Is it possible to define this stochastic process?

Question

My background is in physics, so my apologies in advance for any naivety. I'm studying a physical problem whose mathematical description doesn't appear sensible. In its simplified version, the process is described as follows: There's a stochastic variable $\gamma_i(t)$, with $t$ as the time parameter and $i \in {0,1}$, as a discrete parameter. $\gamma_i(0)= \gamma_0$, with $\gamma_0$ a fixed value. After an interval of time $\Delta t$, that is at $0+\Delta t$, there is a probability $P_1=q \Delta t$ for $i=1$ and a probability $P_0= 1- q \Delta t$ for $i=0$. Furthermore, at time $0+\Delta t$, if $i=0$, $\gamma_0$ gets updated deterministically to $(\gamma_0 + \epsilon \Delta t)_{i=0}$, where $\epsilon$ is a factor that depends on the Hamiltonian of the system. On the other hand, if at time $0+\Delta t$, $i=1$ then $\gamma_0$ gets stochastically updated to $(\gamma_0 + \Delta W)_{i=1}$. Here $\Delta W$ is Gaussian distributed with mean $0$ and variance proportional to $\Delta t$.
Is there a name for this sort of stochastic process? I'm not sure if it's even mathematically sensible to define such a process. But that's what the physics seems to suggest. In the object $\gamma_i(t)$, if I look at the index $i$ and the symbol $\gamma$ separately then the stochastic process seems rather clear: A Poisson and a Wiener process respectively. But I don't know how to study $\gamma_i(t)$ as a whole.

Answer 1

If you want to take the continuous time limit here, it seems like you have to give up some of the interval dependence of the variables in question. The only one that makes sense for Hamiltonian evolution is that jump probability has to be independent of the time interval, $q\Delta t\to v$, which then in the continuous time limit it induces a drift proportional to the length of the interval.

I'll showcase what I mean by assuming $\epsilon$ is a number and not a Hamiltonian operator since there is no more information about the system. Start by defining the following sequences of variables

$$Y_N=\gamma(t+N\Delta t)-\gamma(t)~~~,~~ X_N=\gamma(t+N\Delta t)-\gamma(t+(N-1)\Delta t)$$

From the description of the problem it is reasonable to assume that the increments in the trajectory of the system $X_N$ are iid random variables with pdf

$$f_X(x)=(1-q\Delta t)\delta(x-\epsilon\Delta t)+q\Delta t\frac{e^{-x^2/2\sigma^2\Delta t}}{\sqrt{2\pi\sigma^2\Delta t}}$$ This in turn implies that the generating function of $Y_N$ factorizes:

$$\mathbb{E}(e^{iPY_N})=(\mathbb{E}(e^{iPX}))^N$$

Note $E(Y_N)=NE(X)=N\epsilon \Delta t$, and the generating function of the shifted variable takes on the form

$$\mathbb{E}(e^{iP(Y_N-N\epsilon\Delta t)})=[1+q\Delta t(e^{-(2\sigma^2 P^2+i\epsilon P)\Delta t}-1)]^N$$

To take the continuum limit, we want to take the limit $N\to\infty$ while keeping $N\Delta t=T$ constant. However we notice that the only way for the limit to converge to a value that is not $0$ or $1$, is that a variable absorbs some of the $\Delta t$ dependence. With $q\Delta t=v$, the limit evaluates to

$$\lim_{\substack{N\to \infty\\N\Delta t=T, q\Delta t=v}}\mathbb{E}(e^{iP(Y_N-N\epsilon\Delta t)})=e^{-vT(2\sigma^2 P^2+i\epsilon P)}$$

which in turn implies that the continuum variable is distributed as

$$Y(T;t):=\gamma(t+T)-\gamma(t)\sim \mathcal{N}(\epsilon (1-v)T, v\sigma^2 )$$

It is clear that the parameter $v$ induces a Wiener process with a drift, and hence interpolates between completely deterministic and a completely random motion.

Qualitatively, this means that rare random jolts to an otherwise deterministically evolving system get washed away in the continuum limit in this modelling instance of the problem described in the question. Hence, there is a need to make the motion "more random", at a constant proportion with deterministic; in other words, the system in the short time limit needs to be receiving a non-vanishing fraction $v$ of random jolts for the distribution to change from deterministic to non-deterministic.

Disclaimer: this conversation could be very different in a quantum Hilbert space, which is typically infinite-dimensional. However, I hope this illustrates the potential difficulties that come with taking a continuum limit appropriately.