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I'm interesting to PDE, and I'm asking if the heat kernel with Dirichlet boundary conditions $p_{D}(t,x,y)$ on $[0,1]^{d}$, where $d\geq 1$ satifies

i) $\int_{D}p_{D}(t,x,y)dy =1$ or $c_{0} > 0$ ?

ii) The chapmann Kolmogorov equation $p_{D}(t+s,x,y) = \int_{D}p_{D}(t,x,z)p_{D}(s,y,z)dz$ ?

On $\mathbb{R}^{d}$ this is true, but in a bounded domain $D$ I don't know. Help me please.

Seirios
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user110160
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  • For $D = [0,1]$, I think you can calculate $p_D$ explicitly using Fourier series. So then you can just check the answers for i) and ii) directly. –  Nov 20 '13 at 11:00

1 Answers1

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For the purpose of deciding whether to believe such properties, it helps to think of $p_D(t,x,y)$ in probabilistic terms: as a function of $y$, this is the distribution at time $t$ of Brownian motion that starts at $x$ and gets killed when reaching the boundary (the latter is the Dirichet boundary condition). Then $\int_D p_D(t,x,y)\,dy$ is easy to understand: this is the probability that the Brownian particle is still in $D$ at time $t$. Due to our boundary conditions, this probability is less than $1$ for all $t>0$, and tends to $0$ as $t\to\infty$.

The second property is true. It expresses the lack of memory of Brownian motion: its distribution at time $t+s$ is what we get by starting with distribution at time $t$ and following it for time period $s$. This property is not affected by boundary conditions. To prove it, you can appeal to the corresponding property of the heat equation: $U(\cdot,t+s)$ is $V(\cdot ,s)$ where $V(\cdot,0)=U(\cdot,t)$.

Post No Bulls
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  • Thank you very much, in fact I hope estimate the integrale $\int_{0}^{t}\int_{D}p_{D}(s,x,y)dsdy$ and I hope obtain a lower bound of the form $c t^{\rho}$ where $c$ is a positive constant and $\rho \geq 0 $. I wish that you help me, I thank you very much. – user110160 Nov 28 '13 at 19:27
  • @user110160 This is a new question, and should be asked as such. – Post No Bulls Nov 28 '13 at 19:47