Skip to main content
Log in

The Difference Between a Discrete and Continuous Harmonic Measure

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius h. For a simply connected domain D in the plane, let \(\omega _h(0,\cdot ;D)\) be the discrete harmonic measure at \(0\in D\) associated with this random walk, and \(\omega (0,\cdot ;D)\) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function \(\sigma _D(z)\) on \(\partial D\) such that for functions g which are in \(C^{2+\alpha }(\partial D)\) for some \(\alpha >0\) we have

$$\begin{aligned} \lim _{h\downarrow 0} \frac{\int _{\partial D} g(\xi ) \omega _h(0,|\mathrm{d}\xi |;D) -\int _{\partial D} g(\xi )\omega (0,|\mathrm{d}\xi |;D)}{h} = \int _{\partial D}g(z) \sigma _D(z) |\mathrm{d}z|. \end{aligned}$$

We give an explicit formula for \(\sigma _D\) in terms of the conformal map from D to the unit disk. The proof relies on some fine approximations of the potential kernel and Green’s function of the random walk by their continuous counterparts, which may be of independent interest.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beneš, C., Johansson Viklund, F., Kozdron, M.: On the rate of convergence of loop-erased random walk to SLE\(_2\). Commun. Math. Phys. 318, 307–354 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bramble, J.H., Hubbard, B.E., Zlamal, M.: Discrete analogues of the Dirichlet problem with isolated singularities. SIAM J. Numer. Anal. 5, 1–25 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chelkak, D., Smirnov, S.: Discrete complex analysis on isoradial graphs. Adv. Math. 228, 1590–1630 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100, 32–74 (1928)

    Article  MATH  MathSciNet  Google Scholar 

  5. Forsythe, G., Wasow, W.: Finite-Difference Methods for Partial Differential Equations. Wiley, New York (1960). Applied Mathematics Series

    MATH  Google Scholar 

  6. Fukai, Y., Uchiyama, K.: Potential kernel for two-dimensional random walk. Ann. Probab. 24, 1972–1992 (1996)

    MATH  MathSciNet  Google Scholar 

  7. Garnett, J., Marshall, D.: Harmonic Measure, vol. 2. Cambridge University Press, New York (2005). New Mathematical Monographs Series

    Book  MATH  Google Scholar 

  8. Gerschgorin, S.: Fehlerabschätzung für das Differenzenverfahren zur Lösung partieller Differentialgleichungen. Z. Angew. Math. Mech. 10, 373–382 (1930)

    Article  MATH  Google Scholar 

  9. Kesten, H.: Relations between solutions to a discrete and continuous Dirichlet problem. In Random Walks, Brownian Motion, and Interacting Particle Systems, pp. 309-321. Progr. Probab., 28. Birkhäuser Boston, Boston, MA (1991)

  10. Kozdron, M., Lawler, G.: Estimates of random walk exit probabilities and application to loop-erased random walk. Electon. J. Probab. 44, 1442–1467 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kozma, G., Schreiber, E.: An asymptotic expansion for the discrete harmonic potential. Electon. J. Probab. 9, 1–17 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lawler, G.: Conformally Invariant Processes in the Plane, vol. 114. American Mathematical Society, Providence (2005). Mathematical Surveys and Monographs

    MATH  Google Scholar 

  13. Möters, P., Peres, Y.: Brownian Motion. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  14. Spitzer, F.: Principles of Random Walk. Springer, New York (1976)

    Book  MATH  Google Scholar 

  15. Wasow, W.: The accuracy of difference approximations to plane Dirichlet problems with piecewise analytic boundary values. Quart. Appl. Math 15, 53–63 (1957)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous referee for many valuable comments and suggestions. The research of T. Kennedy was supported in part by NSF Grant DMS-1500850.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jianping Jiang.

Appendix

Appendix

In this appendix, we prove the asymptotics for the potential kernel described in Lemma 4. We follow the methods introduced in Section 12 of [14] and [6]. Let \(\phi (\theta )\) be the characteristic function of the continuous-state random walk with \(h=1\), i.e.,

$$\begin{aligned} \phi (\theta )=E e^{i X\cdot \theta } \end{aligned}$$

where \(\theta =(\theta _1,\theta _2)\) and \(X=(X^{(1)},X^{(2)})\) is uniformly distributed in the disk of radius 1.

Lemma 10

$$\begin{aligned}&\phi (\theta )=1-\frac{|\theta |^2}{8}+\frac{|\theta |^4}{192} +O(|\theta |^6), \theta \rightarrow 0\\&\quad |\phi (\theta )|\le \frac{4}{\pi }\min \{|\theta _1|^{-1}, |\theta _2|^{-1}\}, \theta \rightarrow \infty . \end{aligned}$$

Proof

The first estimate in the lemma follows by Taylor expansion, while the second follows since

$$\begin{aligned} |\phi (\theta )|= & {} |\frac{1}{\pi }\int _{-1}^{1}\int _{-\sqrt{1 -x_1^2}}^{\sqrt{1-x_1^2}}\cos (x_1\theta _1)\cos (x_2\theta _2) \mathrm{d}x_1\mathrm{d}x_2|\le \frac{4}{\pi |\theta _2|} \end{aligned}$$

and the symmetry of \(\theta _1\) and \(\theta _2\). \(\square \)

The following lemma says our potential kernel is well defined.

