Long time asymptotics of unbounded additive functionals of Markov processes

Spectral gap for measure-valued diffusion processes

“…See for instance [8] and references within for the study of the convergence rate. For 0 = θ ∈ M, the Dirichlet distribution D θ with shape θ is the unique probability measure on M 1 such that for any measurable partition {A i } n 1=1 of M, M 1 ∋ µ → (µ(A 1 ), · · · , µ(A n )) obeys the Dirichlet distribution with parameter (θ(A 1 ), · · · , θ(A n )).…”

Moderate deviation principles for unbounded additive functionals of distribution dependent SDEs

“…Introduction. To characterise long time behaviours of stochastic systems, various limit theorems, including LLN(law of large numbers), CLT(central limit theorems), and LDP(large deviation principle) have been intensively investigated in the literature of Markov processes and random sequences, see for instance [1,2,3,4,5,6,9,11,19,20,21]. On the other hand, less is known for limit theorems on nonlinear systems, where a typical model is the distribution dependent SDE (also called McKean-Vlasov or mean-field SDE), which arises from characterizations on nonlinear Fokker-Planck equations and mean-filed particle systems, see [10] and references within.…”

“…The assumptions in [9] was further simplified and improved in [19], so that degenerate situations are also included.…”

Limit theorems for additive functionals of path-dependent SDEs

“…Classical limit theorems include • Strong law of large numbers (SLLN): P-a.s. convergence of A f t to µ ∞ (f ); • Central limit theorem (CLT): The weak convergence of 1 √ t t 0 {f (X s ) − µ ∞ (f )}ds to a normal random variable; • Law of iterated logarithm (LIL): the asymptotic range of 1 √ t log log t t 0 f (X s )ds. Once CLT is established, one may further investigate the large/moderate deviation principles, see for instance [12] and references within.…”