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A Sisyphus Random Walk evolves as follows: With probability (r) you advance the position of the walker by +1. With probability (1-r) the walker resets at x0 = 0.

To simulate the probability of reset I use a Bernoulli Distribution:

t =  Prepend[RandomVariate[BernoulliDistribution[0.7], 9],0]

{0, 1, 1, 0, 1, 0, 1, 1, 1, 1}

According to this distribution the walk should evolve as follows:

srw = {0,1,2,0,1,0,1,2,3,4}

I am unsure what functions I should use to get the desired output.

We can iterate with FoldList:

data = {0, 1, 1, 0, 1, 0, 1, 1, 1, 1};



FoldList[#2 * (#1 + #2)&, data]

{0, 1, 2, 0, 1, 0, 1, 2, 3, 4}

Another approach is to accumulate consecutive runs of 1:

Join @@ Accumulate /@ Split[data]

{0, 1, 2, 0, 1, 0, 1, 2, 3, 4}

WARNING sorry, an error in this solution was just pointed out to me by another user. I am busy doing the necessary corrections.

A nice problem. Why not solve it analytically?

Let us use Latex to formulate the problem and write down the solution, and then move to Mathematica. Finally we discuss the results and generalizations.

Part 1: mathematical formulation of the problem

Let $w(t,k)$ be the probability that at time t(>=0) the walker is at position k.

Then let us derive the evolution equations

For $k>0$ we have the balance

$$w(t+1,k) = w(t,k) text{(old value)} - r w(t,k) text{(loss to k+1)}+ r w(t,k-1) text{(gain from k-1)}-(1-r)w(t,k) text{(loss to 0)}$$

this gives the equation

$$w(t+1,k) = r; w(t,k-1), ;;;;;tge0, kgt0tag{1}$$

For $k=0$ we have

$$w(t+1,0) = w(t,0)text{(old value)}- r w(t,0)text{(loss to 1)} + (1-r) left(w(t,1)+w(t,2)+...right)text{(gain from all others)}$$

This gives

$$w(t+1,0) = (1-r) sum_{k=0}^infty w(t,k)$$

which greatly simplifies to the simple equation

$$w(t+1,0) = (1-r)tag{2}$$

Indeed, since at any time $tge0$ the walker must be at one of the locations $kge 0$ with certainty we must have
$sum_{k=0}^infty w(t,k)=1$.

In order to finalize the formulation of the problem we need initial conditions.

We assume that the walker at $t=0$ is in location $k=nge0$, i.e.

$$w(0,k)=delta_{k,n}tag{3}$$

where $delta_{k,n}$ is the Kronecker symbol defined as unity if $k=n$ and $0$ otherwise.

Part 2: solution with Mathematica

First we calculate w[t,0].

Considering two cases: the walker starts at k=0 or not.

From (4) we have

Case 1: Walker starts at k=0

sol01 = RSolve[w[t + 1, 0] == (1 - r) (1 - w[t, 0]) && w[0, 0] == 1, w[t, 0], 

  t]



Out[11]= {{w[t, 0] -> (-1 - (-1 + r)^t + r)/(-2 + r)}}



Simplify[(-1 - (-1 + r)^t + r)/(-2 + r) == (1 - r)/(

   2 - r) (1 + (-1)^t (1 - r)^(t - 1)), t [Element] Integers]



Out[19]= True

Hence the probability in the origin if the walker starts there is

w01[t_, r_] = (1 - r)/(2 - r) (1 + (-1)^t (1 - r)^(t - 1));

In Latex

$$w_{01}(t,r) = frac{1-r}{2-r}left(1+(-1)^t (1-r)^{t-1}right)tag{6}$$

The first few values are for r=1/2

t01 = Table[w01[t, 1/2], {t, 0, 10}]



(* Out[41]= {1, 0, 1/2, 1/4, 3/8, 5/16, 11/32, 21/64, 43/128, 85/256, 171/512} *)



ListPlot[t01, 

 PlotLabel -> 

  "Sysiphos random walknProbability w(t,0) at origin as a function 

of timencase 1: walker starts at the origin", 

 AxesLabel -> {"t", "w(t,0)"}]

Notice that at t=1 the probability is zero.

The numerators are the so called Jacobsthal numbers (see https://oeis.org/A001045), the denominators are powers of 2.

