Question 1: System with replacement

The lifetime L of a system has an exponential distribution with parameter λ. After a failure, the system is replaced by a new one. A replacement takes a random time Z which is exponentially distributed with parameter µ. All life- and replacement times are assumed to be independent. Define the stochastic process by {Xt : t ≥ 0} with Xt = 1 is the system is operating and Xt = 0 if the system is being replaced.

a) Explain why {Xt : t ≥ 0} is a continuous time Markov chain.

b) Denote with p₁(t) = P (X_t = 1) and p0 = P (X_t = 0). Show that the stochastic process is goverend by the master equations

p´₀(t) = -µp₀(t) + λp₁(t)

p´₁(t) = µp₀(t) - λp₁(t)

c) Find the exact solution p₀(t), p₁(t) to the above equations for the intitial condition p₁(0) = 1.

d) Find the long-time limit p₁^∞ = lim_t→∞ from the above solution. Compare p₁^∞ to the stationary probability measure Π₁.

Question 2: A dying population

Consider the stochastic process {N (t) : t ≥ 0}, where N (t) denotes the size of the population of a certain species at time t. Initially, at time t = 0, the population has M members. For some unknown reasons, no offsprings are born for t ≥ 0. Each member of the population has the same probability γh to die in a small time interval h.

a) What is the state space S of this process?

b) Explain why p_j(t + h) = p_j(t) - jγhp_j(t) + (j + 1)γhpj+1(t) + o(h) for pj (t) = P (N (t) = j) and derive the master equation for p?_j(t).

c) For M = 3, give the matrix of transition rates wij of this process.

d) Find the stationary probability measure Π of this process for general M .

e) Define the time-dependent mean size of the population by µ₁(t) = Σ_n=0^M np_n(t). Show that µ₁(t) = Me^-γt.

f)Check that the long-time limit µ₁(t → ∞) agrees with µ₁ calculated with the invariant probability Π.

Question 3: Swimming algae

The trajectories of swimming microorganisms like algae show random reorientations that in the case of phototaxis can be described by the Fokker-Planck equation

∂/∂t.p(θ, t) = -∂/∂θ [v(θ)p(θ, t)] +D/2.∂²/∂θ²p(θ, t)

for the periodic probability density p(θ, t) with -Π ≤ θ ≤ Π and p(-Π, t) = p(Π, t) with v(θ) = -uθ for -Π < θ < Π and v(θ) = 0 for θ = ±Π.

a) Under which conditions does the Fokker-Planck equation conserve the norm of p?

b)Find the stationary probability density peq(θ) under the conditions found in a).

c)Assume that the algae swim with constant absolute magnitude of velocity v₀ in a two-dimensional plane, V_x = v₀ sin θ, V_y = v₀ cos θ with v₀ > 0 a constant. Define functions K_n(t) = ∫_-Π^Π θⁿsinθ p(θ, t)dθ and L_n(t) = ∫_-Π^Π θⁿcosθ p(θ, t)dθ for n = 0, 1, 2, . . .. Find the time evolution equation for the average velocities E[V_x], E[V_y] in terms of K_n and L_n.