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Potential Theory in Classical Probability Potential Theory in Classical Probability Nicolas Privault Department of Mathematics City University of Hong Kong 83 Tat Chee Avenue Kowloon Tong Hong Kong [email protected] Summary. These notes are an elementary introduction to classical potential the- ory and to its connection with probabilistic tools such as stochastic calculus and the Markov property. In particular we review the probabilistic interpretations of harmonicity, of the Dirichlet problem, and of the Poisson equation using Brownian motion and stochastic calculus. Key words: Potential theory, harmonic functions, Markov processes, stochas- tic calculus, partial differential equations. Mathematics Subject Classification: 31-01, 60J45. 1 Introduction The origins of potential theory can be traced to the physical problem of recon- structing a repartition of electric charges inside a planar or a spatial domain, given the measurement of the electricalfield created on the boundary of this domain. In mathematical analytic terms this amounts to representing the values of a functionh inside a domain given the data of the values ofh on the boundary of the domain. In the simplest case of a domain empty of electric charges, the problem can be formulated as that offinding a harmonic functionh on E (roughly speaking, a function with vanishing Laplacian, see 2.2 below), given its values prescribed by a functionf on the boundary∂E§, i.e. as the Dirichlet problem: Δh(y)=0, y E, ∈ h(y)=f(y), y ∂E. ∈ 2 Nicolas Privault Close connections between the notion of potential and the Markov property have been observed at early stages of the development of the theory, see e.g. [Doo84] and references therein. Thus a number of potential theoretic problems have a probabilistic interpretation or can be solved by probabilistic methods. These notes aim at gathering both analytic and probabilistic aspects of potential theory into a single document. We partly follow the point of view of Chung [Chu95] with complements on analytic potential theory coming from Helms [Hel69], some additions on stochastic calculus, and probabilistic appli- cations found in Bass [Bas98]. More precisely, we proceed as follow. In Section 2 we give a summary of classical analytic potential theory: Green kernels, Laplace and Poisson equations in particular, following mainly Brelot [Bre65], Chung [Chu95] and Helms [Hel69]. Section 3 introduces the Markovian setting of semigroups which will be the main framework for probabilistic interpretations. A sample of ref- erences used in this domain is Ethier and Kurtz [EK86], Kallenberg [Kal02], and also Chung [Chu95]. The probabilistic interpretation of potential theory also makes significant use of Brownian motion and stochastic calculus. They are summarized in Section 4, see Protter [Pro05], Ikeda and Watanabe [IW89], however our presentation of stochastic calculus is given in the framework of normal martingales due to their links with quantum stochastic calculus, cf. Biane [Bia93]. In Section 5 we present the probabilistic connection between po- tential theory and Markov processes, following Bass [Bas98], Dynkin [Dyn65], Kallenberg [Kal02], and Port and Stone [PS78]. Our description of the Martin boundary in discrete time follows that of Revuz [Rev75]. 2 Analytic Potential Theory 2.1 Electrostatic Interpretation LetE denote a closed region ofR n, more precisely a compact subset having a smooth boundary∂E with surface measureσ. Gauss’s law is the main tool for determining a repartition of electric charges insideE, given the values of the electricalfield created on∂E. It states that given a repartition of charges q(dx) theflux of the electricfieldU across the boundary∂E is proportional to the sum of electric charges enclosed inE. Namely we have q(dx) =� n(x),U(x) σ(dx), (2.1) 0 � � �E �∂E whereq(dx) is a signed measure representing the distribution of electric charges,� 0 > 0 is the electrical permittivity constant,U(x) denotes the elec- tricfield atx ∂E, andn(x) represents the outer (i.e. oriented towards the exterior ofE∈) unit vector orthogonal to the surface∂E. Potential Theory in Classical Probability 3 On the other hand the divergence theorem, which can be viewed as a particular case of the Stokes theorem, states that