[Paper Review] The complement value problem for non-local operators

Wei Sun|arXiv (Cornell University)|Mar 31, 2018

Advanced Mathematical Modeling in Engineering35 references3 citations

TL;DR

This paper establishes the existence and uniqueness of a bounded continuous weak solution to the complement value problem for a non-local operator combining the Laplacian, a fractional Laplacian, and drift and potential terms. Using semi-Dirichlet forms and heat kernel estimates, it derives a probabilistic representation of the solution via an expectation involving a Markov process and hitting time on the domain's complement.

ABSTRACT

Let $D$ be a bounded Lipschitz domain of $\mathbb{R}^d$. We consider the complement value problem $$ \left\{\begin{array}{l}(Δ+a^αΔ^{α/2}+b\cdot abla+c)u+f=0\ \ { m in}\ D,\\ u=g\ \ { m on}\ D^c. \end{array} ight.$$ Under mild conditions, we show that there exists a unique bounded continuous weak solution. Moreover, we give an explicit probabilistic representation of the solution. The theory of semi-Dirichlet forms and heat kernel estimates play an important role in our approach.

Motivation & Objective

• To establish the existence and uniqueness of a bounded continuous weak solution to a non-local elliptic integro-differential equation with complement value conditions.
• To provide a probabilistic representation of the solution using a Markov process associated with the non-local operator.
• To extend the theory of non-local operators to cases where coefficients are not necessarily continuous and the maximum principle does not hold.
• To leverage semi-Dirichlet forms and recent heat kernel estimates to overcome analytical challenges in the non-local setting.
• To prove regularity of the solution up to the boundary, including continuity at boundary points when the boundary data is continuous.

Proposed method

• Utilizes the theory of semi-Dirichlet forms to analyze the non-local operator and establish the existence and uniqueness of weak solutions.
• Employs heat kernel estimates from Chen and Hu (2018) to control the transition density of the associated Markov process.
• Constructs a strong Markov process $(X_t, P_x)$ associated with the operator $L = \Delta + a^\alpha \Delta^{\alpha/2} + b\cdot\nabla$ on $\mathbb{R}^d$.
• Derives a probabilistic representation of the solution as $u(x) = \mathbb{E}_x\left[e(\tau)g(X_\tau) + \int_0^\tau e(s)f(X_s)ds\right]$, where $\tau = \inf\{t > 0 : X_t \in D^c\}$.
• Applies the integration by parts formula for semi-martingales to derive a martingale representation and use the dominated convergence theorem to prove uniqueness.
• Uses approximation techniques involving truncations and regularization to handle low-regularity coefficients and establish convergence in the limit.

Experimental results

Research questions

• RQ1 Under what conditions does the complement value problem for a non-local operator with drift and potential terms admit a unique bounded continuous weak solution?
• RQ2 Can a probabilistic representation of the solution be rigorously derived when the coefficients are not continuous and the maximum principle does not hold?
• RQ3 How do heat kernel estimates and semi-Dirichlet form theory facilitate the analysis of non-local operators in bounded domains?
• RQ4 What regularity properties does the solution inherit from the boundary data $g$ on $D^c$, especially continuity at $\partial D$?
• RQ5 What is the role of the positive part of the potential $c^+$ in ensuring the finiteness of the exponential moment $\mathbb{E}_x[\exp(\int_0^\tau c^+(X_s)ds)]$?

Key findings

• There exists a unique bounded continuous weak solution $u \in B_b(\mathbb{R}^d)$ to the complement value problem (1.1) under the condition that $\|c^+\|_{L^{p\vee 1}} \leq M$ for some $M > 0$.
• The solution admits an explicit probabilistic representation: $u(x) = \mathbb{E}_x\left[e(\tau)g(X_\tau) + \int_0^\tau e(s)f(X_s)ds\right]$, where $e(t) = \exp(\int_0^t c(X_s)ds)$ and $\tau = \inf\{t > 0 : X_t \in D^c\}$.
• If $g$ is continuous at $z \in \partial D$, then $\lim_{x \to z} u(x) = u(z)$, ensuring boundary regularity of the solution.
• The solution satisfies the equation in the distributional sense: for all $\phi \in C_c^\infty(D)$, the identity (1.4) holds, involving gradient, non-local, drift, potential, and forcing terms.
• When $c \leq 0$, the solution is continuous on $\overline{D}$, and the solution space is $u \in B_b(\mathbb{R}^d)$ with $u|_D \in W^{1,2}_{\text{loc}}(D) \cap C(\overline{D})$.
• Uniqueness is proven via a martingale argument and the dominated convergence theorem, showing that the only solution to the homogeneous problem is the zero solution under the given moment condition.