Green's Function for Parabolic PDEs¶

A Green's function is the response of a linear PDE to a unit impulse -- a delta source at a single point in space and time. For parabolic equations, the Green's function is the complete solution operator: once you know how the system responds to a point source, arbitrary solutions follow by superposition. This page treats the Green's function through the operator / PDE lens. The probabilistic counterpart -- the transition density of a diffusion -- is developed in Transition Density as Green's Function.

Intuition¶

Inject a unit of heat at point \(y\) at time \(s\). The Green's function \(G(t, x; s, y)\) describes how this heat spreads to point \(x\) at time \(t > s\): it starts as a delta at \(y\), diffuses outward, and (on unbounded domains) dissipates as \(t \to \infty\). Arbitrary initial data are reconstructed by superposing point sources -- the entire content of the superposition principle.

Definition¶

Consider the parabolic operator \(\mathcal{P}u = \partial_t u - \mathcal{L}u\), where \(\mathcal{L}\) is the second-order elliptic operator

The Green's function \(G(t, x; s, y)\) is the fundamental solution:

Here \(\mathcal{L}_x\) acts on the observation variable \(x\). \(G(t, x; s, y)\) is the PDE response at \((t, x)\) to a unit impulse at \((s, y)\).

Superposition Principle¶

This page owns the integral representation: once \(G\) is known, every linear problem reduces to an integral against \(G\).

Initial value problem. For \(\partial_t u = \mathcal{L}_x u\) with \(u(0, x) = f(x)\):

Source problem (Duhamel). For \(\partial_t u - \mathcal{L}_x u = h(t, x)\) with \(u(0, x) = 0\):

Combined. Linearity gives

The response to distributed data is the integral of responses to point sources.

Properties¶

1. Positivity. \(G(t, x; s, y) > 0\) for \(t > s\) -- heat flows everywhere from a positive source.
2. Normalization. \(\int G(t, x; s, y)\,dx = 1\) -- total heat (or probability) is conserved.

3. Semigroup (Chapman-Kolmogorov). For \(s < r < t\),

   \[ G(t, x; s, y) = \int G(t, x; r, z)\,G(r, z; s, y)\,dz \]

   Evolving from \(s\) to \(t\) is the same as evolving from \(s\) to \(r\) and then from \(r\) to \(t\). The full derivation and its probabilistic meaning live in Transition Density as Green's Function.

4. Symmetry. When \(\mathcal{L}\) is self-adjoint (e.g. pure diffusion), \(G(t, x; s, y) = G(t, y; s, x)\). With drift this fails, but \(G\) is related to the Green's function of the adjoint operator -- see the forward/backward discussion in Transition Density as Green's Function.

5. Smoothing. \(f \mapsto \int G(t, x; s, y)\,f(y)\,dy\) sends \(L^\infty\) into \(C^\infty\) for any \(t > s\): parabolic equations regularize instantly.

Construction for Variable Coefficients¶

Parametrix method (Levi). For general \(\mathcal{L} = \mu(x)\partial_x + \tfrac{1}{2}\sigma^2(x)\partial_{xx}\), freeze coefficients at the source \(y\) to get an explicit Gaussian approximation \(G_0\):

and iterate \(G = G_0 + G_1 + G_2 + \cdots\), where each \(G_n\) corrects the frozen-coefficient error. Under Hölder continuity, the series converges to a smooth \(G\), with short-time asymptotics

Canonical Examples (Results Only)¶

The full derivations and probabilistic interpretations are in Transition Density as Green's Function; the spectral/eigenfunction structure is in Spectral Decomposition.

Brownian motion with drift \(dX_t = \mu\,dt + \sigma\,dW_t\): coefficients are constant, so the parametrix \(G_0\) above is exact. See Transition Density as Green's Function for the formula and its probabilistic meaning.

Ornstein-Uhlenbeck. For \(dX_t = -\kappa X_t\,dt + \sigma\,dW_t\),

with \(\tau = t - s\). Mean reverts at rate \(\kappa\); variance saturates at \(\sigma^2/(2\kappa)\).

Summary¶

The Green's function is the fundamental building block for parabolic PDEs: it solves the point-source problem, and arbitrary solutions follow by superposition. In probability it is the transition density; in finance it is the state-price density -- both discussed in Transition Density as Green's Function.

See Also¶

• Transition Density as Green's Function -- probabilistic lens, equivalence proof, forward/backward table
• Spectral Decomposition -- eigenfunction expansion of \(G\)
• Free vs Bounded Domains -- how boundaries modify \(G\)
• Fundamental Solution -- heat kernel as the simplest \(G\)

Exercises¶

Exercise 1. The approximate-identity property \(\lim_{t\to 0^+}\int G(t,x;0,y)f(y)\,dy = f(x)\) for the heat kernel is established in § Fundamental Solution. State why this same property -- recast as \(G \to \delta(x-y)\) -- is exactly what makes the superposition formula \(u(t,x) = \int G\,f\,dy\) recover the initial data \(f\) at \(t = 0\).

Exercise 2. Using the superposition principle, compute \(u(t, x) = \int G(t,x;0,y)\,e^{-y^2}\,dy\) for the free-space heat kernel.

Exercise 3. State the superposition formula for the inhomogeneous problem \(\partial_t u - \tfrac{1}{2}u_{xx} = h(t,x)\), \(u(0,x) = 0\), and explain in one paragraph why it is called Duhamel's principle.

Exercise 4. The parametrix method approximates \(G\) by freezing coefficients at the source. Write down the frozen-coefficient Green's function \(G_0\) for \(\mathcal{L} = \mu(x)\partial_x + \tfrac{1}{2}\sigma^2(x)\partial_{xx}\) and explain why the approximation is good for small \(t - s\).

Exercise 5. Use the superposition formula to express the price of a European derivative with payoff \(g(S_T)\) under a diffusion model as an integral against \(G\). Identify the factor that makes \(G\) the state-price density (Arrow-Debreu price).

Exercise 6. For the constant-coefficient operator \(\mathcal{L} = \tfrac{1}{2}\partial_{xx} - \tfrac{1}{2}\partial_x\) (drift \(-1/2\)), derive \(G(t, x; 0, y)\) from the heat kernel by the change of variable \(u = e^{\alpha x + \beta t}\,v\) that eliminates the first-order term.