# Yukawa Potential, Panharmonic Measure and Brownian Motion

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1.**

## 2. Yukawa Equation and Brownian Motion

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

- (i)
- Then$$u\left(x\right)={\mathbb{E}}^{x}\left[{e}^{-\frac{{\mu}^{2}}{2}{\tau}_{D}}f\left(W\left({\tau}_{D}\right)\right)\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}{\tau}_{D}<\infty \right]$$$$\left\{\begin{array}{ccccc}\hfill \Delta u& =& {\mu}^{2}u\hfill & on\hfill & D,\hfill \\ \hfill u& =& f\hfill & on\hfill & \partial D.\hfill \end{array}\right.$$
- (ii)
- Moreover, if u is bounded and D is small, then Equation (6) is the only solution to the Yukawa–Dirichlet problem.
- (iii)
- As a consequence, the panharmonic measure admits the representation$${H}_{\mu}^{x}(D;\mathrm{d}y)={\int}_{t=0}^{\infty}{e}^{-\frac{{\mu}^{2}}{2}t}\phantom{\rule{0.166667em}{0ex}}{h}^{x}(D;\mathrm{d}y,t)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t,$$

**Proof.**

**Remark**

**1.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Remark**

**2.**

## 3. Equivalence of Harmonic and Panharmonic Measures

**Theorem**

**3.**

**Remark**

**3.**

**Proof**

**of**

**Theorem**

**3**.

**Corollary**

**3.**

**Corollary**

**4.**

- (i)
- Let $D\subset {\mathbb{R}}^{2}$ be a simply connected planar domain bounded by a rectifiable curve. Then ${H}_{\mu}^{x}(D;\xb7)$ and ${\mathcal{H}}^{1}(D;\xb7)$ are equivalent for all $\mu \ge 0$ and $x\in D$.
- (ii)
- Let $D\subset {\mathbb{R}}^{2}$ be a simply connected planar domain. If $E\subset \partial D$ and ${\mathcal{H}}^{s}(D;E)=0$ for some $s<1$, then ${H}_{\mu}^{x}(D;E)=0$ for all $\mu \ge 0$ and $x\in D$. Moreover, ${H}_{\mu}^{x}(D;\xb7)$ and ${\mathcal{H}}^{t}(D;\xb7)$ are singular for all $\mu \ge 0$ and $x\in D$ if $t>1$.
- (iii)
- Let $D\subset {\mathbb{R}}^{n}$ is a bounded Lipschitz domain. Then ${H}_{\mu}^{x}(D;\xb7)$ and ${\mathcal{H}}^{n-1}(D;\xb7)$ are equivalent for all $\mu \ge 0$ and $x\in D$.

## 4. The Average Property for Panharmonic Measures and Functions

**Theorem**

**4.**

**Remark**

**4.**

**Proof**

**of**

**Theorem**

**4**.

**Remark**

**5.**

- (i)
- ${\psi}_{n}\left(\mu \right)$ is continuous in μ,
- (ii)
- ${\psi}_{n}\left(\mu \right)$ is strictly decreasing in μ,
- (iii)
- ${\psi}_{n}\left(\mu \right)\to 0$ as $\mu \to \infty $,
- (iv)
- ${\psi}_{n}\left(\mu \right)\to 1$ as $\mu \to 0$,
- (v)
- ${\psi}_{n}\left(\mu \right)$ is increasing in n.

**Corollary**

**5.**

**Proof.**

## 5. Discussion on Extensions and Simulation

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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Rasila, A.; Sottinen, T. Yukawa Potential, Panharmonic Measure and Brownian Motion. *Axioms* **2018**, *7*, 28.
https://doi.org/10.3390/axioms7020028

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Rasila A, Sottinen T. Yukawa Potential, Panharmonic Measure and Brownian Motion. *Axioms*. 2018; 7(2):28.
https://doi.org/10.3390/axioms7020028

**Chicago/Turabian Style**

Rasila, Antti, and Tommi Sottinen. 2018. "Yukawa Potential, Panharmonic Measure and Brownian Motion" *Axioms* 7, no. 2: 28.
https://doi.org/10.3390/axioms7020028