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A nice question, with probably many possible answers. I'll give it a shot. I think three phenomena should be noted. i) The cumulant function is the Laplace transform of the probability distribution. U …

Yes, the results you quote are general statements on mixing and ergodicity, which can be translated to stochastic process as follows. In the source you mention (and many other sources), mixing is def …

This question is related to the topic of stochastic filtering theory. See e.g. the following monographs * Bucy, Joseph - Filtering for stochastic processes with applications to guidance * Bain, Crisan …

1) Explicit expressions for transition densities In the case of linear systems with additive noise of the form $d X(t) = (A X_t + b) \, d t + \sigma \, d B_t$ it is possible to obtain an explicit …

As formulated, there are some difficulties for the local Lipschitz case. For quadratic $f(x)=x^2$, consider the corresponding ODE for $x$ (i.e. the SDE with $\sigma = 0$). Then there exists a locally …

Let $0 < c < 1$. Consider the Markov chain $(X_i)$ on $\{0, 1, \dots, n\}$, with transition probabilities $$ P(k,k+1) = \left(1 - \tfrac {k}{n} \right)(1-c), \quad k = 0, \dots, n-1, $$ $$ P(k,k-1) …

The key thing is that your diffusion is Feller under the stated conditions, so $x \mapsto \mathbf E_x f(X_t)$ is continuous. Therefore $x \mapsto \int_0^{\infty} \mathbf E_x f(X_t) e^{-rt} \ d t$ is c …

Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \ …

I don't know if continuity of the density functions gives you anything. For example, think about a pure jump process with smooth density function of the jump distribution. You may be familiar with Ko …

Perhaps more a question for math.stackexchange.com. Unless your probability space $(\Omega, \mathcal F)$ is trivial (i.e. $\mathcal F = \{ \emptyset, \Omega \})$, the sets $\mathbf P$ and $\mathbf Q …

You are looking for the Lévy symbol of a 2-dimensional process $(N,D)$, so your symbol will be a function on $\mathbb R^2$. It will be $\phi_{N,D}(u_1, u_2) = \phi_N(u_1) + \phi_D(u_2)$, by the follow …

It would help if you know the marginal of at least one variable $x_i$. Suppose this is the case, i.e. without loss of generality suppose that you know $p(x_1=0)$. Then $p(x_1=i, x_2=j) = \left\{ \beg …

flawed, see Martin Hairer's comment below. Step I) Perform substitution $u_t(x) = \exp(\psi_t(x))$. The SPDE for $\psi$ becomes, after dividing by $u_t(x)$, \begin{equation} \frac{\partial}{\partial …