A representation of a heat equation with initial value

Ito’s formula provides probabilistic representation of PDE with Cauchy terminal data backward in time. By simple change of variable in time, it also provides a useful probabilistic representation of PDE with Cauchy initial data, which recovers Duhamel’s formula.

We will consider the following PDE with initial data:

$\displaystyle \partial_{t} u = \mathcal L u + \ell, \ \forall (x,t) \in \mathbb R^{n} \times (0, \infty), \hbox{ with } u(x,0) = g(x), \forall x\in \mathbb R^{n}, \ \ \ \ \ (1)$

where ${\mathcal L}$ is a time-homogeneous Feller-Dynkin operator. As an example, if ${\mathcal L = \Delta}$ is given, then the above PDE is inhomogeneous heat equation.

We fix ${T>0}$ . Suppose ${u}$ is smooth bounded solution of (1), then the function ${v(x,t) = u(x, T-t)}$ satisfies

$\displaystyle \left\{ \begin{array} {ll} \partial_{t} v (x,t) + \mathcal L v (x,t) + \ell (x, T-t ) = 0, &\ \forall (x,t) \in \mathbb R^{n} \times (0, T), \\ v(x,T) = g(x), & \ \forall x\in \mathbb R^{n}, \end{array} \right. \ \ \ \ \ (2)$

Let ${X}$ be the Markov process with its generator ${\mathcal L}$ , denoted by ${X\sim L}$ . It is straightforward to write ${v}$ in the form of

$\displaystyle v(x,t) = \int_{t}^{T} \mathbb E^{x,t} [\ell(X_{s}, T-s)] ds + \mathbb E^{x,t} [g(X_{T})].$

Since ${\mathcal L}$ is time-homogeneous, there exists time-homogeneous Feller-Dynkin semigroup ${P}$ satisfying ${P_{t} f(x) = \mathbb E^{x,s}[ f(X_{s+t})]}$ for smooth test functions ${f}$ . We can rewrite ${v}$ in terms of ${P}$ as

$\displaystyle v(x,t) = \int_{0}^{T-t} P_{s} \ell(T - t -s, \cdot) (x) ds + P_{T-t} g(x).$

Finally, we could replace ${t}$ in the above by ${T-t}$ and obtain the probabilistic representation of ${u}$ .

Theorem 1 If ${u}$ is a smooth bounded solution of (1), and ${\ell}$ and ${g}$ are all bounded measurable functions. Then

$\displaystyle u(x, t) = \int_{0}^{t} P_{s} \ell(t -s, \cdot) (x) ds + P_{t} g(x).$

Omega

Deterministic Randomness

A representation of a heat equation with initial value

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