinear Fokker-Planck equations for probability measures on path space via path-distribution dependent SDEs. It derives Harnack inequalities, derivative estimates, and Wasserstein continuity under mild coefficient conditions, extending classical results to memory-dependent systems with distributional dependence on the entire path history.

ABSTRACT

By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker--Planck equations for probability measures $(μ_t)_{t \geq 0}$ on the path space $\mathcal C:=C([-r_0,0];\mathbb R^d),$ is analyzed: $$\partial_t μ(t)=L_{t,μ_t}^*μ_t,\ \ t\ge 0,$$ where $μ(t)$ is the image of $μ_t$ under the projection $\mathcal C iξ\mapsto ξ(0)\in\mathbb R^d$, and $$L_{t,μ}(ξ):= \frac 1 2\sum_{i,j=1}^d a_{ij}(t,ξ,μ)\frac{\partial^2} {\partial_{ξ(0)_i} \partial_{ξ(0)_j}} +\sum_{i=1}^d b_i(t,ξ,μ)\frac{\partial}{\partial_{ξ(0)_i}},\ \ t\ge 0, ξ\in \mathcal C, μ\in \mathcal P^{\mathcal C}.$$ Under reasonable conditions on the coefficients $a_{ij}$ and $b_i$, the existence, uniqueness, Lipschitz continuity in Wasserstein distance, total variational norm and entropy, as well as derivative estimates are derived for the martingale solutions.

Motivation & Objective

• To analyze nonlinear Fokker-Planck equations for probability measures on path space driven by path-distribution dependent SDEs.
• To establish existence, uniqueness, and regularity of martingale solutions under mild coefficient conditions.
• To derive quantitative estimates in Wasserstein distance, total variation norm, and entropy for the evolution of measures.
• To extend Harnack and derivative estimates to path-dependent, distribution-affected diffusion processes.

Proposed method

• Formulates a nonlinear Fokker-Planck equation on the path space $\mathscr{C} = C([-r_0,0]; \mathbb{R}^d)$ with coefficients depending on the current path and its law.
• Introduces a path-distribution dependent SDE: $\text{d}X(t) = \bar{b}(t,X_t)\text{d}t + \sigma(t)\text{d}W(t)$, where drift depends on the law of the path segment.
• Applies the coupling by change of measure technique to derive Harnack inequalities and derivative estimates for the associated semigroup.
• Uses the time-marginal evolution of the SDE to define a nonlinear Fokker-Planck equation $\partial_t \mu_t = L_{t,\mu_t}^* \mu_t$ on path measures.
• Imposes conditions on $\sigma^{-1}$ and $\nabla b$ to ensure finite bounds $\Lambda(T)$ and $K(T)$, enabling gradient estimates.
• Applies the Riesz Representation Theorem to derive $L^2$-regularity of the derivative of the law along directions in $\mathbb{H}^1$.

Experimental results

Research questions

• RQ1 Under what conditions does a nonlinear Fokker-Planck equation on path space admit a unique martingale solution for a given initial measure?
• RQ2 How do Wasserstein distance, total variation, and entropy evolve under the nonlinear Fokker-Planck dynamics?
• RQ3 Can Harnack-type inequalities be established for path-distribution dependent SDEs with memory effects?
• RQ4 What are the derivative estimates for the law of the solution process in the path space, and how do they depend on the coefficient structure?
• RQ5 How does the law of the solution process behave under shifts in the initial path, and what regularity properties does it possess?

Key findings

• The paper establishes existence and uniqueness of martingale solutions to the nonlinear Fokker-Planck equation on path space under reasonable coefficient conditions.
• It proves Lipschitz continuity of the solution map in Wasserstein distance, total variation norm, and entropy, with explicit bounds depending on $\Lambda(T)$ and $K(T)$.
• A Harnack inequality is derived: $ (P_T f)^p(\mu_0) \leq P_T f^p(\eta + \cdot)(\mu_0) \exp\left[ \frac{p \Lambda(T)(1 + T^2 K(T))}{(p-1)^2} \left( \frac{|\eta(-r_0)|^2}{T - r_0} + \|\eta\|_{\mathbb{H}^1}^2 \right) \right] $ for $p > 1$.
• An entropy bound is obtained: $ \int \log \frac{d\mu_T(\cdot + \eta)}{d\mu_T} d\mu_T \leq \Lambda(T)(1 + T^2 K(T)) \left( \frac{|\eta(-r_0)|^2}{T - r_0} + \|\eta\|_{\mathbb{H}^1}^2 \right) $.
• The derivative of the law $\mu_T$ along $\eta \in \mathbb{H}^1$ is shown to be absolutely continuous with respect to $\mu_T$, and its Radon-Nikodym derivative satisfies $ \int \left| \frac{d \partial_\eta \mu_T}{d \mu_T} \right|^2 d\mu_T \leq \Lambda(T)(1 + K(T)T^2) \left( \frac{|\eta(-r_0)|^2}{T - r_0} + \|\eta\|_{\mathbb{H}^1}^2 \right) $.
• The coupling by change of measure method yields a representation for the derivative of the semigroup: $ \mathbb{E}(\nabla_\eta f)(X_T) = \mathbb{E}\left[ f(X_T) \int_0^T \big\langle \sigma(t)^{-1}(\Phi(t) - \nabla_{\Theta_t} b(t, \cdot, P_t^* \mu_0)(X_t)), dW(t) \big\rangle \right] $.