ifU:E R n is a 1 vectorfield we have → C divU(x)dx= n(x),U(x) σ(dx), (2.2) � � �E �∂E where the divergence divU is defined as n ∂U divU(x) = i (x). ∂x i=1 i � The divergence theorem (2.2) can be interpreted as a mathematical formula- tion of the Gauss law (2.1). Under this identification, divU(x) is proportional to the density of charges insideE, which leads to the Maxwell equation �0 divU(x)dx=q(dx), (2.3) whereq(dx) is the distribution of electric charge atx andU is viewed as the induced electricfield on the surface∂E. Whenq(dx) has the densityq(x) atx, i.e.q(dx) =q(x)dx, and thefield n U(x) derives from a potentialV:R R +, i.e. when → U(x) = V(x), x E, ∇ ∈ Maxwell’s equation (2.3) takes the form of the Poisson equation: � ΔV(x) =q(x), x E, (2.4) 0 ∈ where the LaplacianΔ = div is given by ∇ n ∂2V ΔV(x) = (x), x E. ∂x2 ∈ j=1 i � In particular, when the domainE is empty of electric charges, the potential V satisfies the Laplace equation ΔV(x) = 0x E. ∈ As mentioned in the introduction, a typical problem in classical potential theory is to recover the values of the potentialV(x) insideE from its values on the boundary∂E, given thatV(x) satisfies the Poisson equation (2.4). This can be achieved in particular by representingV(x),x E, as an integral with respect to the surface measure over the boundary∂E∈, or by direct solution of the Poisson equation forV(x). Consider for example the Newton potential kernel 4 Nicolas Privault q 1 n V(x) = , x R y , � s x y n 2 ∈ \{ } 0 n � − � − n created by a single chargeq aty R , wheres 2 = 2π,s 3 = 4π, and in general ∈ 2πn/2 s = , n 2, n Γ(n/2) ≥ is the surface of the unitn 1-dimensional sphere inR n. − The electricalfield created byV is q x y n U(x) := V(x) = − , x R y , ∇ � s x y n 1 ∈ \{ } 0 n � − � − cf. Figure 2.1. LettingB(y, r), resp.S(y, r), denote the open ball, resp. the sphere, of centery R n and radiusr> 0, we have ∈ ΔV(x)dx= n(x), V(x) σ(dx) � ∇ � �B(y,r) �S(y,r) = n(x),U(x) σ(dx) � � �S(y,r) q = , �0 whereσ denotes the surface measure onS(y, r). -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Fig. 2.1. Electricalfield generated by a point mass aty = 0. From this and the Poisson equation (2.4) we deduce that the repartition of electric charge is Potential Theory in Classical Probability 5 q(dx) =qδ y(dx) i.e. we recover the fact that the potentialV is generated by a single charge located aty. We also obtain a version of the Poisson equation (2.4) in distri- bution sense: 1 Δ =s δ (dx), x x y n 2 n y � − � − where the LaplacianΔ x is taken with respect to the variablex. On the other hand, takingE=B(0, r) B(0,ρ) we have∂E=S(0, r) S(0,ρ) and \ ∪ ΔV(x)dx= n(x), V(x) σ(dx) + n(x), V(x) σ(dx) � ∇ � � ∇ � �E �S(0,r) �S(0,ρ) = cs cs = 0, n − n hence 1 n Δx = 0, x R y . x y n 2 ∈ \{ } � − � − The electrical permittivity� 0 will be set equal to 1 in the sequel. 2.2 Harmonic Functions The notion of harmonic function will befirst introduced from the mean value property. Let x 1 σr (dy)= n 1 σ(dy) snr − denote the normalized surface measure onS(x, r), and recall that ∞ n 1 y f(x)dx=s n r − f(z)σ r (dz)dr. � �0 �S(y,r) Definition 2.1. A continuous real-valued function on an open subsetO ofR n is said to be harmonic, resp. superharmonic, inO if one has x f(x) = f(y)σ r (dy), �S(x,r) resp. f(x) f(y)σ x(dy), ≥ r �S(x,r) for allx O andr>0 such thatB(x, r) O. ∈ ⊂ Next we show the equivalence between the mean value property and the van- ishing of the Laplacian. 6 Nicolas Privault Proposition 2.2.A 2 functionf is harmonic, resp. superharmonic, on an C open subsetO ofR n if and only if it satisfies the Laplace equation Δf(x) = 0, x O, ∈ resp. the partial differential inequality Δf(x) 0, x O. ≤ ∈ Proof. In spherical coordinates, using the divergence formula and the identity d f(y+rx)σ 0(dx) = x, f(y+rx) σ 0(dx) dr 1 � ∇ � 1 �S(0,1) �S(0,1) r = Δf(y+rx)dx, s n �B(0,1) yields n 1 Δf(x)dx=r − Δf(y+rx)dx �B(y,r) �B(0,1) n 2 0 =s r − x, f(y+rx) σ (dx) n � ∇ � 1 �S(0,1) n 2 d 0 =s r − f(y+rx)σ (dx) n dr 1 �S(0,1) n 2 d y =s r − f(x)σ (dx). n dr r �S(y,r) Iff is harmonic, this shows that Δf(x)dx=0, �B(y,r) for ally E andr> 0 such thatB(y, r) O, henceΔf = 0 onO. Conversely, ifΔf =∈ 0 onO then ⊂ y f(x)σ r (dx) �S(y,r) is constant inr, hence y y f(y) = lim f(x)σ ρ (dx) = f(x)σ r (dx), r>0.
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