# *Full proof* references for Markov generators with various boundary conditions

(Note: I've migrated this question from math.stackexchange, as the lack of answers there made me believe it was perhaps too advanced for that forum.)

Consider the one-dimensional heat equation $$\partial_t u(t,x)=\frac12\Delta u(t,x),\qquad t\geq0,~x\in I$$ on some interval $I=(-a,a)$, with some initial condition $u(0,x)=f(x)$ and boundary condition on the interval $I$.

Suppose that we define the following processes

1. $(K_t)_{t\geq0}$, a Brownian motion killed on the boundary of $I$;
2. $(R_t)_{t\geq0}$, a Brownian motion reflected on the boundary of $I$;
3. $(P_t)_{t\geq0}$, a periodic Brownian motion in the interval $I$.

I know from a mix of intuition or folklore that we have the following probabilistic representations the solution $u(t,x)$ for different choices of boundary conditions on $I$: $$u(t,x) =\begin{cases} \mathbb{E}\big[f(x+K_t)\big]&\text{for Dirichlet boundary};\\ \mathbb{E}\big[f(x+R_t)\big]&\text{for Neumann boundary};\\ \mathbb{E}\big[f(x+P_t)\big]&\text{for Periodic boundary}. \end{cases}$$

Question. Are there any references out there with full proofs of these facts?

I've tried extensive researching on this subject to no avail.

• @MateuszKwaśnicki: Concerning your questions: (i) The wording explicit computation is probably a bit misleading; I think the OP wants to know why certain stochastic processes lead to certain boundary conditions for the heat equation. (ii) As far as I can see, Section 5.5 in Dynkin's book does not contain any information about boundary conditions of the differential operator (I hope I haven't overlooked anything). [to be continued.] Commented Jul 27, 2018 at 21:42
• @MateuszKwaśnicki: [continuation] (iii) A killed Brownian motion on an interval $I$ is defined as follows: add a cemetry $c$ to the interval $I$, take a usual Wiener process $(X_t)$ and set it to the value $c$ as soon as it leaves $I$. Intuitively, a reflected Brownian motion is defined similarly, but instead of adding a cemetry, you change the direction of the Brownian motion as soon as it hits the boundary (here, I don't know the technical details, though). [to be continued] Commented Jul 27, 2018 at 21:43
• @MateuszKwaśnicki: [continuation] A periodic Brownian motion on, say, $I = [0,1]$ is simply a usual Brownian motion $(X_t)$ on $\mathbb{R}$ where each value $X_t$ is taken modulo $1$. (iv) In the context of the question the appropriate function space for the solution of the heat equation is certainly $L^1(I)$. (v) In the context of the question I suggest to solve the heat equation (with any of the given boundary conditions) on $L^1(I)$ by means of operator semigroup theory. Commented Jul 27, 2018 at 21:44
• @JochenGlueck: Thanks for your answers. Now I see that the question is "why killing/reflection/periodization correspond to appropriate boundary conditions" rather than "why these stochastic processes correspond to these PDEs". I'll post some references later on. Commented Jul 28, 2018 at 8:00
• @user78270: I gave all references that I am aware of in my answer. However, I do not know the literature on BM well. I suppose some applied textbooks may discuss the BM in an interval with various boundary conditions in more detail. Commented Jul 28, 2018 at 19:18

As I understand, we take for granted that the solution of the heat equation is given in terms of the appropriate heat kernel. This heat kernel is given in terms of the Gauss–Weierstrass kernel $k_t(x) = (2 \pi t)^{-1/2} \exp(-x^2 / (2 t))$; namely, we have $$u(t, x) = \int_{-a}^a p(t, x, y) f(y) dy,$$ where $$p(t, x, y) = \sum_{n \in 2 \mathbb{Z}} k_t(y - x + 2 n a) - \sum_{n \in 2 \mathbb{Z} + 1} k_t(y + x + 2 n a)$$ for the Dirichlet boundary condition, $$p(t, x, y) = \sum_{n \in 2 \mathbb{Z}} k_t(y - x + 2 n a) + \sum_{n \in 2 \mathbb{Z} + 1} k_t(y + x + 2 n a)$$ for the Neumann boundary condition, and $$p(t, x, y) = \sum_{n \in \mathbb{Z}} k_t(y - x + 2 n a)$$ for the periodic boundary condition. The question is: where these kernels are shown to be transition probabilities of the killed, reflected and periodized Brownian motion in $(-a, a)$.

Killed Brownian motion in an interval is discussed, for example, by Karatzas and Shreve, Brownian Motion and Stochastic Calculus. In Section 4.3 killed process in half-line is studied in detail, and killing in an interval is left as an exercise (with an extensive hint); see p. 267.

For extensions to arbitrary domains, see, for example, Theorem 2.13 in Chung and Zhao, From Brownian Motion to Schrodinger's Equation.

The periodic Brownian motion is typically defined as $f(B_t)$, where $B_t$ is the usual Brownian motion on $\mathbb{R}$ and $f(x) = x - 2 a \lfloor(x - a) / (2 a)\rfloor$ is a piecewise linear function. In this case the expression for the heat kernel is a straightforward consequence of the definition. It is quite likely written explicitly in some reference, but I am not aware of any.

The reflected Brownian motion can be defined in a number of equivalent ways:

1. The easiest way is to "fold" the periodic Brownian motion. In other words, define the reflected Brownian motion as $f(B_t)$, where $B_t$ is the usual Brownian motion and $f(x) = \tfrac{2 a}{\pi} \arcsin(\sin(\tfrac{\pi x}{2 a}))$. Then, again, the expression for the heat kernel is a direct consequence of the definition.

2. Another way is to define the reflected Brownian motion as a solution to a Skorokhod problem. This is equivalent to definition 1 as a consequence of Tanaka's formula. Any reference on Tanaka's formula will discuss the case of the half-line. General convex domains in $\mathbb{R}^n$ and general diffusions were already considered by Tanaka. I am not aware of a reference that would discuss the case of the Brownian motion in an interval, but I am not an expert.

3. Feller's work on diffusion processes in an interval defined the reflected diffusion by an appropriate limiting procedure, and linked it directly to an appropriate differential operator. Again, this is not "explicit" and likely too general.

4. Finally, one can define the reflected Brownian motion as a Markov process linked to the usual energy form $\int_D |\nabla u(x)|^2 dx$ for $u$ in the Sobolev space $W^{1,2}(D)$. In this case the process corresponds to the Laplace operator with Neumann boundary conditions by the very definition.

Finally, let me mention that Pitman and Yor provide an extensive list of references. All of them seem to deal with rather general questions, but perhaps some discuss the interval as an example.