Related papers: On time-dependent functionals of diffusions corres…
Recently, functional It\=o calculus has been introduced and developed in finite dimension for functionals of continuous semimartingales. With different techniques, we develop a functional It\=o calculus for functionals of Hilbert…
We consider decompositions of processes of the form $Y=f(t,X_t)$ where $X$ is a semimartingale. The function $f$ is not required to be differentiable, so It\^{o}'s lemma does not apply. In the case where $f(t,x)$ is independent of $t$, it…
This note develops shortly the theory of time-inhomogeneous additive functionals and is a useful support for the analysis of time-dependent Markov processes and related topics. It is a significant tool for the analysis of BSDEs in law. In…
Extending It\^o's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-It\^o, applies to one dimensional semimartingales and convex functions.…
We consider a Markov process $X$ associated to a nonnecessarily symmetric Dirichlet form $\mathcal{E}$. We define a stochastic integral with respect to a class of additive functionals of zero quadratic variation and then we obtain an…
We derive a functional change of variable formula for {\it non-anticipative} functionals defined on the space of right continuous paths with left limits. The functional is only required to possess certain directional derivatives, which may…
For real-valued additive process $(X\_t)\_{t\geq 0}$ a recursive equation is derived for the entire positive moments of functionals $$I\_{s,t}= \int \_s^t\exp(-X\_u)du, \quad 0\leq s<t\leq\infty, $$ in case the Laplace exponent of $X\_t$…
This is a study of a class of nonlocal nonlinear diffusion equations. We present a strong maximum principle for nonlocal time-dependent Dirichlet problems. Results are for bounded functions of space, rather than (semi)-continuous functions.…
We consider sequences of additive functionals of difference approximations for uniformly non-degenerate multidimensional diffusions. The conditions are given, sufficient for such a sequence to converge weakly to a W-functional of the…
We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the…
In this work, we consider a FDE (fractional diffusion equation) $${}^C D_t^\alpha u(x,t)-a(t)\mathcal{L} u(x,t)=F(x,t)$$ with a time-dependent diffusion coefficient $a(t)$. For the direct problem, given an $a(t),$ we establish the…
We derive explicit solutions for time-fractional anomalous diffusion equations with diffusivity coefficients that depend on both space and time variables. These solutions are expressed in Fox-H and generalized Wright functions, which are…
Generalised Ito formulae are proved for time dependent functions of continuous real valued semi-martingales. The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded…
In this work we study partial differential equations defined in a domain that moves in time according to the flow of a given ordinary differential equation, starting out of a given initial domain. We first derive a formulation for a…
We investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined among other…
We study Dirichlet series arising as linear functionals on an inner product space of meromorphic functions and establish a relation between the discontinuities of the former on the boundary and the poles and zeros of the latter on the…
We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to…
We study the diffusion equation with an appropriate change of variables. This equation is in general a partial differential equation (PDE). With the self-similar and related Ansat\"atze we transform the PDE of diffusion to an ordinary…
We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise…
We rigorously derive non-equilibrium space-time fluctuation for the particle density of a system of reflected diffusions in bounded Lipschitz domains in $\mathbb R^d$. The particles are independent and are killed by a time-dependent…