Stochastic Theories and Deterministic Differential Equations

Figure 5

The Liouville solution without friction as a function of space x and time t. 𝜌^𝐿_𝑋(𝑡₀,𝑥)=(1/2)[(2𝜋𝑎²)^−1/2𝑒^{−(𝑥−𝑥01)2/(2𝑎2)}+(2𝜋𝑎²)^−1/2𝑒^{−(𝑥−𝑥02)2/(2𝑎2)}],𝜌^𝐿_𝑌(𝑡₀,𝑦)=(2𝜋𝑏²)^−1/2𝑒^{−𝑦2/(2𝑏2)},𝑏=1/(2𝑎),𝑎²=1/2,𝑥₀₁=1.5,𝑥₀₂=−1.5, and 𝜀=0.