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There are several generalizations of the Brownian case results to general L'evy processes. For instance, under a classical log-concave assumption on $U$ we have a convergence to equilibrium in Wasse …

Under suitable assumptions, this stochastic convolution is well-defined if one undertands the integral in the rough path sense. For the idea of a situation where this applies, see for instance Theore …

The estimate you are interested in has already been studied. The tail bound $ \mathbb{P} \left( \sup_{t \in [0,1]} | J_n^f (t)| \ge K \right) \le C_1 \exp( -C_2 K^{2/n}) $ was proved by C. Borell. T …

The distributions and properties of the maximum of a Bessel processes and Bessel bridges are discussed in details in Jim Pitman: The law of of the maximum of a Bessel bridge

There is a beautiful connection between the area swept out by Brownian motion and the Dirichlet function: $L(s)=\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^s}$. Proposition: Let $A_t$ be the area s …

A good and general reference on stochastic integrals on manifolds is An Invitation to Second-Order Stochastic Differential Geometry See in particular Theorem 1, page 14. The survey also contains …

1) If $M_t$ is a martingale adapted to the Brownian filtration $\mathcal{F}_B$ with $M_0=0$, then $\langle B,M \rangle_t=0$ implies $M=0$. As you point out, this is a consequence of Ito's representati …

A nice quantitative and very general tool to study the speed to convergence of symmetric Markov processes to equilibrium is the Bakry-Emery criterion. More precisely, let $(X_t)_{t \ge 0}$ be a diffus …

These types of questions are treated in great generality in the book Rogers-Williams: Diffusions, Markov processes and martingales, Volume 1 One of the great results is due to Dynkin. Let $(X_t)_{t …

We can define fractional Brownian motion in Lie groups when $H>1/4$ by using the Lyons' rough paths theory Self-similarity and fractional Brownian motion on Lie groups

Some of Malliavin's ideas are well-explained in his own book: Stochastic Analysis. I think the main geometric idea behind his proof of Hormander's theorem is the idea of submersion. More precisely, …

Even in the multidimensional case, from the Wong-Zakai theorem, the sequence of processes which are solutions of the equation $X^N_t=\int_0^t \mu(X^N_s) ds+\int_0^t \sigma(X_s^N)dV^N_s $ will conver …

If you are looking for approximations of the Skorohod integral we can use Wick Riemann sums. Define for $F \in \mathbb{D}^{1,2}$ and $W$ a Wiener process the Wick product $F \star W_t$ by $F \star W …

Actually, it is quite possible to condition Brownian motion to hit a given point at a given time. The process is a called a Brownian bridge and the distribution of the winding number of the Brownian b …

From the skew-product decomposition of the planar Brownian motion (see for instance Revuz-Yor), it is known that $\theta_t=\beta_{C_t}$ where $C_t=\int_0^t \frac{ds}{\rho_s^2}$ with $\rho_s=\| B_s \|$ …