Lemma 11

$$\begin{aligned} a(x)= & {} \lim _{n\rightarrow \infty }\sum _{k=1}^n[p(k,0,0)-p(k,0,x)]\\= & {} \sum _{k=1}^2[p(k,0,0)-p(k,0,x)]+\frac{1}{(2\pi )^2}\int _{\mathbb {R}^2} \frac{1-e^{i\theta \cdot x}}{1-\phi (\theta )}\phi ^3(\theta )\mathrm{d}\theta \end{aligned}$$

Proof

By applying the continuous inversion formula, the proof is similar to the proof of P1 in Section 12 of [14] if one can show \(\frac{1-e^{i\theta \cdot x}}{1-\phi (\theta )}\phi ^3(\theta )\in L^1(\mathbb {R}^2)\). The latter is true because of Lemma 10. \(\square \)

Let \(Q(\theta )=E(X\cdot \theta )^2=\frac{|\theta |^2}{4}\) and \(\psi (\theta )=1/(1-\phi (\theta ))-2/Q(\theta )\). Then Lemma 10 implies

$$\begin{aligned} \psi (\theta )=1/3+O(|\theta |^2) \text{ as } \theta \rightarrow 0; |\psi (\theta )|<2 \text{ as } \theta \rightarrow \infty . \end{aligned}$$
(24)

Now we have all ingredients to prove Lemma 4.

Proof of Lemma 4

By Lemma 11 and the evenness of \(\phi \), we see that

$$\begin{aligned} a(x)= & {} \sum _{k=1}^2[p(k,0,0)-p(k,0,x)]+\frac{2}{\pi ^2} \int _{\mathbb {R}^2}\frac{1-\cos (x\cdot \theta )}{|\theta |^2} \phi ^3(\theta )\mathrm{d}\theta \nonumber \\&+\,\frac{1}{(2\pi )^2}\int _{\mathbb {R}^2}(1-e^{i\theta \cdot x})\psi (\theta )\phi ^3(\theta )\mathrm{d}\theta . \end{aligned}$$
(25)

By the estimates in Lemma 10 and (24), and the Riemann–Lebesgue lemma,

$$\begin{aligned} \frac{1}{(2\pi )^2}\int _{\mathbb {R}^2}(1-e^{i\theta \cdot x})\psi (\theta )\phi ^3(\theta )\mathrm{d}\theta \rightarrow \frac{1}{(2\pi )^2}\int _{\mathbb {R}^2}\psi (\theta )\phi ^3(\theta )\mathrm{d}\theta \text{ as } |x|\rightarrow \infty \end{aligned}$$

which is a constant contributing to \(C_0\) in the lemma.

This gives the first o(1) term

$$\begin{aligned} -\frac{1}{(2\pi )^2}\int _{\mathbb {R}^2}e^{i\theta \cdot x} \psi (\theta )\phi ^3(\theta )\mathrm{d}\theta =-\frac{1}{(2\pi )^2} \int _{\mathbb {R}^2}\cos (x\cdot \theta )\psi (\theta ) \phi ^3(\theta )\mathrm{d}\theta . \end{aligned}$$
(26)

Let \(B:=B(0,\pi ):=\{z:|z|<\pi \}\) and \(B^c=\mathbb {R}^2{\setminus }B\). Then the first integral together with the attached multiplicative term in (25) can be written as the sum of the following two integrals

$$\begin{aligned} I_1(x):= & {} \frac{2}{\pi ^2}\int _B\frac{1-\cos (x\cdot \theta )}{|\theta |^2}\phi ^3(\theta )\mathrm{d}\theta \end{aligned}$$
(27)
$$\begin{aligned} I_2(x):= & {} \frac{2}{\pi ^2}\int _{B^c}\frac{1-\cos (x\cdot \theta )}{|\theta |^2}\phi ^3(\theta )\mathrm{d}\theta . \end{aligned}$$
(28)

By the estimate in Lemma 10 and the Riemann–Lebesgue lemma we have

$$\begin{aligned} I_2(x)\rightarrow \frac{2}{\pi ^2}\int _{B^c}\frac{\phi ^3(\theta )}{|\theta |^2} \text{ as } x\rightarrow \infty , \end{aligned}$$

which leaves the second o(1) term

$$\begin{aligned} -\frac{2}{\pi ^2}\int _{B^c}\frac{\cos (x\cdot \theta )}{|\theta ^2|}\phi ^3(\theta )\mathrm{d}\theta . \end{aligned}$$
(29)

