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We establish pathwise continuity properties of solutions to a stochastic Volterra equation with an additive noise term given by a local martingale. The deterministic part is governed by an operator with an $H^\infty$-calculus and a scalar kernel. The proof relies on the dilation theorem for positive definite operator families on a Hilbert space., Minor revision. Accepted for publication in Journal of Evolution Equations

We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed.

We study the relationship between the Dirichlet problem and the Cauchy problem with inhomogeneous boundary conditions for local operators. Our results are applied to non-autonomous parabolic problems on non-cylindrical domains.

We study the long-term behaviour of the parabolic evolution equation \(\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \]\) If \( A(t) \) converges to a sectorial operator A with \( \sigma(A)\cap i \Bbb R =\emptyset \) as \( t\to\infty \), then the evolution family solving the homogeneous problem has exponential dichotomy. If also \( f(t)\to f_\infty \), then the solution u converges to the 'stationary solution at infinity', i.e., \( \lim_{t\to\infty}u(t)= -A\sp{-1}f_\infty=:u_\infty, \qquad \lim_{t\to\infty}u'(t)=0, \qquad \lim_{t\to\infty}A(t)u(t)=Au_\infty. \).