Case 2: Walker does not start at k=0

sol00 = RSolve[w[t + 1, 0] == (1 - r) (1 - w[t, 0]) && w[0, 0] == 0, w[t, 0], 

  t]



(* Out[31]= {{w[t, 0] -> -(((-1 + (-1 + r)^t) (-1 + r))/(-2 + r))}} *)



Simplify[-(((-1 + (-1 + r)^t) (-1 + r))/(-2 + r)) == (1 - r)/(

   2 - r) (1 - (-1)^t (1 - r)^t), t [Element] Integers]



(* Out[35]= True *)

Hence the probability in the origin if the walker starts not there (but somewhere at k>0) is

w00[t_, r_] = (1 - r)/(2 - r) (1 - (-1)^t (1 - r)^t);

In Latex

$$w_{00}(t,r) = frac{1-r}{2-r}left(1-(-1)^t (1-r)^tright)tag{7}$$

The first few values for r=1/2 are

t00 = Table[w00[t, 1/2], {t, 0, 10}]



(* Out[40]= {0, 1/2, 1/4, 3/8, 5/16, 11/32, 21/64, 43/128, 85/256, 171/512, 341/1024} *)



ListPlot[t00, 

 PlotLabel -> 

  "Sysiphos random walknProbability w(t,0) at origin as a function 

of timencase 2: walker starts not at the origin", 

 AxesLabel -> {"t", "w(t,0)"}, PlotRange -> {-0.1, 1}]

Comparison

t01



(* Out[42]= {1, 0, 1/2, 1/4, 3/8, 5/16, 11/32, 21/64, 43/128, 85/256, 171/512} *)

And in general we have the following relation between the probabilities of the two cases:

w01[t, r] + w00[t, r]/(1 - r)==1 // Simplify



(* True * )

Or in Latex

$$w_{01}(t,r) + frac{w_{00}(t,r)}{1-r}=1tag{8}$$

Now we turn to positions k>0.

From (1) we have

sol = RSolve[w[t + 1, k] == r w[t, k - 1], w[t, k], {t, k}]



(* Out[43]= {{w[t, k] -> r^(-1 + t) C[1][k - t]}} *)

Don't worry about the strange looking expression of the constant C[1] with an argument. This is exactly what we need to apply the initial condition (5).

Defining the auxiliary function

vc[t_, k_] = w[t, k] /. sol[[1]] t_, k]



(* Out[41]= r^(-1 + t) C[1][k - t] *)

condition (5) reads

sol1 = Solve[vc[0, k] == KroneckerDelta[k, n], C[1][k]]



(* Out[43]= {{C[1][k] -> r KroneckerDelta[k, n]}} *)

Hence the probability for the walker to be at location k at time t if he started at location n>0 (with the basic probability r of jumping one step to the the right) is given by

w[t_, k_, r_, n_] = r^t KroneckerDelta[k - t, n]



(* Out[50]= r^t KroneckerDelta[n, k - t] *)

Example 1:

n=1, r=1/2, w(t,k) is only different from 0 if k=t+1.

The first few values of these non vanishing Terms are

With[{n = 1, r = 1/2}, Table[w[t, t + 1, r, n], {t, 0, 10}]]



(* Out[58]= {1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/512, 1/1024} *)

The complete probabilities form a two dimensional array, which might start like this

With[{n = 3, r = 1/2}, 

 tt = Table[w[t, k, r, n], {t, 0, 5}, {k, 1, 10}]]



(* Out[133]= {{0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1/2, 0, 0, 0, 0, 

  0, 0}, {0, 0, 0, 0, 1/4, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1/8, 0, 0, 

  0, 0}, {0, 0, 0, 0, 0, 0, 1/16, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1/

  32, 0, 0}} *)

Visualization

ListPlot3D[tt, 

 PlotLabel -> 

  "Sysiphos random walk for r=1/2 starting at k=3nProbability w(t,k) 

for k>0 as a function of time", PlotRange -> {-0.1, 1}, 

 AxesLabel -> {"k", "t", "w(t,k)"}]

Part 3: discussion

Asymptotic behaviour for large times

The probabilities (6) and (7) at the origin approache a common value

$$w(ttoinfty,0) = frac{1-r}{2-r}tag{9}$$

As the walker must be initially either at the origin or not ("at home or out") the asymptotic value (9) holds for any initial state.

This means that the walker has asymptotically "gone out" with probability

$$w'(ttoinfty,0) = 1- w(ttoinfty,0) = frac{1}{2-r}$$

If the jump probability $r$ is small, it becomes equally probable that the walker is at home or out.

If $rto 1$ the walker is almost certainly out.

Next question: what is the asymptotic shape of the spatial profile? That is what is $w(ttoinfty,k)$ as a function of $k$?

My feeling tells me that is should be exponential ... like the pressure in the atmosphere.

Let's look at it qualitatively: if there were no jumping back, the walker would move away to $ktoinfty$ indefinitely. The jumping back leads to renewed filling from below of, in principle, any position.

What about $w(t,1)$? Let us try to calculate it exactly.