We rewrite \(I_1(x)\) as follows

$$\begin{aligned} I_1(x)= & {} \frac{2}{\pi ^2}\int _B\frac{1-\cos (x\cdot \theta )}{|\theta |^2}\mathrm{d}\theta +\frac{2}{\pi ^2}\int _B\frac{\phi ^3(\theta )-1}{|\theta |^2}\mathrm{d}\theta \nonumber \\&\quad +\,\frac{2}{\pi ^2}\int _B\frac{\cos (x\cdot \theta )}{|\theta |^2}(1-\phi ^3(\theta ))\mathrm{d}\theta . \end{aligned}$$
(30)

By Lemma 10, the second integral in (30) is a constant contributing to \(C_0\) in the lemma, and by the Riemann–Lebesgue lemma the last integral in (30) gives the third o(1) term

$$\begin{aligned} \frac{2}{\pi ^2}\int _B\frac{\cos (x\cdot \theta )}{|\theta |^2}(1 -\phi ^3(\theta ))\mathrm{d}\theta , \end{aligned}$$
(31)

and the first integral together with the attached multiplicative term in (30) is equal to (using the proof of P3 in Section 12 of [14])

$$\begin{aligned} \frac{8}{\pi ^2}\int _0^{\pi /2}\left[ \gamma +\ln \pi +\ln |x| +\ln (\sin \alpha )+\int _{\pi |x|\sin \alpha }^{\infty }\frac{\cos u}{u}\mathrm{d}u\right] \mathrm{d}\alpha \end{aligned}$$
(32)

where \(\gamma \) is Euler’s constant.

It is clear that \(\gamma +\ln \pi +\ln |x|+\ln (\sin \alpha )\) in (32) as a function of \(\alpha \) is integrable from 0 to \(\pi /2\), so the fourth o(1) term is

$$\begin{aligned} \frac{8}{\pi ^2}\int _0^{\pi /2}\int _{\pi |x|\sin \alpha }^{\infty }\frac{\cos u}{u}\mathrm{d}u\mathrm{d}\alpha =\frac{2}{\pi ^2}\int _{B^c}\frac{\cos (x\cdot \theta )}{|\theta |^2}\mathrm{d}\theta . \end{aligned}$$
(33)

where the equality follows by reversing the procedure which led to (32).

Adding the four o(1) terms, i.e., (26) + (29) + (31) + (33), we get

$$\begin{aligned}&-\,\frac{1}{(2\pi )^2}\int _{{\mathbb {R}}^2}\cos (x\cdot \theta ) \psi (\theta )\phi ^3(\theta )\mathrm{d}\theta +\frac{2}{\pi ^2} \int _{{\mathbb {R}}^2}\frac{\cos (x\cdot \theta )}{|\theta |^2}(1 -\phi ^3(\theta ))\mathrm{d}\theta \nonumber \\&\quad =\frac{1}{4\pi ^2}\int _{{\mathbb {R}}^2}\cos (x\cdot \theta ) \left[ \frac{8}{|\theta |^2}-\frac{\phi ^3(\theta )}{1 -\phi (\theta )}\right] \mathrm{d}\theta . \end{aligned}$$
(34)

Noting that \(\cos (x\cdot \theta )=\nabla \cdot {\mathbf {b}}(\theta )\) where \(\mathbf {b}(\theta )=\sin (x\cdot \theta )(x_1/|x|^2,x_2/|x|^2)\), the divergence theorem gives

$$\begin{aligned}&\int _{\mathbb {R}^2}\cos (x\cdot \theta )\left[ \frac{8}{|\theta |^2} -\frac{\phi ^3(\theta )}{1-\phi (\theta )}\right] \mathrm{d}\theta \nonumber \\= & {} \lim _{N\rightarrow \infty }\int _{B(0,N)}\left[ \frac{8}{|\theta |^2} -\frac{\phi ^3(\theta )}{1-\phi (\theta )}\right] \nabla \cdot \mathbf {b}(\theta )\mathrm{d}\theta \nonumber \\= & {} \lim _{N\rightarrow \infty }|x|^{-1}\left( -\int _{B(0, N)}\left( \frac{x_1}{|x|},\frac{x_2}{|x|}\right) \cdot \nabla \left[ \frac{8}{|\theta |^2} -\frac{\phi ^3(\theta )}{1-\phi (\theta )}\right] \sin (x\cdot \theta ) \mathrm{d}\theta \right) .\nonumber \\ \end{aligned}$$
(35)

We can apply the divergence theorem again to (35). As a result, we see that (34) has order \(O(|x|^{-2})\).

Therefore, the proof of Lemma 4 is complete if one can show \(\Delta _{h} a(x/h)=\frac{1}{\pi }I_{B(0,h)}(x)\). But the latter is easy to verify (note that \(a(x/h)=h^2a_h(x)\)). \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, J., Kennedy, T. The Difference Between a Discrete and Continuous Harmonic Measure. J Theor Probab 30, 1424–1444 (2017). https://doi.org/10.1007/s10959-016-0695-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-016-0695-3

Keywords

Mathematics Subject Classification (2010)

Navigation