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Compute Moments Symbolically

    
Compute moments of the Jacobi diffusion process symbolically:
Wolfram Language code: j𝒫[α_, x0_] = ItoProcess[ⅆx[t] == (α - x[t])ⅆt + Sqrt[x[t](1 - x[t])]ⅆw[t], x[t], {x, x0}, t, wWienerProcess[]];
Wolfram Language code: moments = Table[Moment[j𝒫[a, x0][t], r], {r, 1, 4}];
Wolfram Language code: Column[moments, Background -> {{LightOrange, LightGreen}}, Dividers -> All, Spacings -> {1, 2}]
Average over the initial value, sampled from BetaDistribution. Time independence of the expected value indicates that the beta distribution is the stationary process of the Jacobi diffusion process :
Wolfram Language code: Expectation[moments, x0BetaDistribution[2a, 2(1 - a)]]
Compute the three-point correlation function:
Wolfram Language code: Expectation[ϕ[Subscript[t, 1]]ϕ[Subscript[t, 2]]ϕ[Subscript[t, 3]], ϕj𝒫[a, x0]]
Wolfram Language code: j𝒫[α_, x0_] = ItoProcess[ⅆx[t] == (α - x[t])ⅆt + Sqrt[x[t](1 - x[t])]ⅆw[t], x[t], {x, x0}, t, wWienerProcess[]]; Framed[Column[Table[Moment[j𝒫[a, x0][t], r], {r, 1, 4}], Background -> {{LightOrange, LightGreen}}, Dividers -> All, Spacings -> {1, 2}, FrameStyle -> Directive[Thick, White]], FrameStyle -> None, RoundingRadius -> 10, Background -> Hue[.15, .15, .95]]