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).


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